# Patent

## Title
Method for managing production from a hydrocarbon producing reservoir in real-time

## Abstract

The invention relates to a method of performing an oilfield operation of an oilfield having at least one wellsite, each wellsite having a wellbore penetrating a subterranean formation for extracting fluid from an underground reservoir therein. The method steps include obtaining a plurality of real-time parameters from a plurality of sensors disposed about the oilfield, wherein the plurality of real-time parameters comprise at least one selected from a group consisting of real-time flow rate data and real-time pressure data of the wellbore, configuring a gridless analytical simulator for simulating the underground reservoir based on the plurality of real-time parameters, generating real-time simulation results of the underground reservoir and the at least one wellsite in real-time using the gridless analytical simulator, and performing the oilfield operation based on the real-time simulation results.

## Background
CROSS REFERENCE TO RELATED APPLICATIONS
This application claims priority under 35 U.S.C. §119(e) from Provisional Patent Application Nos. 60/953,449 filed Aug. 1, 2007, 60/956,070 filed Aug. 15, 2007, and 61/027,801 filed Feb. 11, 2008. This is a continuation-in-part (CIP) application of and claims priority under 35 U.S.C. §120 to U.S. patent application Ser. No. 11/924,560 filed on Oct. 25, 2007.


BACKGROUND
1. Field of the Invention
The present invention relates to techniques for performing oilfield operations relating to subterranean formations having reservoirs therein. More particularly, the invention relates to techniques for performing oilfield operations involving an analysis of reservoir operations, and their impact on such oilfield operations.
2. Background of the Related Art
Oilfield operations, such as surveying, drilling, wireline testing, completions, production, planning and oilfield analysis, are typically performed to locate and gather valuable downhole fluids. Various aspects of the oilfield and its related operations are shown in FIGS. 1A-1D. As shown in FIG. 1A, surveys are often performed using acquisition methodologies, such as seismic scanners to generate maps of underground structures. These structures are often analyzed to determine the presence of subterranean assets, such as valuable fluids or minerals. This information is used to assess the underground structures and locate the formations containing the desired subterranean assets. Data collected from the acquisition methodologies may be evaluated and analyzed to determine whether such valuable items are present, and if they are reasonably accessible.
As shown in FIG. 1B-1D, one or more wellsites may be positioned along the underground structures to gather valuable fluids from the subterranean reservoirs. The wellsites are provided with tools capable of locating and removing hydrocarbons from the subterranean reservoirs. As shown in FIG. 1B, drilling tools are typically advanced from the oil rigs and into the earth along a given path to locate the valuable downhole fluids. During the drilling operation, the drilling tool may perform downhole measurements to investigate downhole conditions. In some cases, as shown in FIG. 1C, the drilling tool is removed and a wireline tool is deployed into the wellbore to perform additional downhole testing.
After the drilling operation is complete, the well may then be prepared for production. As shown in FIG. 1D, wellbore completions equipment is deployed into the wellbore to complete the well in preparation for the production of fluid therethrough. Fluid is then drawn from downhole reservoirs, into the wellbore and flows to the surface. Production facilities are positioned at surface locations to collect the hydrocarbons from the wellsite(s). Fluid drawn from the subterranean reservoir(s) passes to the production facilities via transport mechanisms, such as tubing. Various equipment may be positioned about the oilfield to monitor oilfield parameters and/or to manipulate the oilfield operations.
During the oilfield operations, data is typically collected for analysis and/or monitoring of the oilfield operations. Such data may include, for example, subterranean formation, equipment, historical and/or other data. Data concerning the subterranean formation is collected using a variety of sources. Such formation data may be static or dynamic. Static data relates to, for example, formation structure and geological stratigraphy that define the geological structure of the subterranean formation. Dynamic data relates to, for example, fluids flowing through the geologic structures of the subterranean formation over time. Such static and/or dynamic data may be collected to learn more about the formations and the valuable assets contained therein.
Sources used to collect static data may be seismic tools, such as a seismic truck that sends compression waves into the earth as shown in FIG. 1A. These waves are measured to characterize changes in the density of the geological structure at different depths. This information may be used to generate basic structural maps of the subterranean formation. Other static measurements may be gathered using core sampling and well logging techniques. Core samples may be used to take physical specimens of the formation at various depths as shown in FIG. 1B. Well logging typically involves deployment of a downhole tool into the wellbore to collect various downhole measurements, such as density, resistivity, etc., at various depths. Such well logging may be performed using, for example, the drilling tool of FIG. 1B and/or the wireline tool of FIG. 1C. Once the well is formed and completed, fluid flows to the surface using production tubing as shown in FIG. 1D. As fluid passes to the surface, various dynamic measurements, such as fluid flow rates, pressure, and composition may be monitored. These parameters may be used to determine various characteristics of the subterranean formation.
Sensors may be positioned about the oilfield to collect data relating to various oilfield operations. For example, sensors in the drilling equipment may monitor drilling conditions, sensors in the wellbore may monitor fluid composition, sensors located along the flow path may monitor flow rates, and sensors at the processing facility may monitor fluids collected. Other sensors may be provided to monitor downhole, surface, equipment or other conditions.
The monitored data is often used to make decisions at various locations of the oilfield at various times. Data collected by these sensors may be further analyzed and processed. Data may be collected and used for current or future operations. When used for future operations at the same or other locations, such data may sometimes be referred to as historical data.
The processed data may be used to predict downhole conditions, and make decisions concerning oilfield operations. Such decisions may involve well planning, well targeting, well completions, operating levels, production rates and other operations and/or conditions. Often this information is used to determine when to drill new wells, re-complete existing wells, or alter wellbore production.
Data from one or more wellbores may be analyzed to plan or predict various outcomes at a given wellbore. In some cases, the data from neighboring wellbores or wellbores with similar conditions or equipment may be used to predict how a well will perform. There are usually a large number of variables and large quantities of data to consider in analyzing oilfield operations. It is, therefore, often useful to model the behavior of the oilfield operation to determine the desired course of action. During the ongoing operations, the operating conditions may need adjustment as conditions change and new information is received.
Techniques have been developed to model the behavior of various aspects of the oilfield operations, such as geological structures, downhole reservoirs, wellbores, surface facilities as well as other portions of the oilfield operation. Typically, there are different types of simulators for different purposes. For example, there are simulators that focus on reservoir properties, wellbore production, or surface processing. Examples of simulators that may be used at the wellsite are described in U.S. Pat. No. 5,992,519 and WO2004049216. Other examples of these modeling techniques are shown in U.S. Pat. Nos. 5,992,519, 6,313,837, WO1999/064896, WO2005/122001, US2003/0216897, US2003/0132934, US2005/0149307, and US2006/0197759.
Recent attempts have been made to consider a broader range of data in oilfield operations. For example, U.S. Pat. No. 6,842,700 to Poe describes a method for evaluating a well and a reservoir without the need for well pressure history. In another example, US2006/0069511 to Thambynayagam discloses a gas reservoir evaluation and assessment tool. Other examples of such recent attempts are disclosed in U.S. Pat. Nos. 6,018,497, 6,078,869, 6,106,561, 6,230,101, 6,980,940, 7,164,990, GB2336008, US2004/0220846, US2006/0129366, US2006/0184329, U.S. Ser. No. 10/586,283, and WO04049216.
Despite the development and advancement of wellbore modeling and/or simulation techniques, many of which employ finite difference numerical methods to construct reservoir models, there remains a need to provide techniques capable of performing real-time simulations for the oilfield operation. It would be desirable to have a system that performs simulations that consider data throughout the oilfield operation. In some cases, it may be desirable to continuously monitor and analyze oilfield data, anticipate and identify events, and to perform real-time diagnostics and interpretation of the oilfield data. In other cases, it may be desirable to support real-time decision making for performing oilfield operations. It is further desirable that such techniques be capable of one of more of the following, among others: taking into consideration the effects of production from other wells in the same reservoir; updating the reservoir model based on history matching; and automatic workflow with real-time plotting of key parameters against time and real-time alarms based on pre-determined criteria.
SUMMARY
In general, in one aspect, the invention relates to a method of performing an oilfield operation of an oilfield having at least one wellsite, each wellsite having a wellbore penetrating a subterranean formation for extracting fluid from an underground reservoir therein. The method steps include obtaining a plurality of real-time parameters from a plurality of sensors disposed about the oilfield, wherein the plurality of real-time parameters comprise at least one selected from a group consisting of real-time flow rate data and real-time pressure data of the wellbore, configuring a gridless analytical simulator for simulating the underground reservoir based on the plurality of real-time parameters, generating real-time simulation results of the underground reservoir and the at least one wellsite in real-time using the gridless analytical simulator, and performing the oilfield operation based on the real-time simulation results.
In general, in one aspect, the invention relates to a method of performing an oilfield operation of an oilfield having a plurality of wellsites, each wellsite having a wellbore penetrating a subterranean formation for extracting fluid from an underground reservoir therein. The method steps include obtaining real-time pressure data from a permanent down-hole pressure gauge, identifying a reservoir model for a gridless analytical simulator based on a rate of change of the real-time pressure data using a neural network method, generating real-time simulation results of the underground reservoir and the plurality of wellsites in real-time using the gridless analytical simulator, and performing the oilfield operation based on the real-time simulation results.
In general, in one aspect, the invention relates to a method of performing an oilfield operation of an oilfield having a plurality of gas wells, each gas well having a wellbore penetrating a subterranean formation for extracting gas from an underground reservoir therein. The method steps include obtaining real-time flow rate data from a flow meter, obtaining at least one selected from a group consisting of real-time pressure data and offline pressure data, generating a first simulation result of the underground reservoir and the plurality of gas wells using a non-linear regression model with the real-time flow rate data, and the real-time pressure data, and the offline pressure data if the real-time pressure data is not available, identifying a reservoir model for a gridless analytical simulator using a neural network method if the real-time pressure data is available, generating a second simulation result of the reservoir and the plurality of gas wells in real-time using the gridless analytical simulator, and performing the oilfield operation based on at least one selected from a group consisting of the first simulation result and the second simulation result.
In general, in one aspect, the invention relates to a computer readable medium, embodying instructions executable by a computer to perform method steps for an oilfield operation, the oilfield having at least one wellsite, each of the at least one wellsite having a wellbore penetrating a subterranean formation for extracting fluid from an underground reservoir therein. The instructions include functionality to obtain a plurality of real-time parameters from a plurality of sensors disposed about the oilfield, wherein the plurality of real-time parameters comprise at least one selected from a group consisting of flow rate and pressure of the wellbore, configure a gridless analytical simulator for simulating the reservoir based on the plurality of real-time parameters, and generate real-time simulation results of the reservoir and the at least one wellsite in real-time using the gridless analytical simulator, wherein the oilfield operation is performed based on the real-time simulation results.
In general, in one aspect, the invention relates to a computer readable medium, embodying instructions executable by a computer to perform method steps for an oilfield operation, the oilfield having a plurality of wellsites, each of the plurality of wellsites having a wellbore penetrating a subterranean formation for extracting fluid from an underground reservoir therein. The instructions include functionality to obtain real-time pressure data from a permanent down-hole pressure gauge, identify a reservoir model for a gridless analytical simulator based on a rate of change of the real-time pressure data using a neural network method, generate real-time simulation results of the reservoir and the plurality of wellsites in real-time using the gridless analytical simulator, and perform the oilfield operation based on the real-time simulation results.
In general, in one aspect, the invention relates to a computer readable medium, embodying instructions executable by a computer to perform method steps for an oilfield operation, the oilfield having a plurality of gas wells, each of the plurality of gas wells having a wellbore penetrating a subterranean formation for extracting gas from an underground reservoir therein. The instructions include functionality to obtain real-time flow rate data from a flow meter, obtain at least one selected from a group consisting of real-time pressure data and offline pressure data, generate a first simulation result of the underground reservoir and the plurality of gas wells using a non-linear regression model with the real-time flow rate data, and the real-time pressure data, and the offline pressure data if the real-time pressure data is not available, identify a reservoir model for a gridless analytical simulator using a neural network method if the real-time pressure data is available, generate a second simulation result of the reservoir and the plurality of gas wells in real-time using the gridless analytical simulator, and perform the oilfield operation based on at least one selected from a group consisting of the first simulation result and the second simulation result.
Other aspects and advantages of the invention will be apparent from the following description and the appended claims.



BRIEF DESCRIPTION OF THE DRAWINGS
So that the above recited features and advantages of the present invention can be understood in detail, a more particular description of the invention, briefly summarized above, may be had by reference to the embodiments thereof that are illustrated in the appended drawings. It is to be noted, however, that the appended drawings illustrate only typical embodiments of this invention and are therefore not to be considered limiting of its scope, for the invention may admit to other equally effective embodiments.
FIGS. 1A-1D show exemplary schematic views of an oilfield having subterranean structures including reservoirs therein and various oilfield operations being performed on the oilfield. FIG. 1A depicts an exemplary survey operation being performed by a seismic truck. FIG. 1B depicts an exemplary drilling operation being performed by a drilling tool suspended by a rig and advanced into the subterranean formation. FIG. 1C depicts an exemplary wireline operation being performed by a wireline tool suspended by the rig and into the wellbore of FIG. 1B. FIG. 1D depicts an exemplary production operation being performed by a production tool being deployed from the rig and into a completed wellbore for drawing fluid from the downhole reservoir into a surface facility.
FIGS. 2A-2D are exemplary graphical depictions of data collected by the tools of FIGS. 1A-1D, respectively. FIG. 2A depicts an exemplary seismic trace of the subterranean formation of FIG. 1A. FIG. 2B depicts exemplary core sample of the formation shown in FIG. 1B. FIG. 2C depicts an exemplary well log of the subterranean formation of FIG. 1C. FIG. 2D depicts an exemplary production decline curve of fluid flowing through the subterranean formation of FIG. 1D.
FIG. 3 shows an exemplary schematic view, partially in cross section, of an oilfield having a plurality of data acquisition tools positioned at various locations along the oilfield for collecting data from the subterranean formation.
FIG. 4 shows an exemplary schematic view of an oilfield having a plurality of wellsites for producing hydrocarbons from the subterranean formation.
FIG. 5 shows an exemplary schematic diagram of a portion of the oilfield of FIG. 4 depicting the production operation in detail.
FIG. 6 is a flow chart of a permanent downhole pressure gauge (PDG) workflow in an oilfield.
FIG. 7 is a flow chart of a gas rate workflow in a gas field.
FIG. 8 shows an exemplary schematic diagram of a reservoir modeled in a gridless analytical simulator.
FIG. 9 is a flow chart of a method to perform an oilfield operation using the real-time analytical simulator.



DETAILED DESCRIPTION
Presently preferred embodiments of the invention are shown in the above-identified figures and described in detail below. In describing the preferred embodiments, like or identical reference numerals are used to identify common or similar elements. The figures are not necessarily to scale and certain features and certain views of the figures may be shown exaggerated in scale or in schematic in the interest of clarity and conciseness.
FIGS. 1A-D show an oilfield (100) having geological structures and/or subterranean formations therein. As shown in these figures, various measurements of the subterranean formation are taken by different tools at the same location. These measurements may be used to generate information about the formation and/or the geological structures and/or fluids contained therein.
FIGS. 1A-1D depict schematic views of an oilfield (100) having subterranean formations (102) containing a reservoir (104) therein and depicting various oilfield operations being performed on the oilfield (100). FIG. 1A depicts a survey operation being performed by a seismic truck (106a) to measure properties of the subterranean formation. The survey operation is a seismic survey operation for producing sound vibration(s) (112). In FIG. 1A, one such sound vibration (112) is generated by a source (110) and reflects off a plurality of horizons (114) in an earth formation (116). The sound vibration(s) (112) is (are) received in by sensors (S), such as geophone-receivers (118), situated on the earth's surface, and the geophone-receivers (118) produce electrical output signals, referred to as data received (120) in FIG. 1.
In response to the received sound vibration(s) (112) representative of different parameters (such as amplitude and/or frequency) of the sound vibration(s) (112). The data received (120) is provided as input data to a computer (122a) of the seismic recording truck (106a), and responsive to the input data, the recording truck computer (122a) generates a seismic data output record (124). The seismic data may be further processed as desired, for example by data reduction.
FIG. 1B depicts a drilling operation being performed by a drilling tool (106b) suspended by a rig (128) and advanced into the subterranean formation (102) to form a wellbore (136). A mud pit (130) is used to draw drilling mud into the drilling tool (106b) via flow line (132) for circulating drilling mud through the drilling tool (106b) and back to the surface. The drilling tool (106b) is advanced into the formation to reach reservoir (104). The drilling tool (106b) is preferably adapted for measuring downhole properties. The drilling tool (106b) may also be adapted for taking a core sample (133), as shown, or removed so that a core sample (133) may be taken using another tool.
A surface unit (134) is used to communicate with the drilling tool (106b) and offsite operations. The surface unit (134) is capable of communicating with the drilling tool (106b) to send commands to drive the drilling tool (106b), and to receive data therefrom. The surface unit (134) is preferably provided with computer facilities for receiving, storing, processing, and analyzing data from the oilfield (100). The surface unit (134) collects data output (135) generated during the drilling operation. Computer facilities, such as those of the surface unit (134), may be positioned at various locations about the oilfield (100) and/or at remote locations.
Sensors (S), such as gauges, may be positioned throughout the reservoir, rig, oilfield equipment (such as the downhole tool), or other portions of the oilfield for gathering information about various parameters, such as surface parameters, downhole parameters, and/or operating conditions. These sensors (S) preferably measure oilfield parameters, such as weight on bit, torque on bit, pressures, temperatures, flow rates, compositions and other parameters of the oilfield operation.
The information gathered by the sensors (S) may be collected by the surface unit (134) and/or other data collection sources for analysis or other processing. The data collected by the sensors (S) may be used alone or in combination with other data. The data may be collected in a database and all or select portions of the data may be selectively used for analyzing and/or predicting oilfield operations of the current and/or other wellbores.
Data outputs from the various sensors (S) positioned about the oilfield may be processed for use. The data may be historical data, real time data, or combinations thereof. The real time data may be used in real time, or stored for later use. The data may also be combined with historical data or other inputs for further analysis. The data may be housed in separate databases, or combined into a single database.
The collected data may be used to perform analysis, such as modeling operations. For example, the seismic data output may be used to perform geological, geophysical, reservoir engineering, and/or production simulations. The reservoir, wellbore, surface and/or process data may be used to perform reservoir, wellbore, or other production simulations. The data outputs from the oilfield operation may be generated directly from the sensors (S), or after some preprocessing or modeling. These data outputs may act as inputs for further analysis.
The data is collected and stored at the surface unit (134). One or more surface units (134) may be located at the oilfield (100), or linked remotely thereto. The surface unit (134) may be a single unit, or a complex network of units used to perform the necessary data management functions throughout the oilfield (100). The surface unit (134) may be a manual or automatic system. The surface unit (134) may be operated and/or adjusted by a user.
The surface unit (134) may be provided with a transceiver (137) to allow communications between the surface unit (134) and various portions (or regions) of the oilfield (100) or other locations. The surface unit (134) may also be provided with or functionally linked to a controller for actuating mechanisms at the oilfield (100). The surface unit (134) may then send command signals to the oilfield (100) in response to data received. The surface unit (134) may receive commands via the transceiver or may itself execute commands to the controller. A processor may be provided to analyze the data (locally or remotely) and make the decisions to actuate the controller. In this manner, the oilfield (100) may be selectively adjusted based on the data collected to optimize fluid recovery rates, or to maximize the longevity of the reservoir and its ultimate production capacity. These adjustments may be made automatically based on computer protocol, or manually by an operator. In some cases, well plans may be adjusted to select optimum operating conditions, or to avoid problems.
FIG. 1C depicts a wireline operation being performed by a wireline tool (106c) suspended by the rig (128) and into the wellbore (136) of FIG. 1B. The wireline tool (106c) is preferably adapted for deployment into a wellbore (136) for performing well logs, performing downhole tests and/or collecting samples. The wireline tool (106c) may be used to provide another method and apparatus for performing a seismic survey operation. The wireline tool (106c) of FIG. 1C may have an explosive or acoustic energy source (143) that provides electrical signals to the surrounding subterranean formations (102).
The wireline tool (106c) may be operatively linked to, for example, the geophones (118) stored in the computer (122a) of the seismic recording truck (106a) of FIG. 1A. The wireline tool (106c) may also provide data to the surface unit (134). As shown, data output (135) is generated by the wireline tool (106c) and collected at the surface. The wireline tool (106c) may be positioned at various depths in the wellbore (136) to provide a survey of the subterranean formation.
FIG. 1D depicts a production operation being performed by a production tool (106d) deployed from a production unit or christmas tree (129) and into the completed wellbore (136) of FIG. 1C for drawing fluid from the downhole reservoirs into the surface facilities (142). Fluid flows from reservoir (104) through perforations in the casing (not shown) and into the production tool (106d) in the wellbore (136) and to the surface facilities (142) via a gathering network (146).
Sensors (S), such as gauges, may be positioned about the oilfield to collect data relating to various oilfield operations as described previously. As shown, the sensor (S) may be positioned in the production tool (106d) or associated equipment, such as the Christmas tree, gathering network, surface facilities and/or the production facility, to measure fluid parameters, such as fluid composition, flow rates, pressures, temperatures, and/or other parameters of the production operation.
While only simplified wellsite configurations are shown, it will be appreciated that the oilfield may cover a portion of land, sea, and/or water locations that hosts one or more wellsites. Production may also include injection wells (not shown) for added recovery. One or more gathering facilities may be operatively connected to one or more of the wellsites for selectively collecting downhole fluids from the wellsite(s).
During the production process, data output (135) may be collected from various sensors (S) and passed to the surface unit (134) and/or processing facilities. This data may be, for example, reservoir data, wellbore data, surface data, and/or process data.
While FIGS. 1A-1D depict monitoring tools used to measure properties of an oilfield (100), it will be appreciated that the tools may be used in connection with non-oilfield operations, such as mines, aquifers or other subterranean facilities. Also, while certain data acquisition tools are depicted, it will be appreciated that various measurement tools capable of sensing properties, such as seismic two-way travel time, density, resistivity, production rate, etc., of the subterranean formation and/or its geological structures may be used. Various sensors (S) may be located at various positions along the subterranean formation and/or the monitoring tools to collect and/or monitor the desired data. Other sources of data may also be provided from offsite locations.
The oilfield configuration in FIGS. 1A-1D is not intended to limit the scope of the invention. Part, or all, of the oilfield (100) may be on land and/or sea. Also, while a single oilfield at a single location is depicted, the present invention may be used with any combination of one or more oilfields (100), one or more processing facilities and one or more wellsites. Additionally, while only one wellsite is shown, it will be appreciated that the oilfield (100) may cover a portion of land that hosts one or more wellsites. One or more gathering facilities may be operatively connected to one or more of the wellsites for selectively collecting downhole fluids from the wellsite(s).
FIGS. 2A-2D are graphical depictions of data collected by the tools of FIGS. 1A-D, respectively. FIG. 2A depicts a seismic trace (202) of the subterranean formation of FIG. 1A taken by survey tool (106a). The seismic trace measures a two-way response over a period of time. FIG. 2B depicts a core sample (133) taken by the drilling tool (106b). The core test typically provides a graph of the density, resistivity, or other physical property of the core sample (133) over the length of the core. Tests for density and viscosity are often performed on the fluids in the core at varying pressures and temperatures. FIG. 2C depicts a well log (204) of the subterranean formation of FIG. 1C taken by the wireline tool (106c). The wireline log typically provides a resistivity measurement of the formation at various depths. FIG. 2D depicts a production decline curve (206) of fluid flowing through the subterranean formation of FIG. 1D taken by the production tool (106d). The production decline curve (206) typically provides the production rate Q as a function of time t.
The respective graphs of FIGS. 2A-2C contain static measurements that describe the physical characteristics of the formation. These measurements may be compared to determine the accuracy of the measurements and/or for checking for errors. In this manner, the plots of each of the respective measurements may be aligned and scaled for comparison and verification of the properties.
FIG. 2D provides a dynamic measurement of the fluid properties through the wellbore. As the fluid flows through the wellbore, measurements are taken of fluid properties, such as flow rates, pressures, composition, etc. As described below, the static and dynamic measurements may be used to generate models of the subterranean formation to determine characteristics thereof.
FIG. 3 is a schematic view, partially in cross section of an oilfield (300) having data acquisition tools (302a), (302b), (302c), and (302d) positioned at various locations along the oilfield for collecting data of a subterranean formation (304). The data acquisition tools (302a-302d) may be the same as data acquisition tools (106a-106d) of FIG. 1, respectively. As shown, the data acquisition tools (302a-302d) generate data plots or measurements (308a-308d), respectively.
Data plots (308a-308c) are examples of static data plots that may be generated by the data acquisition tools (302a-302d), respectively. Static data plot (308a) is a seismic two-way response time and may be the same as the seismic trace (202) of FIG. 2A. Static plot (308b) is core sample data measured from a core sample of the formation (304), similar to the core sample (133) of FIG. 2B. Static data plot (308c) is a logging trace, similar to the well log (204) of FIG. 2C. Data plot (308d) is a dynamic data plot of the fluid flow rate over time, similar to the graph (206) of FIG. 2D. Other data may also be collected, such as historical data, user inputs, economic information, other measurement data, and other parameters of interest.
The subterranean formation (304) has a plurality of geological structures (306a-306d). As shown, the formation has a sandstone layer (306a), a limestone layer (306b), a shale layer (306c), and a sand layer (306d). A fault line (307) extends through the formation. The static data acquisition tools are preferably adapted to measure the formation and detect the characteristics of the geological structures of the formation.
While a specific subterranean formation (304) with specific geological structures are depicted, it will be appreciated that the formation may contain a variety of geological structures. Fluid may also be present in various portions of the formation. Each of the measurement devices may be used to measure properties of the formation and/or its underlying structures. While each acquisition tool is shown as being in specific locations along the formation, it will be appreciated that one or more types of measurement may be taken at one or more location across one or more oilfields or other locations for comparison and/or analysis.
The data collected from various sources, such as the data acquisition tools of FIG. 3, may then be evaluated. Typically, seismic data displayed in the static data plot (308a) from the data acquisition tool (302a) is used by a geophysicist to determine characteristics of the subterranean formation (304). Core data shown in static plot (308b) and/or log data from the well log (308c) is typically used by a geologist to determine various characteristics of the geological structures of the subterranean formation (304). Production data from the production graph (308d) is typically used by the reservoir engineer to determine fluid flow reservoir characteristics.
FIG. 4 shows an oilfield (400) for performing production operations. As shown, the oilfield has a plurality of wellsites (402) operatively connected to a central processing facility (454). The oilfield configuration of FIG. 4 is not intended to limit the scope of the invention. Part or all of the oilfield may be on land and/or sea. Also, while a single oilfield with a single processing facility and a plurality of wellsites is depicted, any combination of one or more oilfields, one or more processing facilities and one or more wellsites may be present.
Each wellsite (402) has equipment that forms a wellbore (436) into the earth. The wellbores extend through subterranean formations (406) including reservoirs (404). These reservoirs (404) contain fluids, such as hydrocarbons. The wellsites draw fluid from the reservoirs and pass them to the processing facilities via surface networks (444). The surface networks (444) have tubing and control mechanisms for controlling the flow of fluids from the wellsite to the processing facility (454).
FIG. 5 shows a schematic view of a portion (or region) of the oilfield (400) of FIG. 4, depicting a producing wellsite (402) and surface network (444) in detail. The wellsite (402) of FIG. 5 has a wellbore (436) extending into the earth therebelow. As shown, the wellbores (436) has already been drilled, completed, and prepared for production from reservoir (404).
Wellbore production equipment (564) extends from a wellhead (566) of wellsite (402) and to the reservoir (404) to draw fluid to the surface. The wellsite (402) is operatively connected to the surface network (444) via a transport line (561). Fluid flows from the reservoir (404), through the wellbore (436), and onto the surface network (444). The fluid then flows from the surface network (444) to the process facilities (454).
As further shown in FIG. 5, sensors (S) are located about the oilfield (400) to monitor various parameters during oilfield operations. The sensors (S) may measure, for example, pressure, temperature, flow rate, composition, and other parameters of the reservoir, wellbore, surface network, process facilities and/or other portions (or regions) of the oilfield operation. These sensors (S) are operatively connected to a surface unit (534) for collecting data therefrom. The surface unit may be, for example, similar to the surface unit (134) of FIGS. 1A-D.
One or more surface units (534) may be located at the oilfield (400), or linked remotely thereto. The surface unit (534) may be a single unit, or a complex network of units used to perform the necessary data management functions throughout the oilfield (400). The surface unit may be a manual or automatic system. The surface unit may be operated and/or adjusted by a user. The surface unit is adapted to receive and store data. The surface unit may also be equipped to communicate with various oilfield equipment. The surface unit may then send command signals to the oilfield in response to data received or modeling performed.
As shown in FIG. 5, the surface unit (534) has computer facilities, such as memory (520), controller (522), processor (524), and display unit (526), for managing the data. The data is collected in memory (520), and processed by the processor (524) for analysis. Data may be collected from the oilfield sensors (S) and/or by other sources. For example, oilfield data may be supplemented by historical data collected from other operations, or user inputs.
The analyzed data (e.g., based on modeling performed) may then be used to make decisions. A transceiver (not shown) may be provided to allow communications between the surface unit (534) and the oilfield (400). The controller (522) may be used to actuate mechanisms at the oilfield (400) via the transceiver and based on these decisions. In this manner, the oilfield (400) may be selectively adjusted based on the data collected. These adjustments may be made automatically based on computer protocol and/or manually by an operator. In some cases, well plans are adjusted to select optimum operating conditions or to avoid problems.
To facilitate the processing and analysis of data, simulators may be used to process the data for modeling various aspects of the oilfield operation. Specific simulators are often used in connection with specific oilfield operations, such as reservoir or wellbore simulation. Data fed into the simulator(s) may be historical data, real time data or combinations thereof. Simulation through one or more of the simulators may be repeated or adjusted based on the data received.
As shown, the oilfield operation is provided with wellsite and non-wellsite simulators. The wellsite simulators may include a reservoir simulator (340), a wellbore simulator (342), and a surface network simulator (344). The reservoir simulator (340) solves for hydrocarbon flow through the reservoir rock and into the wellbores. The wellbore simulator (342) and surface network simulator (344) solves for hydrocarbon flow through the wellbore and the surface network (444) of pipelines. As shown, some of the simulators may be separate or combined, depending on the available systems.
The non-wellsite simulators may include process (346) and economics (348) simulators. The processing unit has a process simulator (346). The process simulator (346) models the processing plant (e.g., the process facilities (454)) where the hydrocarbon(s) is/are separated into its constituent components (e.g., methane, ethane, propane, etc.) and prepared for sales. The oilfield (400) is provided with an economics simulator (348). The economics simulator (348) models the costs of part or the entire oilfield (400) throughout a portion or the entire duration of the oilfield operation. Various combinations of these and other oilfield simulators may be provided.
While high quality petroleum reservoirs have been successfully explored and exploited for producing oil and gas. Large reservoirs are increasingly difficult to find and producing reservoirs have problems that need to be quickly diagnosed and remedied. Therefore, honoring all relevant measurements to enable on-time decision making is necessary for oilfield operations. The oilfield operations generates a large amount of pressure and production rate data (e.g., data generated from sensors (S) and/or data acquisition tools disposed throughout the oilfield as described with respect to FIGS. 1A-D and 2-5 above), some of which can be measured continuously in real-time. In addition, there are data acquired sporadically, such as well logs and formation test data (e.g., the well log (308c) and seismic trace (308d) of FIG. 3). Timely and methodical interpretation of this data can provide insight into the status of the well and the reservoir as well as advanced notice to potentially detrimental events.
A workflow is a sequence of steps, organized into routines or subroutines—some of which may be quite complex—that are carried out to achieve a particular goal. Each step receives input in various formats, ranging from digital files or spreadsheets to expert commentary. This input is then processed using a predefined mode, such as a reservoir simulator, spreadsheet analysis, or structured discussions and meetings. The resulting output is utilized in subsequent steps. The goal for most oilfield asset teams is to arrive at an answer that will be used as input for another process, or which will be used to drive a decision. Repetitive workflows can often be automated, freeing personnel to attend to non-routine tasks.
The present invention relates to simulating oilfield workflows using a gridless analytical simulator. In one or more embodiments of the invention, the computation efficiency of the gridless analytical simulator enables the integration of various sources of data at different frequencies in one integrated application, which allows user to step from a single well evaluation & interpretation to multi-well, multi-phase, and/or multi-event diagnostic in a synchronized mode. In one or more embodiments of the invention, oilfield workflows may be simulated by this fast gridless analytical simulator for handling pressure transient data and performing interpretation of key performance indicators during the well/field production life. In one or more embodiments of the invention, these capabilities allow oilfield workflows to monitor and analyze data, anticipate and identify events, and to perform real-time diagnostics and interpretation during the entire life of producing wells.
In one or more embodiments of the invention, the gridless analytical simulator, described below, supports several well configurations and reservoir conditions including vertical, deviated, horizontal, and fractured wells, single and multiple layer heterogeneous reservoir, single phase and multi-phase flowing conditions, and is capable of taking into account superposition effect in multi-well and multi rate scenarios. In one or more embodiments of the invention, special reservoir condition, such as interference effects of multiple wells at different events, may be simulated including surface constrains, pressure transient or rate transient events, etc.
In one or more embodiments of the invention, the gridless analytical simulator may be used either in automatic history matching mode or in prediction mode. The automatic history matching mode aims to compute in real time, key reservoir and well parameters such as reservoir pressure, well skin, effective permeability and well productivity. Subsequently, the prediction mode predicts well and reservoir performance in real-time. The prediction mode is a component to integrate more common production engineer analysis that is used to manage a reservoir, such as well test validation and back allocation correction and forecast in real time, among others.
In one or more embodiments of the invention, the gridless analytical simulator may be used to integrate and keep alive the interaction of the multiple oilfield workflow sub-processes, such as data integration (sources, frequency, etc.), data preparation using techniques such as wavelets transforms to reduce data, remove noise & outliers and transient identification, alarm management system to monitor and control KPI, pressure transient interpretation, automatic model identification using neural networks and systems identification including the use of deconvolution, back allocation, rate reconstruction and well test validation, production (rate and pressure) forecast, reporting and visualization, and/or other suitable oilfield workflow sub-processes.
FIGS. 6 and 7 show exemplary oilfield workflows modeled using the gridless analytical simulator. FIG. 6 is a flow chart of a permanent downhole pressure gauge (PDG) workflow in an oilfield (e.g., the oilfield (300) of FIG. 3). One of the objectives of the PDG workflow is to enable a lifecycle process to maximize hydrocarbon producing performance of the reservoir over its full life cycle. This is achieved by using a gridless analytical simulator (e.g., a version of the reservoir simulator (340) of FIG. 5), which is described in detail below and can be configured to simulate an interference effect, for example from multiple wellsites of the oilfield (300) in FIG. 3. In the PDG workflow, real-time pressure data is obtained for the gridless analytical simulator from a permanent down-hole pressure gauge (e.g., the data acquisition tool (302d) of FIG. 3) (Step 601). The real-time pressure data is filtered, for example, by using a wavelet decomposition technique to remove outlier(s), noise, and identify transients (Step 613). The transients may result from a changing oil production rate or shutting down and turning up the production. The identified transients may be used to mark a time interval for simulation sessions. The large amount of real-time raw data may be sampled to reduce to the filtered data to a manageable amount, while retaining all the relevant characteristics of the original larger data set.
The flow rate data may be obtained for the gridless analytical simulator using a variety of methods. In some examples, the flow rate data is obtained through real-time measurement (e.g., the fluid flow rate data plot (308d) of FIG. 3) using sensors (e.g., data acquisition tool (302d) of FIG. 3) disposed throughout the oilfield (Step 603). In some examples, missing periods of the real-time measurement may exist, which may be supplemented with flow rate re-construction, for example based on tubing head or bottom hole pressure measurement (Step 604). The real-time flow rate data (if available) is also filtered in a similar fashion as filtering of the real-time pressure data (Step 605). In other examples, the real-time flow rate measurement may not be available (Step 602). In such cases, the offline flow rate data is obtained, for example by a method of back allocation using total volume at the point of sales, well test data, and/or downtime measurement at a well (Step 606). The offline flow rate data may also be supplemented with flow rate re-construction, for example, based on tubing head or bottom hole pressure measurement (Step 612).
A set of alarm conditions are calculated based on the real-time data after filtering (Step 607). The alarms may include, for example drawdown alarm, downtime alarm, etc. If the alarm is triggered, detailed diagnostics are performed thereafter.
Within the gridless analytical simulator, many parameters may be used to configure an appropriate model for simulating the oilfield (e.g., the oilfield (300) of FIG. 3). In some examples, the model may be determined manually. The model may be identified by using a neural network method based on, for example, rate of change of the real-time pressure data (Step 608). The model may be further configured based on static parameters obtained through geological surveys (e.g. as depicted in FIG. 1 and FIG. 3 above).
Once the model is identified and the simulator is configured, real-time simulation results are then generated (Step 609). The real-time simulation may include a history matching of key parameters and a prediction of the production rates and reservoir pressure over time. The history matching may be performed as a calibration step at the beginning of a simulation session marked by an identified transient from a change of production rate and/or shutting down and turning up of the production. The real-time simulation results may be delivered in an automatic workflow (i.e., the PDG workflow) with real-time plotting of the key parameters and alarm setting based on pre-determined criteria. The key parameters for the history matching and the real-time plotting may include the reservoir pressure, well skin, effective permeability, and well productivity, etc. The model is automatically updated if the predicted performance diverges from the actual performance by more than a pre-determined limit (Step 610).
In Step 611, the oilfield operation is performed based on the real-time simulation results. For example, the real-time plotting in the simulation results may be analyzed to determine a trend of a wellbore skin, and the oilfield operation performed includes scheduling a workover operation to reduce the wellbore skin. In another example, the real-time plotting in the simulation results may be analyzed to determine a trend of effective permeability, and the oilfield operation performed includes determining a re-completion strategy, such as scheduling an artificial lift operation.
FIG. 7 is a flow chart of a gas field workflow in a gas field, for example, gas may be produced in the oilfield operations depicted in FIGS. 1A-1D and 2-5 above. Initially, the flow rate data is obtained through real-time measurement (e.g., the fluid flow rate data plot (308d) of FIG. 3) using sensors (e.g., data acquisition tool (302d) of FIG. 3) disposed throughout the oilfield (Step 701). In some examples, missing periods of the real-time measurement may exist. These missing periods may be supplemented with flow rate re-construction, for example, based on tubing head or bottom hole pressure measurement. The real-time flow rate data is also filtered. The filtering functionality includes, for example de-noising using wavelets decomposition, outlier removal, transient identification, data reduction, etc.
As gas wells often may not be equipped with a permanent down hole pressure gauge. A set of first level alarm conditions are calculated based on the real-time flow rate data and basic historic bottom hole or tubing head pressure measurements (Step 702). The alarms may include, for example, drawdown alarm, downtime alarm, etc. If the alarm is triggered, detailed diagnostics are performed thereafter.
Next, a determination is made as to whether real-time measurement is available for bottom hole or tubing head pressure (Step 703). If neither bottom hole nor tubing head pressure measurements is available, offline pressure data is obtained (if available), for example, using historical data and/or by spot measurement (Step 708). The processed real-time flow rate data, and the offline pressure data (if available) are then used to compute key reservoir parameters such as total skin factor, permeability, drainage area, etc. using evaluation method without real-time pressure data, for example, a non-linear regression model (Step 710).
If real-time pressure measurement is available (Step 703), the reliability of the analysis may increase by obtaining pressure data from either bottom hole or tubing head (Step 704). The real-time pressure data obtained this way also involve a filtering step, which includes de-noising, outlier removal, transient identification, and sampling for data reduction.
The reservoir model for a gridless analytical simulator is then identified (Step 705). The model may be identified by using a neural network method based on, for example, hydraulic flow units obtained from pre-processed logs containing information such as layer thickness, porosity, effective permeability, and saturation dependent petro-physical properties. In this step, the model may be further configured based on a history matching method of these key parameters.
Once the model is identified and the simulator is configured, real-time simulation results are then generated (Step 706). The real-time simulation includes a history matching of key parameters and a prediction of the production rates and reservoir pressure over time. The history matching may be performed as a calibration step at the beginning of a simulation session marked by an identified transient from a change of production rate and/or a shutting down and a turning up of the production. The real-time simulation results can be delivered in an automatic workflow (i.e., the gas field workflow) with real-time plotting of the key parameters and alarm setting based on pre-determined criteria. The key parameters for the history matching and the real-time plotting may include the reservoir pressure, well skin, effective permeability, and well productivity, etc. The model is automatically updated if the predicted performance diverges from the actual performance by more than a pre-determined limit (Step 707).
In Step 711, the oilfield operation is performed based on the real-time simulation results.
FIG. 8 shows an exemplary schematic diagram of a reservoir modeled in a gridless analytical simulator. In FIG. 8, the reservoir (800) (a portion of which may correspond to the reservoir (404) depicted in FIG. 4 and FIG. 5 above) is represented as a series of N vertically stacked cuboids (or layers) (801), where each of the N cuboids is indexed from 1 through N by an index j. The reservoir (800) is bounded by the planes passing through x=0, x=a; y=0, y=b; z=0, z=dN. Layer j has porosity φj and permeability kxj, kyj, kzj in the x, y and z directions respectively. The scale of the reservoir (800) drawn in FIG. 8 may be substantially larger than the scale used in FIG. 3, FIG. 4, and FIG. 5.
For example, portions of these cuboids (801) may correspond to the geological structures (306a-306d) of FIG. 3. The reservoir (800) may be penetrated by multiple wells such as vertical wells (802), horizontal wells (803), and deviated wells (804). The wells (802, 803, 804) may be fractured or un-fractured, the fracture(s) may be naturally occurring or induced by hydraulic fracturing process (not shown). The hydraulic fractures may have finite or infinite conductivity. The reservoir boundary may be modeled as no-flow, constant pressure, or a combination thereof. Even though the wells (802, 803, 804) are represented as a line, suitable corrections may be applied in the model to account for wellbore storage effects and finite wellbore radius. Interference (or superposition) effects from multiple wells in the oilfield are accounted for in the model.
In one or more embodiments of the invention, a gridless analytical simulator may be developed for the vertically stacked system of layers described above. Specifically, an analytic solution within each layer can be derived using a method of integral transforms. In one or more embodiments of the invention, the crossflow between layers are accounted for by coupling these analytic solutions together and solving Fredholm integral equations to obtain the flux field at the layer interfaces. The time evolution of these fluxes is governed by a Volterra integral equation. In one or more embodiments of the invention, the form of these equations allows for stopping a model execution and then restarting from the exact terminated state.
In one or more embodiments of the invention, a general solution for hydrocarbon production can be formulated based on initial and boundary conditions and the governing equations listed in TABLE 1.







TABLE 1











  
    
      
        
          
            
              
                
                  
                    ∂
                    
                      
                        p
                        j
                      
                      ⁡
                      
                        (
                        
                          0
                          ,
                          y
                          ,
                          z
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ∂
                    x
                  
                
                =
                
                  
                    -
                    
                      
                        (
                        
                          μ
                          
                            k
                            x
                          
                        
                        )
                      
                      j
                    
                  
                  ⁢
                  
                    
                      ψ
                      
                        0
                        ⁢
                        yzj
                      
                    
                    ⁡
                    
                      (
                      
                        y
                        ,
                        z
                        ,
                        t
                      
                      )
                    
                  
                
              
              ,
              
                
                  
                    ∂
                    
                      
                        p
                        j
                      
                      ⁡
                      
                        (
                        
                          a
                          ,
                          y
                          ,
                          z
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ∂
                    x
                  
                
                =
                
                  
                    -
                    
                      
                        (
                        
                          μ
                          
                            k
                            x
                          
                        
                        )
                      
                      j
                    
                  
                  ⁢
                  
                    
                      ψ
                      ayzj
                    
                    ⁡
                    
                      (
                      
                        y
                        ,
                        z
                        ,
                        t
                      
                      )
                    
                  
                
              
              ,
            
          
        
        
          
            
              
                
                  
                    ∂
                    
                      
                        p
                        j
                      
                      ⁡
                      
                        (
                        
                          x
                          ,
                          0
                          ,
                          z
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ∂
                    y
                  
                
                =
                
                  
                    -
                    
                      
                        (
                        
                          μ
                          
                            k
                            x
                          
                        
                        )
                      
                      j
                    
                  
                  ⁢
                  
                    
                      ψ
                      x0zj
                    
                    ⁡
                    
                      (
                      
                        x
                        ,
                        z
                        ,
                        t
                      
                      )
                    
                  
                
              
              ,
              
                
                  
                    ∂
                    
                      
                        p
                        j
                      
                      ⁡
                      
                        (
                        
                          x
                          ,
                          b
                          ,
                          z
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ∂
                    y
                  
                
                =
                
                  
                    -
                    
                      
                        (
                        
                          μ
                          
                            k
                            y
                          
                        
                        )
                      
                      j
                    
                  
                  ⁢
                  
                    
                      ψ
                      xbzj
                    
                    ⁡
                    
                      (
                      
                        x
                        ,
                        z
                        ,
                        t
                      
                      )
                    
                  
                
              
              ,
              
                
                  d
                  j
                
                <
                z
                <
                
                  d
                  
                    j
                    +
                    1
                  
                
              
            
          
        
        
          
            
              
                
                  ∀
                  j
                
                =
                0
              
              ,
              1
              ,
              ...
              ,
              
                
                  א
                  -
                  
                    1.
                    ⁢
                    
                      
                    
                    ⁢
                    At
                    ⁢
                    
                      
                    
                    ⁢
                    z
                  
                
                =
                
                  d
                  0
                
              
              ,
              
                
              
              ⁢
              
                
                  
                    ∂
                    
                      p
                      ⁡
                      
                        (
                        
                          x
                          ,
                          y
                          ,
                          
                            d
                            0
                          
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ∂
                    z
                  
                
                =
                
                  
                    -
                    
                      
                        (
                        
                          μ
                          
                            k
                            z
                          
                        
                        )
                      
                      0
                    
                  
                  ⁢
                  
                    
                      ψ
                      
                        xy
                        ⁢
                        
                          
                        
                        ⁢
                        0
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                    ⁡
                    
                      (
                      
                        x
                        ,
                        y
                        ,
                        t
                      
                      )
                    
                  
                
              
              ,
              
                
                  and
                  ⁢
                  
                    
                  
                  ⁢
                  at
                  ⁢
                  
                    
                  
                  ⁢
                  z
                
                =
                
                  d
                  א
                
              
              ,
            
          
        
        
          
            
              
                
                  
                    ∂
                    
                      p
                      ⁡
                      
                        (
                        
                          x
                          ,
                          y
                          ,
                          
                            d
                            א
                          
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ∂
                    z
                  
                
                =
                
                  
                    -
                    
                      
                        (
                        
                          μ
                          
                            k
                            z
                          
                        
                        )
                      
                      א
                    
                  
                  ⁢
                  
                    
                      ψ
                      
                        xyd
                        ⁢
                        א
                      
                    
                    ⁡
                    
                      (
                      
                        x
                        ,
                        y
                        ,
                        t
                      
                      )
                    
                  
                
              
              ,
              
                0
                <
                x
                <
                a
              
              ,
              
                
                  0
                  <
                  y
                  <
                  
                    
                      b
                      .
                      
                        
                      
                      ⁢
                      At
                    
                    ⁢
                    
                      
                    
                    ⁢
                    the
                    ⁢
                    
                      
                    
                    ⁢
                    interface
                    ⁢
                    
                      
                    
                    ⁢
                    z
                  
                
                =
                
                  d
                  j
                
              
              ,
            
          
        
      
    
    
      
        (
        0.1
        )
      
    
  






 






  
    
  
  ⁢
  
    
      
        
          
            
              ψ
              j
            
            ⁡
            
              (
              
                x
                ,
                y
                ,
                t
              
              )
            
          
          =
          
            
              
                -
                
                  
                    (
                    
                      
                        k
                        z
                      
                      μ
                    
                    )
                  
                  j
                
              
              ⁢
              
                (
                
                  
                    ∂
                    
                      
                        p
                        j
                      
                      ⁡
                      
                        (
                        
                          x
                          ,
                          y
                          ,
                          
                            d
                            j
                          
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ∂
                    y
                  
                
                )
              
            
            =
            
              
                -
                
                  
                    (
                    
                      
                        k
                        z
                      
                      μ
                    
                    )
                  
                  
                    j
                    -
                    1
                  
                
              
              ⁢
              
                (
                
                  
                    ∂
                    
                      
                        p
                        
                          j
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          x
                          ,
                          y
                          ,
                          
                            d
                            j
                          
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ∂
                    y
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
            
          
        
      
      
        
          
        
      
    
    
      
        
          
            
              
                
                  λ
                  ⋓
                
                j
              
              ⁢
              
                
                  ψ
                  j
                
                ⁡
                
                  (
                  
                    x
                    ,
                    y
                    ,
                    t
                  
                  )
                
              
            
            =
            
              {
              
                
                  
                    p
                    
                      
                        j
                        ⁢
                        ...
                      
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      x
                      ,
                      y
                      ,
                      
                        d
                        j
                      
                      ,
                      t
                    
                    )
                  
                
                -
                
                  
                    p
                    j
                  
                  ⁡
                  
                    (
                    
                      x
                      ,
                      y
                      ,
                      
                        d
                        j
                      
                      ,
                      t
                    
                    )
                  
                
              
              }
            
          
          ,
          
            
              ∀
              j
            
            ⁢
            
              =
              0
            
          
          ,
          1
          ,
          ...
          ⁢
          
            
          
          ,
          
            א
            -
            
              1.
              ⁢
              
                
              
              ⁢
              The
              ⁢
              
                
              
              ⁢
              initial
              ⁢
              
                
              
              ⁢
              pressure
            
          
        
      
      
        
          
        
      
    
    
      
        
          
            
              
                p
                j
              
              ⁡
              
                (
                
                  x
                  ,
                  y
                  ,
                  z
                  ,
                  0
                
                )
              
            
            =
            
              
                
                  
                    
                      φ
                      j
                    
                    ⁡
                    
                      (
                      
                        x
                        ,
                        y
                        ,
                        z
                      
                      )
                    
                  
                  .
                  
                    
                  
                  ⁢
                  In
                
                ⁢
                
                  
                
                ⁢
                the
                ⁢
                
                  
                
                ⁢
                interval
                ⁢
                
                  
                
                ⁢
                
                  d
                  j
                
              
              ≤
              z
              ≤
              
                d
                
                  j
                  +
                  1
                
              
            
          
          ,
          
            j
            =
            0
          
          ,
          1
          ,
          ...
          ⁢
          
            
          
          ,
          
            א
            -
            1
          
          ,
          
            we
            ⁢
            
              
            
            ⁢
            find
            ⁢
            
              
            
            ⁢
            
              p
              j
            
          
          ,
          
            the
            ⁢
            
              
            
            ⁢
            pressure
            ⁢
            
              
            
            ⁢
            response
          
        
      
      
        
          
        
      
    
    
      
        
          
            corresponding
            ⁢
            
              
            
            ⁢
            any
            ⁢
            
              
            
            ⁢
            perturbation
          
          ,
          
            from
            ⁢
            
              
            
            ⁢
            the
            ⁢
            
              
            
            ⁢
            partial
            ⁢
            
              
            
            ⁢
            differential
            ⁢
            
              
            
            ⁢
            equation
          
        
      
      
        
          
        
      
    
    
      
        
          
            
              ∂
              
                p
                j
              
            
            
              ∂
              t
            
          
          =
          
            
              
                η
                xj
              
              ⁢
              
                
                  
                    ∂
                    2
                  
                  ⁢
                  
                    p
                    j
                  
                
                
                  ∂
                  
                    x
                    2
                  
                
              
            
            +
            
              
                η
                yj
              
              ⁢
              
                
                  
                    
                      ∂
                      2
                    
                    ⁢
                    
                      p
                      j
                    
                  
                  
                    ∂
                    
                      y
                      2
                    
                  
                
                ++
              
              ⁢
              
                
              
              ⁢
              
                η
                zj
              
              ⁢
              
                
                  
                    ∂
                    2
                  
                  ⁢
                  
                    p
                    j
                  
                
                
                  ∂
                  
                    z
                    2
                  
                
              
            
            +
            
              
                U
                ⁡
                
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        j
                      
                    
                  
                  )
                
              
              ⁢
              
                
                  
                    q
                    j
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          j
                        
                      
                    
                    )
                  
                
                
                  
                    (
                    
                      φc
                      t
                    
                    )
                  
                  j
                
              
              ⁢
              
                δ
                ⁡
                
                  (
                  
                    x
                    -
                    
                      x
                      
                        0
                        ⁢
                        j
                      
                    
                  
                  )
                
              
              ⁢
              
                δ
                ⁡
                
                  (
                  
                    y
                    -
                    
                      y
                      
                        0
                        ⁢
                        j
                      
                    
                  
                  )
                
              
              ⁢
              
                δ
                ⁡
                
                  (
                  
                    z
                    -
                    
                      z
                      
                        0
                        ⁢
                        j
                      
                    
                  
                  )
                
              
            
          
        
      
      
        
          
        
      
    
  













In the general solution, the hydrocarbon production occurs through multiple vertical or horizontal wells (e.g., vertical wells (802) and horizontal wells (803)), multiple deviated wells (e.g., deviated wells (804)), and fractures.
The multiple vertical or horizontal wells are modeled as line sources of finite lengths [y02ij−y01ij], [z02ij−z01ij], [x02ij−x01ij] passing through:
(x0ij, y0ij) for t=1, 2 . . . , Li 
(y0ij, z0ij) for t=Li+2 . . . , Mi 
(x0ij, z0ij) for t=Mi+1 . . . , Ni 
The multiple deviated wells are modeled as [(z02ij−z01ij) sin θ0tj], which passes through (x0ij, y0ij, z0ij) for t=Nt+1, . . . , Nd.
The fractures are modeled as rectangle sources of finite area [x02tj−x01tj][y02tj−y01tj], [y02tj−y01tj][z02tj−z01tj], and [x02tj−x01tj][z02tj−z01tj], which passes through:
z0tj for t=Nd+1, . . . , Lr 
x0tj for t=Lr+1, . . . , Mr 
y0tj for t=Mr+1, . . . , Nr 
(Lt<Mt<Nt<Nd<Lr<Mr<Nr)
The pressure solution at any given point [x, y, z] in the reservoir at time t and the derivation to arrive at a set of general expressions is given as the equations (0.2) through (0.8) listed in TABLE 2 below.







TABLE 2











  
    
      
        
          
            
              
                p
                j
              
              =
              
              ⁢
              
                
                  1
                  
                    4
                    ⁢
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      j
                    
                    ⁢
                    ab
                  
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      1
                    
                    
                      L
                      t
                    
                  
                  ⁢
                  
                    
                      U
                      (
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        ∫
                        0
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                      
                      ⁢
                      
                        
                          
                            q
                            
                              i
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                t
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              τ
                            
                            )
                          
                        
                        ×
                      
                    
                  
                
              
            
          
        
        
          
            
               
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  -
                                  
                                    x
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  +
                                  
                                    x
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  -
                                  
                                    y
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                yj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  +
                                  
                                    y
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                yj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    
                      -
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      i
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ⁢
                
                  
                
                ⁢
                d
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
    
    
      
        (
        0.2
        )
      
    
  






 






  
    
      
        
                 
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    b
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  ι
                  =
                  
                    
                      L
                      l
                    
                    +
                    1
                  
                
                
                  M
                  l
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  j
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                x
                                -
                                
                                  x
                                  
                                    
                                      0
                                      ⁢
                                      2
                                    
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            dτ
          
          +
        
      
    
  
   






 






  
    
      
        
                  
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  ι
                  =
                  
                    
                      M
                      l
                    
                    +
                    1
                  
                
                
                  N
                  l
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                -
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              yj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            dτ
          
          +
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  8
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    
                      a
                      ⁢
                      b
                    
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      l
                    
                    +
                    1
                  
                
                
                  N
                  d
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        j
                      
                    
                  
                  )
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  θ
                  
                    0
                    ⁢
                    
                      i
                      ⁢
                      j
                    
                  
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ⁢
                    
                      
                        ∫
                        
                          z
                          ⁢
                          
                            
                          
                          ⁢
                          01
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                        
                          z
                          
                            02
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                      ⁢
                      
                        [
                        
                          {
                          
                            
                              
                                Θ
                                3
                              
                              ⁡
                              
                                (
                                
                                  
                                    
                                      π
                                      
                                        2
                                        ⁢
                                        a
                                      
                                    
                                    ⁢
                                    
                                      {
                                      
                                        x
                                        -
                                        
                                          
                                            (
                                            
                                              
                                                z
                                                
                                                  0
                                                  ⁢
                                                  
                                                    i
                                                    ⁢
                                                    j
                                                  
                                                
                                              
                                              -
                                              
                                                γ
                                                
                                                  0
                                                  ⁢
                                                  
                                                    i
                                                    ⁢
                                                    j
                                                  
                                                
                                              
                                            
                                            )
                                          
                                          ⁢
                                          cot
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            θ
                                            
                                              0
                                              ⁢
                                              
                                                i
                                                ⁢
                                                j
                                              
                                            
                                          
                                          ⁢
                                          cos
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            θ
                                            
                                              0
                                              ⁢
                                              
                                                i
                                                ⁢
                                                j
                                              
                                            
                                          
                                        
                                      
                                      }
                                    
                                  
                                  ,
                                  
                                    e
                                    
                                      
                                        -
                                        
                                          
                                            (
                                            
                                              π
                                              a
                                            
                                            )
                                          
                                          2
                                        
                                      
                                      ⁢
                                      
                                        η
                                        ⁢
                                        xj
                                        ⁢
                                        τ
                                      
                                    
                                  
                                
                                )
                              
                            
                            +
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              +
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          a
                        
                      
                      ⁢
                      
                        {
                        
                          x
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    j
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  ι
                                  ⁢
                                  j
                                
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          ⁢
                          xj
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            b
                          
                        
                        ⁢
                        
                          {
                          
                            y
                            -
                            
                              
                                (
                                
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      i
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      j
                                    
                                  
                                
                                )
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                          
                          }
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            ⁢
                            yj
                            ⁢
                            τ
                          
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              +
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        {
                        
                          y
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                
                                  
                                
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          ⁢
                          yj
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      ij
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      ij
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
              
              ]
            
            ⁢
            
              dz
              
                0
                ⁢
                ij
              
            
            ⁢
            
              d
              ⁢
              τ
            
          
          +
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      d
                    
                    +
                    1
                  
                
                
                  L
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            x
                            +
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            y
                            +
                            
                              y
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          yj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
              
              ]
            
            ⁢
            
              dz
              
                0
                ⁢
                ij
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  a
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      r
                    
                    +
                    1
                  
                
                
                  M
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            y
                            +
                            
                              y
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          yj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            
              d
              ⁢
              τ
            
          
          +
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  b
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      r
                    
                    +
                    1
                  
                
                
                  N
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            x
                            +
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            
              d
              ⁢
              τ
            
          
          +
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    ab
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                    ∫
                    0
                    b
                  
                  ⁢
                  
                    [
                    
                      
                        
                          
                            ψ
                            j
                          
                          ⁡
                          
                            (
                            
                              u
                              ,
                              v
                              ,
                              τ
                            
                            )
                          
                        
                        ⁢
                        
                          Θ
                          3
                        
                        ⁢
                        
                          {
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  
                                    η
                                    zj
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      τ
                                    
                                    )
                                  
                                
                              
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    ψ
                    
                      j
                      +
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      u
                      ,
                      v
                      ,
                      τ
                    
                    )
                  
                
              
              ⁢
              
                Θ
                4
              
              ⁢
              
                {
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          z
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      
                        (
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        
                          η
                          zj
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                    
                  
                
                }
              
            
            ]
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              xj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              xj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              ]
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            
              ×
              
                [
                
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                -
                                v
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                yj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                +
                                v
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                yj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                
                ]
              
              ⁢
              dudvdτ
            
            +
          
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    ab
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  b
                
                ⁢
                
                  
                    ∫
                    0
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                  
                  ⁢
                  
                    
                      [
                      
                        
                          
                            
                              ψ
                              
                                0
                                ⁢
                                yzj
                              
                            
                            ⁡
                            
                              (
                              
                                v
                                ,
                                w
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            Θ
                            3
                          
                          ⁢
                          
                            {
                            
                              
                                
                                  π
                                  ⁢
                                  x
                                
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    
                                      η
                                      xj
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                            
                            )
                          
                        
                        -
                        
                          
                            
                              ψ
                              ayzj
                            
                            ⁡
                            
                              (
                              
                                v
                                ,
                                w
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            Θ
                            4
                          
                          ⁢
                          
                            {
                            
                              
                                
                                  π
                                  ⁢
                                  x
                                
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    
                                      η
                                      xj
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                            
                            )
                          
                        
                      
                      }
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              yj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              yj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              ]
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            
              ×
              
                [
                
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  d
                                  j
                                
                                -
                                w
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  d
                                  j
                                
                                +
                                w
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                
                ]
              
              ⁢
              dvdwd
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    ab
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                    ∫
                    0
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                  
                  ⁢
                  
                    
                      [
                      
                        
                          
                            
                              ψ
                              
                                x
                                ⁢
                                0
                                ⁢
                                zj
                              
                            
                            ⁡
                            
                              (
                              
                                u
                                ,
                                w
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            Θ
                            3
                          
                          ⁢
                          
                            {
                            
                              
                                
                                  π
                                  ⁢
                                  y
                                
                                
                                  2
                                  ⁢
                                  b
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          b
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    
                                      η
                                      yj
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                            
                            )
                          
                        
                        -
                        
                          
                            
                              ψ
                              xbzj
                            
                            ⁡
                            
                              (
                              
                                u
                                ,
                                w
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            Θ
                            4
                          
                          ⁢
                          
                            {
                            
                              
                                
                                  π
                                  ⁢
                                  y
                                
                                
                                  2
                                  ⁢
                                  b
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          b
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    
                                      η
                                      yj
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                            
                            )
                          
                        
                      
                      }
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              xj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              xj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              ]
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            
              ×
              
                [
                
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  d
                                  j
                                
                                -
                                w
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  d
                                  j
                                
                                +
                                w
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                
                ]
              
              ⁢
              dudwd
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  8
                  ⁢
                  
                    ab
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                
                  
                    d
                    
                      j
                      +
                      1
                    
                  
                  -
                  
                    d
                    j
                  
                
              
              ⁢
              
                
                  ∫
                  0
                  b
                
                ⁢
                
                  
                    ∫
                    0
                    a
                  
                  ⁢
                  
                    
                      
                        φ
                        j
                      
                      ⁡
                      
                        (
                        
                          u
                          ,
                          v
                          ,
                          
                            w
                            ⁢
                            
                              
                            
                            +
                            
                              d
                              j
                            
                          
                        
                        )
                      
                    
                    [
                    
                      
                        {
                        
                          
                            
                              Θ
                              3
                            
                            ⁢
                            
                              {
                              
                                
                                  
                                    π
                                    (
                                    
                                      x
                                      -
                                      u
                                    
                                    )
                                  
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            a
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      η
                                      xj
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              )
                            
                          
                          +
                          
                            
                              Θ
                              3
                            
                            ⁢
                            
                              {
                              
                                
                                  
                                    π
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        +
                                        u
                                      
                                      )
                                    
                                  
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            a
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      η
                                      xj
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              )
                            
                          
                        
                        }
                      
                      ×
                    
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
            
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  d
                                  j
                                
                                -
                                w
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            t
                          
                        
                      
                      }
                    
                  
                  +
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  d
                                  j
                                
                                +
                                w
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            t
                          
                        
                      
                      }
                    
                  
                
                }
              
            
            ]
          
          ⁢
          dudvdw
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          where
          ⁢
          
            
          
          ⁢
          
            η
            xj
          
        
        =
        
          
            (
            
              
                k
                x
              
              
                φ
                ⁢
                
                  
                
                ⁢
                
                  c
                  t
                
                ⁢
                μ
              
            
            )
          
          j
        
      
      ,
      
        
          η
          yj
        
        =
        
          
            
              
                (
                
                  
                    k
                    y
                  
                  
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      c
                      t
                    
                    ⁢
                    μ
                  
                
                )
              
              j
            
            ⁢
            
              
            
            ⁢
            and
            ⁢
            
              
            
            ⁢
            
              η
              zj
            
          
          =
          
            
              
                
                  (
                  
                    
                      k
                      z
                    
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      
                        c
                        t
                      
                      ⁢
                      μ
                    
                  
                  )
                
                j
              
              .
              
                
              
              ⁢
              We
            
            ⁢
            
              
            
            ⁢
            employ
          
        
      
      ,
      
        
      
      ⁢
      
        in
        ⁢
        
          
        
        ⁢
        time
        ⁢
        
          
        
        ⁢
        domain
      
      ,
      
        the
        ⁢
        
          
        
        ⁢
        interfacial
        ⁢
        
          

        
        ⁢
        boundary
        ⁢
        
          
        
        ⁢
        
          condition
          .
          
            
          
          ⁢
          Substituting
        
        ⁢
        
          
        
        ⁢
        for
        ⁢
        
          
        
        ⁢
        
          
            p
            j
          
          ⁡
          
            (
            
              x
              ,
              y
              ,
              
                d
                j
              
              ,
              t
            
            )
          
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        
          
            p
            
              j
              -
              1
            
          
          ⁡
          
            (
            
              x
              ,
              y
              ,
              
                d
                j
              
              ,
              t
            
            )
          
        
        ⁢
        
          
        
        ⁢
        from
        ⁢
        
          
        
        ⁢
        equation
        ⁢
        
          
        
        ⁢
        
          (
          0.2
          )
        
        ⁢
        
          
        
        ⁢
        in
      
    
  






 






  
    
      
        
          
            
              
                
                  λ
                  ⋓
                
                j
              
              ⁢
              
                
                  ψ
                  j
                
                ⁡
                
                  (
                  
                    x
                    ,
                    y
                    ,
                    t
                  
                  )
                
              
            
            =
            
              {
              
                
                  
                    p
                    
                      j
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      x
                      ,
                      y
                      ,
                      
                        d
                        j
                      
                      ,
                      t
                    
                    )
                  
                
                -
                
                  
                    p
                    j
                  
                  ⁡
                  
                    (
                    
                      x
                      ,
                      y
                      ,
                      
                        d
                        j
                      
                      ,
                      t
                    
                    )
                  
                
              
              }
            
          
          ,
          
            
          
          ⁢
          
            
              ∀
              j
            
            ⁢
            
              =
              1
            
          
          ,
          2
          ,
          ...
          ⁢
          
            
          
          ,
          
            א
            -
            1
          
        
        ⁢
        
          

        
        ⁢
        
          we
          ⁢
          
            
          
          ⁢
          obtain
          ⁢
          
            
          
          ⁢
          a
          ⁢
          
            
          
          ⁢
          three
          ⁢
          
            -
          
          ⁢
          point
          ⁢
          
            
          
          ⁢
          
            recurrence
          
          ⁢
          
            
          
          ⁢
          integral
          ⁢
          
            
          
          ⁢
          equation
          ⁢
          
            
          
          ⁢
          relationship
          ⁢
          
            
          
          ⁢
          in
          ⁢
          
            
          
          ⁢
          time
          ⁢
          
            
          
          ⁢
          and
          ⁢
          
            
          
          ⁢
          space
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  
                    λ
                    ⋓
                  
                  j
                
                ⁢
                
                  
                    ψ
                    j
                  
                  ⁡
                  
                    (
                    
                      x
                      ,
                      y
                      ,
                      t
                    
                    )
                  
                
              
              =
              
              ⁢
              
                
                  
                    ∫
                    0
                    t
                  
                  ⁢
                  
                    
                      ∫
                      0
                      a
                    
                    ⁢
                    
                      
                        ∫
                        0
                        b
                      
                      ⁢
                      
                        
                          
                            A
                            j
                          
                          ⁡
                          
                            (
                            
                              x
                              ,
                              u
                              ,
                              
                                y
                                .
                                v
                              
                              ,
                              
                                t
                                -
                                τ
                              
                            
                            )
                          
                        
                        ⁢
                        
                          
                            ψ
                            j
                          
                          ⁡
                          
                            (
                            
                              u
                              ,
                              v
                              ,
                              τ
                            
                            )
                          
                        
                        ⁢
                        dvdud
                        ⁢
                        
                          
                        
                        ⁢
                        τ
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  +
                  
                    
                      ∫
                      0
                      t
                    
                    ⁢
                    
                      
                        ∫
                        0
                        a
                      
                      ⁢
                      
                        
                          ∫
                          0
                          b
                        
                        ⁢
                        
                          
                            
                              B
                              j
                            
                            ⁡
                            
                              (
                              
                                x
                                ,
                                u
                                ,
                                y
                                ,
                                v
                                ,
                                
                                  t
                                  -
                                  τ
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              ψ
                              
                                j
                                +
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                u
                                ,
                                v
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          dvdud
                          ⁢
                          
                            
                          
                          ⁢
                          τ
                        
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  +
                  
                    
                      ∫
                      0
                      t
                    
                    ⁢
                    
                      
                        ∫
                        0
                        a
                      
                      ⁢
                      
                        
                          ∫
                          0
                          b
                        
                        ⁢
                        
                          
                            
                              C
                              j
                            
                            ⁡
                            
                              (
                              
                                x
                                ,
                                u
                                ,
                                y
                                ,
                                v
                                ,
                                
                                  t
                                  -
                                  τ
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              ψ
                              
                                j
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                u
                                ,
                                v
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          dvdud
                          ⁢
                          
                            
                          
                          ⁢
                          τ
                        
                      
                    
                  
                
                +
                
                  
                    Ω
                    j
                  
                  ⁡
                  
                    (
                    
                      x
                      ,
                      y
                      ,
                      t
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        0.3
        )
      
    
  
  
    
      
        
          
            The
            ⁢
            
              
            
            ⁢
            coefficients
            ⁢
            
              
            
            ⁢
            of
            ⁢
            
              
            
            ⁢
            the
            ⁢
            
              
            
            ⁢
            recurrence
            ⁢
            
              
            
            ⁢
            integral
            ⁢
            
              
            
            ⁢
            equation
            ⁢
            
              
            
            ⁢
            
              (
              0.3
              )
            
            ⁢
            
              
            
            ⁢
            for
            ⁢
            
              
            
            ⁢
            
              d
              j
            
          
          <
          z
          <
          
            d
            
              j
              +
              1
            
          
        
        ,
        
          
            ∀
            j
          
          ⁢
          
            =
            1
          
        
        ,
        2
        ,
        ...
        ⁢
        
          
        
        ,
        
          א
          -
          1
        
        ,
        
          are
          ⁢
          
            
          
          ⁢
          given
          ⁢
          
            
          
          ⁢
          by
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            A
            j
          
          ⁡
          
            (
            
              x
              ,
              u
              ,
              y
              ,
              v
              ,
              t
            
            )
          
        
        =
        
          
            
              g
              j
            
            ⁡
            
              (
              
                x
                ,
                u
                ,
                y
                ,
                v
                ,
                t
              
              )
            
          
          +
          
            
              g
              
                j
                -
                1
              
            
            ⁡
            
              (
              
                x
                ,
                u
                ,
                y
                ,
                v
                ,
                t
              
              )
            
          
        
      
    
    
      
        (
        0.4
        )
      
    
  






 






  
    
      
        
          
            
              
                
                  g
                  j
                
                ⁡
                
                  (
                  
                    x
                    ,
                    u
                    ,
                    y
                    ,
                    v
                    ,
                    t
                  
                  )
                
              
              =
              
              ⁢
              
                
                  -
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          0
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  zj
                                
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                        }
                      
                    
                    
                      4
                      ⁢
                      
                        
                          (
                          
                            φc
                            t
                          
                          )
                        
                        j
                      
                      ⁢
                      
                        ab
                        ⁡
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                  
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  [
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  -
                                  u
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    x
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  +
                                  u
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    x
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                  
                  ]
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  [
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  -
                                  v
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  yj
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  +
                                  v
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  yj
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        (
        0.5
        )
      
    
  
  
    
      
        
          
            
              
                
                  B
                  j
                
                ⁡
                
                  (
                  
                    x
                    ,
                    u
                    ,
                    y
                    ,
                    v
                    ,
                    t
                  
                  )
                
              
              =
              
              ⁢
              
                
                  
                    
                      Θ
                      4
                    
                    ⁢
                    
                      {
                      
                        0
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  z
                                  ⁢
                                  j
                                
                              
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                      }
                    
                  
                  
                    4
                    ⁢
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      j
                    
                    ⁢
                    
                      ab
                      ⁡
                      
                        (
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                    
                  
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  [
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  -
                                  u
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    x
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  +
                                  u
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    x
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                  
                  ]
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  [
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  -
                                  v
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  yj
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  +
                                  v
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  yj
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        (
        0.6
        )
      
    
  
  
    
      
        
          
            C
            j
          
          ⁡
          
            (
            
              x
              ,
              u
              ,
              y
              ,
              v
              ,
              t
            
            )
          
        
        =
        
          
            B
            
              j
              -
              1
            
          
          ⁡
          
            (
            
              x
              ,
              u
              ,
              y
              ,
              v
              ,
              t
            
            )
          
        
      
    
    
      
        (
        0.7
        )
      
    
  






 






  
    
      
        
          
            
              
                
                  Ω
                  j
                
                ⁡
                
                  (
                  
                    x
                    ,
                    y
                    ,
                    t
                  
                  )
                
              
              =
              
              ⁢
              
                
                  1
                  
                    4
                    ⁢
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      
                        j
                        -
                        1
                      
                    
                    ⁢
                    
                      a
                      ⁢
                      b
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      ι
                      =
                      1
                    
                    
                      L
                      t
                    
                  
                  ⁢
                  
                    
                      U
                      (
                      
                        t
                        -
                        
                          t
                          
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                            -
                            1
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        ∫
                        0
                        
                          t
                          -
                          
                            t
                            
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                              -
                              1
                            
                          
                        
                      
                      ⁢
                      
                        
                          
                            q
                            
                              
                                ι
                                ⁢
                                j
                              
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                t
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                              -
                              τ
                            
                            )
                          
                        
                        ×
                      
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  -
                                  
                                    x
                                    
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  xj
                                  -
                                  1
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  +
                                  
                                    x
                                    
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  xj
                                  -
                                  1
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  -
                                  
                                    y
                                    
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  yj
                                  -
                                  1
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  +
                                  
                                    y
                                    
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  yj
                                  -
                                  1
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ×
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
                                          
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    z
                                    
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          j
                                        
                                        -
                                        
                                          d
                                          
                                            j
                                            -
                                            1
                                          
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  zj
                                  -
                                  1
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                +
                                
                                  z
                                  
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  d
                                  
                                    j
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        j
                                      
                                      -
                                      
                                        d
                                        
                                          j
                                          -
                                          1
                                        
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                zj
                                -
                                1
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  z
                                  
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  d
                                  
                                    j
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        j
                                      
                                      -
                                      
                                        d
                                        
                                          j
                                          -
                                          1
                                        
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                zj
                                -
                                1
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  +
                                  
                                    z
                                    
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      
                                        j
                                        -
                                        1
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          j
                                        
                                        -
                                        
                                          d
                                          
                                            j
                                            -
                                            1
                                          
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  zj
                                  -
                                  1
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ⁢
                d
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
    
    
      
        
      
    
  






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    
                      j
                      -
                      1
                    
                  
                  ⁢
                  
                    b
                    ⁡
                    
                      (
                      
                        
                          d
                          j
                        
                        -
                        
                          d
                          
                            j
                            -
                            1
                          
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      t
                    
                    +
                    1
                  
                
                
                  M
                  t
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                        -
                        1
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                          -
                          1
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          
                            i
                            ⁢
                            j
                          
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                              -
                              1
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                z
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      j
                                    
                                    -
                                    
                                      d
                                      
                                        j
                                        -
                                        1
                                      
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              zj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              +
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              d
                              j
                            
                            +
                            
                              z
                              
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              j
                            
                            -
                            
                              d
                              
                                j
                                -
                                1
                              
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            zj
                            -
                            1
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            x
                            +
                            
                              x
                              
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            xj
                            -
                            1
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              }
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   






 






  
    
      
        
                                
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    
                      j
                      -
                      1
                    
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        
                          d
                          j
                        
                        -
                        
                          d
                          
                            j
                            -
                            1
                          
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      t
                    
                    +
                    1
                  
                
                
                  N
                  t
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                        -
                        1
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                          -
                          1
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          
                            i
                            ⁢
                            j
                          
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                              -
                              1
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                z
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      j
                                    
                                    -
                                    
                                      d
                                      
                                        j
                                        -
                                        1
                                      
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              zj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              +
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              d
                              j
                            
                            +
                            
                              z
                              
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              j
                            
                            -
                            
                              d
                              
                                j
                                -
                                1
                              
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            zj
                            -
                            1
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                -
                                
                                  y
                                  
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                yj
                                -
                                1
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  8
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    
                      j
                      -
                      1
                    
                  
                  ⁢
                  
                    ab
                    ⁡
                    
                      (
                      
                        
                          d
                          j
                        
                        -
                        
                          d
                          
                            j
                            -
                            1
                          
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      t
                    
                    +
                    1
                  
                
                
                  N
                  d
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                        -
                        1
                      
                    
                  
                  )
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  θ
                  
                    
                      0
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                    -
                    1
                  
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                          -
                          1
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          
                            i
                            ⁢
                            j
                          
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                              -
                              1
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                
                  z
                  
                    
                      01
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                    -
                    1
                  
                
                
                  z
                  
                    
                      02
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                    -
                    1
                  
                
              
              ⁢
              
                [
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              
                                2
                                ⁢
                                a
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              {
                              
                                x
                                -
                                
                                  
                                    (
                                    
                                      
                                        z
                                        
                                          
                                            0
                                            ⁢
                                            
                                              i
                                              ⁢
                                              j
                                            
                                          
                                          -
                                          1
                                        
                                      
                                      -
                                      
                                        γ
                                        
                                          
                                            0
                                            ⁢
                                            
                                              i
                                              ⁢
                                              j
                                            
                                          
                                          -
                                          1
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  cot
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                
                              
                              }
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  xj
                                  -
                                  1
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              +
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          a
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        {
                        
                          x
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                                -
                                
                                  γ
                                  
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                              
                              )
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            zj
                            -
                            1
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            b
                          
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          {
                          
                            y
                            -
                            
                              
                                (
                                
                                  
                                    z
                                    
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                  -
                                  
                                    γ
                                    
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                              ⁢
                              
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                          
                          }
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              +
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        {
                        
                          y
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                                -
                                
                                  γ
                                  
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                              
                              )
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            yj
                            -
                            1
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                z
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      j
                                    
                                    -
                                    
                                      d
                                      
                                        j
                                        -
                                        1
                                      
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
              
                
                ⁢
                
                  +
                  
                    
                      Θ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                +
                                
                                  z
                                  
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  d
                                  
                                    j
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        j
                                      
                                      -
                                      
                                        d
                                        
                                          j
                                          -
                                          1
                                        
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                yj
                                -
                                1
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ]
            
            ⁢
            
              
            
            ⁢
            
              dz
              
                
                  0
                  ⁢
                  
                    i
                    ⁢
                    j
                  
                
                -
                1
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    
                      j
                      -
                      1
                    
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        j
                      
                      -
                      
                        d
                        
                          j
                          -
                          1
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      d
                    
                    +
                    1
                  
                
                
                  L
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                        -
                        1
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                          -
                          1
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          
                            i
                            ⁢
                            j
                          
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                              -
                              1
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            x
                            +
                            
                              x
                              
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            xj
                            -
                            1
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            y
                            +
                            
                              y
                              
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            yj
                            -
                            1
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                z
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      j
                                    
                                    -
                                    
                                      d
                                      
                                        j
                                        -
                                        1
                                      
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              zj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  +
                  
                    Θ
                    3
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              d
                              j
                            
                            +
                            
                              z
                              
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              j
                            
                            -
                            
                              d
                              
                                j
                                -
                                1
                              
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            zj
                            -
                            1
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              }
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    
                      j
                      -
                      1
                    
                  
                  ⁢
                  a
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      r
                    
                    +
                    1
                  
                
                
                  M
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                        -
                        1
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                          -
                          1
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          
                            i
                            ⁢
                            j
                          
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                ι
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            y
                            +
                            
                              y
                              
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            yj
                            -
                            1
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
            
                                       
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    z
                                    
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          j
                                        
                                        -
                                        
                                          d
                                          
                                            j
                                            -
                                            1
                                          
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  zj
                                  -
                                  1
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                +
                                
                                  z
                                  
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  d
                                  
                                    j
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        j
                                      
                                      -
                                      
                                        d
                                        
                                          j
                                          -
                                          1
                                        
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                zj
                                -
                                1
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  z
                                  
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  d
                                  
                                    j
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        j
                                      
                                      -
                                      
                                        d
                                        
                                          j
                                          -
                                          1
                                        
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                zj
                                -
                                1
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  +
                                  
                                    z
                                    
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      
                                        j
                                        -
                                        1
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          j
                                        
                                        -
                                        
                                          d
                                          
                                            j
                                            -
                                            1
                                          
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  zj
                                  -
                                  1
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ⁢
                
                  
                
                ⁢
                d
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
       
    
  






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    
                      j
                      -
                      1
                    
                  
                  ⁢
                  b
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      r
                    
                    +
                    1
                  
                
                
                  N
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                        -
                        1
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                          -
                          1
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          
                            ι
                            ⁢
                            j
                          
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                              -
                              1
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              xj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            x
                            +
                            
                              x
                              
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                -
                                1
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            xj
                            -
                            1
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
            
                                       
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    z
                                    
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          j
                                        
                                        -
                                        
                                          d
                                          
                                            j
                                            -
                                            1
                                          
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  zj
                                  -
                                  1
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                +
                                
                                  z
                                  
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  d
                                  
                                    j
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        j
                                      
                                      -
                                      
                                        d
                                        
                                          j
                                          -
                                          1
                                        
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                zj
                                -
                                1
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  z
                                  
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  d
                                  
                                    j
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        j
                                      
                                      -
                                      
                                        d
                                        
                                          j
                                          -
                                          1
                                        
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                zj
                                -
                                1
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  +
                                  
                                    z
                                    
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                      -
                                      1
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      
                                        j
                                        -
                                        1
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          j
                                        
                                        -
                                        
                                          d
                                          
                                            j
                                            -
                                            1
                                          
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  zj
                                  -
                                  1
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ⁢
                
                  
                
                ⁢
                d
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
       
    
  






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    
                      j
                      -
                      1
                    
                  
                  ⁢
                  
                    ab
                    (
                    
                      
                        d
                        j
                      
                      -
                      
                        d
                        
                          j
                          -
                          1
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  b
                
                ⁢
                
                  
                    ∫
                    0
                    
                      (
                      
                        
                          d
                          j
                        
                        -
                        
                          d
                          
                            j
                            -
                            1
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    [
                    
                      
                        
                          
                            ψ
                            
                              
                                0
                                ⁢
                                yzj
                              
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              v
                              ,
                              w
                              ,
                              τ
                            
                            )
                          
                        
                        ⁢
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  ⁢
                                  x
                                
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    
                                      η
                                      
                                        xj
                                        -
                                        1
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                            
                            )
                          
                        
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    ψ
                    
                      ayzj
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      v
                      ,
                      w
                      ,
                      τ
                    
                    )
                  
                
              
              ⁢
              
                Θ
                4
              
              ⁢
              
                {
                
                  
                    
                      π
                      ⁢
                      x
                    
                    
                      2
                      ⁢
                      a
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        
                          η
                          
                            xj
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                    
                  
                
                )
              
            
            ]
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                yj
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                yj
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              ]
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      j
                                    
                                    -
                                    
                                      d
                                      
                                        j
                                        -
                                        1
                                      
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                zj
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
                
              ⁢
              
                
                  +
                  
                    Θ
                    3
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              d
                              j
                            
                            -
                            
                              d
                              
                                j
                                -
                                1
                              
                            
                            +
                            w
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              j
                            
                            -
                            
                              d
                              
                                j
                                -
                                1
                              
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              zj
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              ]
            
            ⁢
            dvdwd
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    
                      j
                      -
                      1
                    
                  
                  ⁢
                  
                    ab
                    (
                    
                      
                        d
                        j
                      
                      -
                      
                        d
                        
                          j
                          -
                          1
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                    ∫
                    0
                    
                      
                        d
                        j
                      
                      -
                      
                        d
                        
                          j
                          -
                          1
                        
                      
                    
                  
                  ⁢
                  
                    [
                    
                      
                        
                          
                            ψ
                            
                              
                                x
                                ⁢
                                0
                                ⁢
                                zj
                              
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              u
                              ,
                              w
                              ,
                              τ
                            
                            )
                          
                        
                        ⁢
                        
                          Θ
                          3
                        
                        ⁢
                        
                          {
                          
                            
                              
                                π
                                ⁢
                                y
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  
                                    η
                                    
                                      yj
                                      -
                                      1
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      τ
                                    
                                    )
                                  
                                
                              
                            
                          
                          )
                        
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    ψ
                    
                      xbzj
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      u
                      ,
                      w
                      ,
                      τ
                    
                    )
                  
                
              
              ⁢
              
                Θ
                4
              
              ⁢
              
                {
                
                  
                    
                      π
                      ⁢
                      y
                    
                    
                      2
                      ⁢
                      b
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        
                          η
                          
                            yj
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                    
                  
                
                )
              
            
            ]
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                xj
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                xj
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              ]
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      j
                                    
                                    -
                                    
                                      d
                                      
                                        j
                                        -
                                        1
                                      
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                zj
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
                
              ⁢
              
                
                  +
                  
                    Θ
                    3
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              d
                              j
                            
                            -
                            
                              d
                              
                                j
                                -
                                1
                              
                            
                            +
                            w
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              j
                            
                            -
                            
                              d
                              
                                j
                                -
                                1
                              
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    j
                                  
                                  -
                                  
                                    d
                                    
                                      j
                                      -
                                      1
                                    
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              zj
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              ]
            
            ⁢
            dudwd
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  8
                  ⁢
                  
                    ab
                    ⁡
                    
                      (
                      
                        
                          d
                          j
                        
                        -
                        
                          d
                          
                            j
                            -
                            1
                          
                        
                      
                      )
                    
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                
                  
                    d
                    j
                  
                  -
                  
                    d
                    
                      j
                      -
                      1
                    
                  
                
              
              ⁢
              
                
                  ∫
                  0
                  b
                
                ⁢
                
                  
                    ∫
                    0
                    a
                  
                  ⁢
                  
                    
                      
                        φ
                        ⁡
                        
                          (
                          
                            u
                            ,
                            v
                            ,
                            
                              w
                              +
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                          
                          )
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              Θ
                              3
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    (
                                    
                                      x
                                      -
                                      u
                                    
                                    )
                                  
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            a
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      η
                                      
                                        zj
                                        -
                                        1
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              )
                            
                          
                          +
                          
                            
                              Θ
                              3
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        +
                                        u
                                      
                                      )
                                    
                                  
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            a
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      η
                                      
                                        zj
                                        -
                                        1
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              yj
                              -
                              1
                            
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
              
              ]
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      j
                                    
                                    -
                                    
                                      d
                                      
                                        j
                                        -
                                        1
                                      
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              zj
                              -
                              1
                            
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                d
                                
                                  j
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      j
                                    
                                    -
                                    
                                      d
                                      
                                        j
                                        -
                                        1
                                      
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              zj
                              -
                              1
                            
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
              
              ]
            
            ⁢
            dudvdw
          
          -
        
      
    
  
   






 





  
    
      
        
          
            
          
          ⁢
          
            
              -
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  ab
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  1
                
                
                  L
                  t
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        ij
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          ι
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ⁢
              
                d
                ⁢
                τ
              
            
            -
          
        
      
    
  




  
    
      
        
          
            
              
            
            ⁢
            
              
                -
                
                ⁢
                
                  1
                  
                    2
                    ⁢
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      j
                    
                    ⁢
                    
                      b
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                
              
              ×
            
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                
                  ∑
                  
                    i
                    =
                    
                      
                        L
                        t
                      
                      +
                      1
                    
                  
                  
                    M
                    t
                  
                
                ⁢
                
                  
                    U
                    (
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      ∫
                      0
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                    
                    ⁢
                    
                      
                        
                          q
                          
                            i
                            ⁢
                            j
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              t
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            τ
                          
                          )
                        
                      
                      ⁢
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      d
                                      j
                                    
                                    -
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      ×
                    
                  
                
              
            
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                -
                                
                                  y
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              yj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  +
                  
                    
                      Θ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                +
                                
                                  y
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ×
            
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                x
                                -
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                x
                                +
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
        
      
      
        
          
            
              
                
                ⁢
                
                  
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  -
                                  
                                    x
                                    
                                      02
                                      ⁢
                                      ij
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                x
                                +
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              τ
            
            -
          
        
      
    
     
  





 






  
    
  
  ⁢
  
    
      
        
          
            
              -
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    a
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                
                  ∑
                  
                    ι
                    =
                    
                      
                        M
                        r
                      
                      +
                      1
                    
                  
                  
                    N
                    t
                  
                
                ⁢
                
                  
                    U
                    (
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      ∫
                      0
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                    
                    ⁢
                    
                      
                        
                          q
                          
                            i
                            ⁢
                            j
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              t
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            τ
                          
                          )
                        
                      
                      ⁢
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      d
                                      j
                                    
                                    -
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      ×
                    
                  
                
              
            
          
        
      
    
    ⁢
    
       
      
        

      
      ⁢
      
        
          
            
              
                                          
                ⁢
                
                  ×
                  
                    {
                    
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    x
                                    -
                                    
                                      x
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                a
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zxj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    x
                                    +
                                    
                                      x
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                a
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  xj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                  ×
                
              
            
          
          
            
              
                
                ⁢
                
                  ×
                  
                    {
                    
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    -
                                    
                                      y
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  yj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      -
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    +
                                    
                                      y
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  yj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      -
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                    ⁢
                    
                      
                        -
                        
                          
                            Θ
                            3
                            ∫
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      y
                                      -
                                      
                                        y
                                        
                                          02
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  b
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          b
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    yj
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    +
                                    
                                      y
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  yj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                  ⁢
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  τ
                
                -
              
            
          
        
         
      
    
  






 






  
    
      
        
                                     
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            b
                          
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          {
                          
                            y
                            -
                            
                              
                                (
                                
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      i
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      j
                                    
                                  
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                          
                          }
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              +
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        {
                        
                          y
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          yj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
            
              
              ⁢
              
                ×
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                d
                                j
                              
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              ]
            
            ⁢
            
              dz
              
                0
                ⁢
                
                  ι
                  ⁢
                  j
                
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          -
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              -
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  ι
                  =
                  
                    
                      N
                      d
                    
                    +
                    1
                  
                
                
                  L
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ⁢
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            x
                            +
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                -
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              yj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          -
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              -
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    ab
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      t
                    
                    +
                    1
                  
                
                
                  N
                  d
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  θ
                  
                    0
                    ⁢
                    
                      i
                      ⁢
                      j
                    
                  
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                
                  z
                  
                    01
                    ⁢
                    
                      i
                      ⁢
                      j
                    
                  
                
                
                  z
                  
                    02
                    ⁢
                    
                      i
                      ⁢
                      j
                    
                  
                
              
              ⁢
              
                [
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              
                                2
                                ⁢
                                a
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              {
                              
                                x
                                -
                                
                                  
                                    (
                                    
                                      
                                        z
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                      -
                                      
                                        γ
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  cot
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                              
                              }
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              +
              
                
                  Θ
                  3
                
                (
                
                  {
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          a
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        {
                        
                          x
                          -
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          zj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
                }
              
            
            ×
          
        
      
    
  
   






 






  
    
      
        
          
            
          
          ⁢
          
            
              -
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  a
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      r
                    
                    +
                    1
                  
                
                
                  M
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            y
                            +
                            
                              y
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          yj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ⁢
              
                d
                ⁢
                τ
              
            
            -
          
        
      
    
  
   






 





  
    
      
        
          
            
          
          ⁢
          
            
              -
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  b
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      r
                    
                    +
                    1
                  
                
                
                  N
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            x
                            +
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    w
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  d
                                  j
                                
                                -
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ⁢
              
                d
                ⁢
                τ
              
            
            -
          
        
      
    
  




  
    
      
        
          
            
          
          ⁢
          
            
              -
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    ab
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  b
                
                ⁢
                
                  
                    ∫
                    0
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                  
                  ⁢
                  
                    [
                    
                      
                        
                          
                            ψ
                            
                              0
                              ⁢
                              yzj
                            
                          
                          ⁡
                          
                            (
                            
                              v
                              ,
                              w
                              ,
                              τ
                            
                            )
                          
                        
                        ⁢
                        
                          Θ
                          3
                        
                        ⁢
                        
                          {
                          
                            
                              
                                π
                                ⁢
                                x
                              
                              
                                2
                                ⁢
                                a
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  
                                    η
                                    xj
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      τ
                                    
                                    )
                                  
                                
                              
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    ψ
                    ayzj
                  
                  ⁡
                  
                    (
                    
                      v
                      ,
                      w
                      ,
                      τ
                    
                    )
                  
                
              
              ⁢
              
                Θ
                4
              
              ⁢
              
                {
                
                  
                    
                      π
                      ⁢
                      x
                    
                    
                      2
                      ⁢
                      a
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        
                          η
                          xj
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                    
                  
                
                )
              
            
            ]
          
          ⁢
          
            
              Θ
              3
            
            ⁡
            
              (
              
                
                  πw
                  
                    2
                    ⁢
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                
                ,
                
                  e
                  
                    
                      -
                      
                        
                          (
                          
                            π
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                          
                          )
                        
                        2
                      
                    
                    ⁢
                    
                      
                        η
                        zj
                      
                      (
                      
                        t
                        -
                        τ
                      
                      )
                    
                  
                
              
              )
            
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            
              ×
              
                [
                
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                -
                                v
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                yj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                +
                                v
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                yj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                
                ]
              
              ⁢
              dvdwd
              ⁢
              
                
              
              ⁢
              τ
            
            -
          
        
      
    
  





 






  
    
      
        
          
            
              
                
                  
                
                ⁢
                
                  
                    -
                    
                    ⁢
                    
                      1
                      
                        2
                        ⁢
                        
                          
                            (
                            
                              φc
                              t
                            
                            )
                          
                          j
                        
                        ⁢
                        
                          ab
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                  
                  ⁢
                  
                    
                      ∫
                      0
                      t
                    
                    ⁢
                    
                      
                        ∫
                        0
                        a
                      
                      ⁢
                      
                        
                          ∫
                          0
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                        
                        ⁢
                        
                          [
                          
                            
                              
                                
                                  ψ
                                  
                                    x
                                    ⁢
                                    0
                                    ⁢
                                    zj
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    u
                                    ,
                                    w
                                    ,
                                    τ
                                  
                                  )
                                
                              
                              ⁢
                              
                                Θ
                                3
                              
                              ⁢
                              
                                {
                                
                                  
                                    
                                      π
                                      ⁢
                                      y
                                    
                                    
                                      2
                                      ⁢
                                      b
                                    
                                  
                                  ,
                                  
                                    e
                                    
                                      
                                        -
                                        
                                          
                                            (
                                            
                                              
                                                π
                                                ⁢
                                                y
                                              
                                              b
                                            
                                            )
                                          
                                          2
                                        
                                      
                                      ⁢
                                      
                                        
                                          η
                                          
                                            y
                                            ⁢
                                            j
                                          
                                        
                                        ⁡
                                        
                                          (
                                          
                                            t
                                            -
                                            τ
                                          
                                          )
                                        
                                      
                                    
                                  
                                
                                }
                              
                            
                            -
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                  ⁢
                  
                    
                      -
                      
                        
                          ψ
                          xbzj
                        
                        ⁡
                        
                          (
                          
                            u
                            ,
                            w
                            ,
                            τ
                          
                          )
                        
                      
                    
                    ⁢
                    
                      
                        Θ
                        4
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁢
                              y
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  yj
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        )
                      
                    
                  
                  ]
                
                ⁢
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        πw
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                    )
                  
                
                ×
              
            
          
          
            
              
                
                ⁢
                
                  
                    ×
                    
                      [
                      
                        
                          
                            Θ
                            3
                          
                          ⁢
                          
                            {
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      -
                                      u
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    
                                      η
                                      
                                        x
                                        ⁢
                                        j
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                            
                            }
                          
                        
                        +
                        
                          
                            Θ
                            3
                          
                          ⁢
                          
                            {
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      +
                                      u
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    
                                      η
                                      xj
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                            
                            }
                          
                        
                      
                      ]
                    
                    ⁢
                    dudwd
                    ⁢
                    
                      
                    
                    ⁢
                    τ
                  
                  -
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  
                
                ⁢
                
                  
                    -
                    
                    ⁢
                    
                      1
                      
                        4
                        ⁢
                        
                          ab
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                  
                  ⁢
                  
                    
                      ∫
                      0
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                    
                    ⁢
                    
                      
                        ∫
                        0
                        b
                      
                      ⁢
                      
                        
                          ∫
                          0
                          a
                        
                        ⁢
                        
                          
                            
                              φ
                              j
                            
                            ⁡
                            
                              (
                              
                                u
                                ,
                                v
                                ,
                                
                                  w
                                  +
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          ⁢
                          
                            Θ
                            3
                          
                          ⁢
                          
                            {
                            
                              
                                
                                  π
                                  ⁢
                                  w
                                
                                
                                  2
                                  ⁢
                                  
                                    (
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                    )
                                  
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          
                                            
                                              d
                                              
                                                j
                                                +
                                                1
                                              
                                            
                                            -
                                            
                                              d
                                              j
                                            
                                          
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    zj
                                  
                                  ⁢
                                  t
                                
                              
                            
                            )
                          
                          ×
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  ×
                  
                    [
                    
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    x
                                    -
                                    u
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                a
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  xj
                                
                                ⁢
                                t
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    x
                                    +
                                    u
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                a
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  xj
                                
                                ⁢
                                t
                              
                            
                          
                          )
                        
                      
                    
                    ]
                  
                  ×
                
              
            
          
          
            
              
                
                ⁢
                
                  ×
                  
                    [
                    
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    -
                                    v
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  yj
                                
                                ⁢
                                t
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    +
                                    v
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  yj
                                
                                ⁢
                                t
                              
                            
                          
                          )
                        
                      
                    
                    ]
                  
                  ⁢
                  dudvd
                  ⁢
                  w
                
              
            
          
        
      
    
    
      
        (
        0.8
        )
      
    
  






 






  j
  ×
  
    {
    
      
        
          Θ
          3
        
        ⁡
        
          (
          
            
              
                π
                ⁡
                
                  (
                  
                    y
                    -
                    
                      y
                      
                        0
                        ⁢
                        ij
                      
                    
                  
                  )
                
              
              
                2
                ⁢
                b
              
            
            ,
            
              e
              
                
                  -
                  
                    
                      (
                      
                        π
                        b
                      
                      )
                    
                    2
                  
                
                ⁢
                
                  η
                  yj
                
                ⁢
                τ
              
            
          
          )
        
      
      +
      
        
          Θ
          3
        
        ⁡
        
          (
          
            
              
                π
                ⁡
                
                  (
                  
                    y
                    -
                    
                      y
                      
                        0
                        ⁢
                        ij
                      
                    
                  
                  )
                
              
              
                2
                ⁢
                b
              
            
            ,
            
              e
              
                
                  -
                  
                    
                      (
                      
                        π
                        b
                      
                      )
                    
                    2
                  
                
                ⁢
                
                  η
                  yj
                
                ⁢
                τ
              
            
          
          )
        
      
    
    }
  
  ×
  ×
  
    {
    
      
        
          
            Θ
            3
            ∫
          
          (
          
            
              
                π
                ⁡
                
                  (
                  
                    z
                    -
                    
                      z
                      
                        01
                        ⁢
                        ij
                      
                    
                  
                  )
                
              
              
                2
                ⁢
                
                  (
                  
                    
                      d
                      
                        j
                        +
                        1
                      
                    
                    -
                    
                      d
                      j
                    
                  
                  )
                
              
            
            ,
            
              e
              
                
                  -
                  
                    
                      (
                      
                        π
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                      
                      )
                    
                    2
                  
                
                ⁢
                
                  η
                  zj
                
                ⁢
                τ
              
            
          
          )
        
        -
        
          
            
              Θ
              3
              ∫
            
            (
            
              
                
                  π
                  ⁡
                  
                    (
                    
                      z
                      +
                      
                        z
                        
                          01
                          ⁢
                          ij
                        
                      
                      -
                      
                        2
                        ⁢
                        
                          d
                          j
                        
                      
                    
                    )
                  
                
                
                  2
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                
              
              ,
              
                e
                
                  
                    -
                    
                      
                        (
                        
                          π
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                        
                        )
                      
                      2
                    
                  
                  ⁢
                  
                    η
                    zj
                  
                  ⁢
                  τ
                
              
            
            )
          
          ⁢
          
            --
            
              
                Θ
                3
                ∫
              
              (
              
                
                  
                    π
                    ⁡
                    
                      (
                      
                        z
                        -
                        
                          z
                          
                            02
                            ⁢
                            ij
                          
                        
                      
                      )
                    
                  
                  
                    2
                    ⁢
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                
                ,
                
                  e
                  
                    
                      -
                      
                        
                          (
                          
                            π
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                          
                          )
                        
                        2
                      
                    
                    ⁢
                    
                      η
                      zj
                    
                    ⁢
                    τ
                  
                
              
              )
            
          
        
        +
        
          Θ3
          ⁢
          
            ∫
            
              π
              ⁢
              z
            
          
        
        +
        z02ij
        -
        
          2
          ⁢
          dj2dj
        
        +
        1
        -
        dj
      
      ,
      
        e
        -
        πdj
        +
        1
        -
        dj2ηzjτdτ
        +
        
          ×
          Θ3πy
        
        -
        y0ij2b
      
      ,
      
        e
        -
        πb2ηyjτ
        +
        Θ3πy
        +
        y0ij2b
      
      ,
      
        e
        -
        
          πb2ηyjτ
          ×
        
      
    
  






 






  ×
  
    {
    
      
        
          Θ
          3
        
        (
        
          
            
              π
              ⁡
              
                (
                
                  z
                  -
                  
                    z
                    
                      0
                      ⁢
                      ij
                    
                  
                
                )
              
            
            
              2
              ⁢
              
                (
                
                  
                    d
                    
                      j
                      +
                      1
                    
                  
                  -
                  
                    d
                    j
                  
                
                )
              
            
          
          ,
          
            e
            
              
                -
                
                  
                    (
                    
                      π
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                    
                    )
                  
                  2
                
              
              ⁢
              
                η
                zj
              
              ⁢
              τ
            
          
        
        )
      
      +
      
        
          Θ
          3
        
        (
        
          
            
              π
              ⁡
              
                (
                
                  z
                  +
                  
                    z
                    
                      0
                      ⁢
                      ij
                    
                  
                  -
                  
                    2
                    ⁢
                    
                      d
                      j
                    
                  
                
                )
              
            
            
              2
              ⁢
              
                (
                
                  
                    d
                    
                      j
                      +
                      1
                    
                  
                  -
                  
                    d
                    j
                  
                
                )
              
            
          
          ,
          
            e
            
              
                -
                
                  
                    (
                    
                      π
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                    
                    )
                  
                  2
                
              
              ⁢
              
                η
                zj
              
              ⁢
              τ
            
          
        
        )
      
    
    }
  
  ×
  ×
  
    {
    
      
        
          Θ
          3
          ∫
        
        ⁡
        
          (
          
            
              
                π
                ⁡
                
                  (
                  
                    x
                    -
                    
                      x
                      
                        01
                        ⁢
                        ij
                      
                    
                  
                  )
                
              
              
                2
                ⁢
                a
              
            
            ,
            
              e
              
                
                  -
                  
                    
                      (
                      
                        π
                        a
                      
                      )
                    
                    2
                  
                
                ⁢
                
                  η
                  xj
                
                ⁢
                τ
              
            
          
          )
        
      
      -
      
        
          Θ
          3
          ∫
        
        ⁡
        
          (
          
            
              
                π
                ⁡
                
                  (
                  
                    x
                    +
                    
                      x
                      
                        01
                        ⁢
                        ij
                      
                    
                  
                  )
                
              
              
                2
                ⁢
                a
              
            
            ,
            
              e
              
                
                  -
                  
                    
                      (
                      
                        π
                        a
                      
                      )
                    
                    2
                  
                
                ⁢
                
                  η
                  xj
                
                ⁢
                τ
              
            
          
          )
        
      
      -
      
        ×
        
           
          
            
          
          ⁢
          
             
            
              
                [
                
                  
                    
                      Θ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            (
                            
                              x
                              -
                              u
                            
                            )
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            t
                          
                        
                      
                      )
                    
                  
                  +
                  
                    
                      Θ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            (
                            
                              x
                              +
                              u
                            
                            )
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            t
                          
                        
                      
                      )
                    
                  
                
                ]
              
              ×
              ×
              
                 
                
                  [
                  
                    
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    -
                                    v
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  yj
                                
                                ⁢
                                t
                              
                            
                          
                          )
                        
                      
                      +
                      Θ3πy
                      +
                      v2b
                    
                    ,
                    
                      e
                      -
                      πb2ηyjtdudvbdw
                    
                  
                
              
            
          
        
      
    
  









* For i = 1, 2 . . . Nlqij is flux per unit length in layer j and for i = Nl + 1, . . . Nr qij is flux per unit area in layer






Based on the derivation shown in TABLE 2, the general expressions for the hydrocarbon production occurring through multiple vertical wells, horizontal wells, multiple deviated wells, and multiple fractures in a reservoir (e.g., the reservoir (800)) are shown in TABLEs 3 through 7.







TABLE 3





 






  
    The
    ⁢
    
      
    
    ⁢
    spatial
    ⁢
    
      
    
    ⁢
    average
    ⁢
    
      
    
    ⁢
    pressure
    ⁢
    
      
    
    ⁢
    response
    ⁢
    
      
    
    ⁢
    of
    ⁢
    
      
    
    ⁢
    the
    ⁢
    
      
    
    ⁢
    
      line
      ⁢
      
        
      
      [
      
        
          z
          
            02
            ⁢
            
              ◇
              ⁢
              j
            
          
        
        -
        
          z
          
            01
            ⁢
            
              ◇
              ⁢
              j
            
          
        
      
      ]
    
  
  ,
  
    ι
    =
    ◇
  
  ,
  
    1
    ≤
    ◇
    ≤
    
      
        L
        l
      
      ⁢
      
        
      
      ⁢
      is
      ⁢
      
        
      
      ⁢
      given
      ⁢
      
        
      
      ⁢
      
        by
        .
      
    
  















  
    
      
         
        
          
            
              
                
                  
                    p
                    j
                  
                  =
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                      
                        2
                        ⁢
                        
                          
                            (
                            
                              φc
                              t
                            
                            )
                          
                          j
                        
                        ⁢
                        
                          ab
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    jj
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          1
                        
                        
                          L
                          ι
                        
                      
                      ⁢
                      
                        
                          U
                          (
                          
                            t
                            -
                            
                              t
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                            ∫
                            0
                            
                              t
                              -
                              
                                t
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                          
                          ⁢
                          
                            
                              
                                q
                                
                                  i
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    t
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  τ
                                
                                )
                              
                            
                            ×
                          
                        
                      
                    
                  
                
              
            
            
              
                
                  
                  ⁢
                  
                    ×
                    
                      {
                      
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      -
                                      
                                        x
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    xj
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                        +
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      +
                                      
                                        x
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    xj
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                      
                      }
                    
                    ×
                  
                
              
            
            
              
                
                  
                  ⁢
                  
                    ×
                    
                      {
                      
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      y
                                      -
                                      
                                        y
                                        
                                          01
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  b
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          b
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    
                                      y
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                        +
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      y
                                      +
                                      
                                        y
                                        
                                          01
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  b
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          b
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    
                                      y
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                      
                      }
                    
                    ×
                  
                
              
            
          
          ⁢
          
             
            
              

            
            ⁢
            
              
                
                  
                                
                    ⁢
                    
                      ×
                      
                        {
                        
                          
                            
                              Θ
                              3
                              
                                ∫
                                
                                  
                                
                                ∫
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁡
                                    
                                      (
                                      
                                        
                                          z
                                          
                                            02
                                            ⁢
                                            
                                              ◇
                                              ⁢
                                              j
                                            
                                          
                                        
                                        -
                                        
                                          z
                                          
                                            01
                                            ⁢
                                            
                                              i
                                              ⁢
                                              j
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  
                                    2
                                    ⁢
                                    
                                      (
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            
                                              
                                                d
                                                
                                                  j
                                                  +
                                                  1
                                                
                                              
                                              -
                                              
                                                d
                                                j
                                              
                                            
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      η
                                      zj
                                    
                                    ⁢
                                    τ
                                  
                                
                              
                              )
                            
                          
                          -
                          
                            
                              Θ
                              3
                              
                                ∫
                                
                                  
                                
                                ∫
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁡
                                    
                                      (
                                      
                                        
                                          z
                                          
                                            01
                                            ⁢
                                            
                                              ◇
                                              ⁢
                                              j
                                            
                                          
                                        
                                        -
                                        
                                          z
                                          
                                            01
                                            ⁢
                                            
                                              i
                                              ⁢
                                              j
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  
                                    2
                                    ⁢
                                    
                                      (
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            
                                              
                                                d
                                                
                                                  j
                                                  +
                                                  1
                                                
                                              
                                              -
                                              
                                                d
                                                j
                                              
                                            
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      η
                                      zj
                                    
                                    ⁢
                                    τ
                                  
                                
                              
                              )
                            
                          
                          -
                        
                      
                    
                  
                
              
              
                
                  
                    
                    ⁢
                    
                      
                        -
                        
                          
                            Θ
                            3
                            
                              ∫
                              
                                
                              
                              ∫
                            
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      
                                        z
                                        
                                          02
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                      
                                      +
                                      
                                        z
                                        
                                          01
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                      -
                                      
                                        2
                                        ⁢
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  
                                    (
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                    )
                                  
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          
                                            
                                              d
                                              
                                                j
                                                +
                                                1
                                              
                                            
                                            -
                                            
                                              d
                                              j
                                            
                                          
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    zj
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          Θ
                          3
                          
                            ∫
                            
                              
                            
                            ∫
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      z
                                      
                                        01
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    +
                                    
                                      z
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      -
                    
                  
                
              
              
                
                  
                    
                    ⁢
                    
                      
                        -
                        
                          
                            Θ
                            3
                            
                              ∫
                              
                                
                              
                              ∫
                            
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      
                                        z
                                        
                                          02
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                      
                                      -
                                      
                                        2
                                        
                                          02
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  
                                    (
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                    )
                                  
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          
                                            
                                              d
                                              
                                                j
                                                +
                                                1
                                              
                                            
                                            -
                                            
                                              d
                                              j
                                            
                                          
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    zj
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          Θ
                          3
                          
                            ∫
                            
                              
                            
                            ∫
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      
                        
                        ⁢
                        
                          
                            +
                            
                              
                                Θ
                                3
                                
                                  ∫
                                  
                                    
                                  
                                  ∫
                                
                              
                              ⁡
                              
                                (
                                
                                  
                                    
                                      π
                                      ⁡
                                      
                                        (
                                        
                                          
                                            z
                                            
                                              02
                                              ⁢
                                              
                                                ◇
                                                ⁢
                                                j
                                              
                                            
                                          
                                          +
                                          
                                            z
                                            
                                              02
                                              ⁢
                                              
                                                i
                                                ⁢
                                                j
                                              
                                            
                                          
                                          -
                                          
                                            2
                                            ⁢
                                            
                                              d
                                              j
                                            
                                          
                                        
                                        )
                                      
                                    
                                    
                                      2
                                      ⁢
                                      
                                        (
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ,
                                  
                                    e
                                    
                                      
                                        -
                                        
                                          
                                            (
                                            
                                              π
                                              
                                                
                                                  d
                                                  
                                                    j
                                                    +
                                                    1
                                                  
                                                
                                                -
                                                
                                                  d
                                                  j
                                                
                                              
                                            
                                            )
                                          
                                          2
                                        
                                      
                                      ⁢
                                      
                                        η
                                        zj
                                      
                                      ⁢
                                      τ
                                    
                                  
                                
                                )
                              
                            
                          
                          -
                          
                            
                              Θ
                              3
                              
                                ∫
                                
                                  
                                
                                ∫
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁡
                                    
                                      (
                                      
                                        
                                          z
                                          
                                            01
                                            ⁢
                                            
                                              ◇
                                              ⁢
                                              j
                                            
                                          
                                        
                                        +
                                        
                                          z
                                          
                                            02
                                            ⁢
                                            
                                              i
                                              ⁢
                                              j
                                            
                                          
                                        
                                        -
                                        
                                          2
                                          ⁢
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                  
                                  
                                    2
                                    ⁢
                                    
                                      (
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            
                                              
                                                d
                                                
                                                  j
                                                  +
                                                  1
                                                
                                              
                                              -
                                              
                                                d
                                                j
                                              
                                            
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      η
                                      zj
                                    
                                    ⁢
                                    τ
                                  
                                
                              
                              )
                            
                          
                        
                        }
                      
                      ⁢
                      d
                      ⁢
                      
                        
                      
                      ⁢
                      τ
                    
                    +
                  
                
              
            
          
        
      
    
    
      
        (
        0.9
        )
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    b
                    ⁡
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  ι
                  =
                  
                    
                      L
                      ι
                    
                    +
                    1
                  
                
                
                  M
                  ι
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                ⁢
                
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                
                  +
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              z
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                d
                                j
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            zj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              z
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                d
                                j
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            zj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                x
                                -
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   










   
  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      ι
                    
                    +
                    1
                  
                
                
                  N
                  ι
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                ⁢
                
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                
                  +
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              z
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                d
                                j
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            zj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              z
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                d
                                j
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            zj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                -
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                    
                    j
                  
                  ⁢
                  
                    ab
                    ⁡
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      ι
                    
                    +
                    1
                  
                
                
                  N
                  d
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  θ
                  
                    0
                    ⁢
                    
                      i
                      ⁢
                      j
                    
                  
                
                ×
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                
                  t
                  -
                  
                    t
                    
                      0
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                  
                
              
              ⁢
              
                
                  
                    q
                    j
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                      -
                      τ
                    
                    )
                  
                
                ⁢
                
                  
                    ∫
                    
                      z
                      ⁢
                      
                        
                      
                      ⁢
                      01
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                    
                      z
                      
                        02
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  ⁢
                  
                    [
                    
                      {
                      
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ⁢
                                
                                  {
                                  
                                    x
                                    -
                                    
                                      
                                        (
                                        
                                          
                                            z
                                            
                                              0
                                              ⁢
                                              
                                                i
                                                ⁢
                                                j
                                              
                                            
                                          
                                          -
                                          
                                            γ
                                            
                                              0
                                              ⁢
                                              
                                                i
                                                ⁢
                                                j
                                              
                                            
                                          
                                        
                                        )
                                      
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      cot
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                      ⁢
                                      cos
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                  
                                  }
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    ⁢
                                    xj
                                    ⁢
                                    τ
                                  
                                
                              
                            
                            )
                          
                        
                        +
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              +
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          a
                        
                      
                      ⁢
                      
                        {
                        
                          x
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          ⁢
                          xj
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            b
                          
                        
                        ⁢
                        
                          {
                          
                            y
                            -
                            
                              
                                (
                                
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                          
                          }
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            ⁢
                            yj
                            ⁢
                            τ
                          
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              +
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        {
                        
                          y
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          ⁢
                          yj
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
                      
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    ◇
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      j
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  
                                    η
                                    zj
                                  
                                  ⁡
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                
                                2
                              
                            
                            ⁢
                            t
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  
                                    η
                                    zj
                                  
                                  ⁡
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                
                                2
                              
                            
                            ⁢
                            t
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
                
                  
                  ⁢
                  
                    
                      +
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    +
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      
                                        η
                                        zj
                                      
                                      ⁡
                                      
                                        (
                                        
                                          π
                                          
                                            
                                              d
                                              
                                                j
                                                +
                                                1
                                              
                                            
                                            -
                                            
                                              d
                                              j
                                            
                                          
                                        
                                        )
                                      
                                    
                                    2
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          )
                        
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    
                                      η
                                      zj
                                    
                                    ⁡
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                  
                                  2
                                
                              
                              ⁢
                              t
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ]
              
              ⁢
              
                
              
              ⁢
              
                dz
                
                  0
                  ⁢
                  
                    i
                    ⁢
                    j
                  
                
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      d
                    
                    +
                    1
                  
                
                
                  L
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            x
                            +
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          zj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            y
                            +
                            
                              y
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          yj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
                      
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                      }
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
                
                ⁢
                
                  
                    ×
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                
                }
              
              ⁢
              
                
              
              ⁢
              dτ
            
            +
          
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            ◇
                            ⁢
                            
                              
                            
                            ⁢
                            
                              j
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      r
                    
                    +
                    1
                  
                
                
                  M
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            y
                            +
                            
                              y
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          yj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
                     
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      
                        ∫
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
        
      
      
        
          
             
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  
                    ∫
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              z
                              
                                01
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                d
                                j
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          zj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              -
            
          
        
      
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  
                    ∫
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              z
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          zj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              +
            
          
        
      
      
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      
                        Θ
                        3
                        
                          ∫
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ⁢
              
                
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    b
                    ⁡
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      r
                    
                    +
                    1
                  
                
                
                  N
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            x
                            +
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
                       
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      
                        ∫
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  +
                  
                    
                      Θ
                      3
                      
                        ∫
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
        
      
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  
                    ∫
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              z
                              
                                01
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                d
                                j
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          zj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              -
            
          
        
      
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  
                    ∫
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              z
                              
                                01
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          zj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              +
            
          
        
      
      
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      
                        Θ
                        3
                        
                          ∫
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ⁢
              
                
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  










  
    
  
  ⁢
  
    
      
        
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    ab
                    ⁡
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
            
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                    ∫
                    0
                    b
                  
                  ⁢
                  
                    [
                    
                      
                        
                          ψ
                          j
                        
                        ⁡
                        
                          (
                          
                            u
                            ,
                            v
                            ,
                            τ
                          
                          )
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              Θ
                              3
                              ∫
                            
                            ⁢
                            
                              {
                              
                                
                                  
                                    π
                                    ⁡
                                    
                                      (
                                      
                                        
                                          z
                                          
                                            02
                                            ⁢
                                            
                                              ◇
                                              ⁢
                                              j
                                            
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                      )
                                    
                                  
                                  
                                    2
                                    ⁢
                                    
                                      (
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            
                                              
                                                d
                                                
                                                  j
                                                  +
                                                  1
                                                
                                              
                                              -
                                              
                                                d
                                                j
                                              
                                            
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        zj
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              }
                            
                          
                          -
                        
                      
                    
                  
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            zj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              }
            
            -
          
        
      
      
        
          
            
            ⁢
            
              
                -
                
                  
                    ψ
                    
                      j
                      +
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      u
                      ,
                      v
                      ,
                      τ
                    
                    )
                  
                
              
              ⁢
              
                {
                
                  
                    
                      Θ
                      4
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                      }
                    
                  
                  -
                
              
            
          
        
      
      
        
          
            
              
                
                ⁢
                
                  
                    -
                    
                      Θ
                      4
                      ∫
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                }
              
              ]
            
            ×
          
        
      
    
    ⁢
    
       
      
        

      
      ⁢
      
        
          
            
                       
              ⁢
              
                ×
                
                  [
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  -
                                  u
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  xj
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  +
                                  u
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  xj
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                  
                  ]
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                
                  ×
                  
                    [
                    
                      
                        
                          Θ
                          3
                        
                        ⁢
                        
                          {
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    -
                                    v
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  
                                    η
                                    yj
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      τ
                                    
                                    )
                                  
                                
                              
                            
                          
                          }
                        
                      
                      +
                      
                        
                          Θ
                          3
                        
                        ⁢
                        
                          {
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    +
                                    v
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  
                                    η
                                    yj
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      τ
                                    
                                    )
                                  
                                
                              
                            
                          
                          }
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    dudvd
                    ⁢
                    τ
                  
                
                +
              
            
          
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    ab
                    ⁡
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  b
                
                ⁢
                
                  
                    ∫
                    0
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                  
                  ⁢
                  
                    
                      {
                      
                        
                          
                            
                              ψ
                              
                                0
                                ⁢
                                yzj
                              
                            
                            ⁡
                            
                              (
                              
                                v
                                ,
                                w
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            
                              Θ
                              3
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁢
                                    x
                                  
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            a
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        xj
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                        
                        -
                        
                          
                            
                              ψ
                              ayzj
                            
                            ⁡
                            
                              (
                              
                                v
                                ,
                                w
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            
                              Θ
                              4
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁢
                                    x
                                  
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            a
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        xj
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                        
                      
                      }
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          (
                          
                            y
                            -
                            v
                          
                          )
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              yj
                            
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                    }
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          (
                          
                            y
                            +
                            v
                          
                          )
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              yj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    jj
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    jj
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  ×
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    jj
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              }
            
            ⁢
            dvdwdτ
          
          +
        
      
    
  
   










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    ab
                    ⁡
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                    ∫
                    0
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                  
                  ⁢
                  
                    
                      {
                      
                        
                          
                            
                              ψ
                              
                                x
                                ⁢
                                
                                  
                                
                                ⁢
                                0
                                ⁢
                                zj
                              
                            
                            ⁡
                            
                              (
                              
                                v
                                ,
                                w
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            
                              Θ
                              3
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁢
                                    y
                                  
                                  
                                    2
                                    ⁢
                                    b
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            b
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        yj
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                        
                        -
                        
                          
                            
                              ψ
                              xbzj
                            
                            ⁡
                            
                              (
                              
                                u
                                ,
                                w
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            
                              Θ
                              4
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁢
                                    y
                                  
                                  
                                    2
                                    ⁢
                                    b
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            b
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        yj
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                        
                      
                      }
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          (
                          
                            x
                            -
                            u
                          
                          )
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                    }
                  
                
                +
                
                  
                    Θ
                    3
                  
                  (
                  
                    
                      
                        π
                        (
                        
                          x
                          +
                          u
                        
                        )
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              ]
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    jj
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    jj
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  ×
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    jj
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              }
            
            ⁢
            dvdwdτ
          
          +
        
      
    
  
   










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    ab
                    ⁡
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                
                  
                    d
                    
                      j
                      +
                      1
                    
                  
                  -
                  
                    d
                    j
                  
                
              
              ⁢
              
                
                  ∫
                  0
                  b
                
                ⁢
                
                  
                    ∫
                    0
                    a
                  
                  ⁢
                  
                    
                      φ
                      j
                    
                    ⁢
                    
                      {
                      
                        
                          (
                          
                            u
                            ,
                            v
                            ,
                            
                              w
                              +
                              
                                d
                                j
                              
                            
                          
                          )
                        
                        [
                        
                          
                            {
                            
                              
                                
                                  Θ
                                  3
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        π
                                        (
                                        
                                          x
                                          -
                                          u
                                        
                                        )
                                      
                                      
                                        2
                                        ⁢
                                        a
                                      
                                    
                                    ,
                                    
                                      e
                                      
                                        
                                          -
                                          
                                            
                                              (
                                              
                                                π
                                                a
                                              
                                              )
                                            
                                            2
                                          
                                        
                                        ⁢
                                        
                                          η
                                          xj
                                        
                                        ⁢
                                        t
                                      
                                    
                                  
                                  )
                                
                              
                              +
                              
                                
                                  Θ
                                  3
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        π
                                        (
                                        
                                          x
                                          +
                                          u
                                        
                                        )
                                      
                                      
                                        2
                                        ⁢
                                        a
                                      
                                    
                                    ,
                                    
                                      e
                                      
                                        
                                          -
                                          
                                            
                                              (
                                              
                                                π
                                                a
                                              
                                              )
                                            
                                            2
                                          
                                        
                                        ⁢
                                        
                                          η
                                          xj
                                        
                                        ⁢
                                        t
                                      
                                    
                                  
                                  )
                                
                              
                            
                            }
                          
                          ×
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          (
                          
                            y
                            -
                            v
                          
                          )
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          (
                          
                            y
                            +
                            v
                          
                          )
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    jj
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  jj
                                
                              
                            
                            -
                            
                              d
                              j
                            
                            -
                            w
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          zj
                        
                        ⁢
                        t
                      
                    
                  
                  }
                
                +
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    jj
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  jj
                                
                              
                            
                            -
                            
                              d
                              j
                            
                            +
                            w
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          zj
                        
                      
                    
                  
                  }
                
              
              }
            
            ⁢
            dudvdw
          
        
      
    
  
   




















TABLE 4





 






  
    The
    ⁢
    
      
    
    ⁢
    spatial
    ⁢
    
      
    
    ⁢
    average
    ⁢
    
      
    
    ⁢
    pressure
    ⁢
    
      
    
    ⁢
    response
    ⁢
    
      
    
    ⁢
    of
    ⁢
    
      
    
    ⁢
    the
    ⁢
    
      
    
    ⁢
    
      line
      ⁢
      
        
      
      [
      
        
          x
          
            02
            ⁢
            
              ◇
              ⁢
              j
            
          
        
        -
        
          x
          
            01
            ⁢
            
              ◇
              ⁢
              j
            
          
        
      
      ]
    
  
  ,
  
    ι
    =
    ◇
  
  ,
  
    
      L
      
        l
        +
        1
      
    
    ≤
    ◇
    ≤
    
      
        M
        l
      
      ⁢
      
        
      
      ⁢
      is
      ⁢
      
        
      
      ⁢
      given
      ⁢
      
        
      
      ⁢
      
        by
        .
      
    
  















  
    
      
         
        
          
            
              
                
                  p
                  j
                
                =
                
                ⁢
                
                  
                    1
                    
                      2
                      ⁢
                      
                        
                          (
                          
                            φc
                            t
                          
                          )
                        
                        j
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        b
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  jj
                                
                              
                            
                          
                          )
                        
                      
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        1
                      
                      
                        L
                        ι
                      
                    
                    ⁢
                    
                      
                        U
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        
                          ∫
                          0
                          
                            t
                            -
                            
                              t
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        ⁢
                        
                          
                            
                              q
                              
                                i
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  t
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                τ
                              
                              )
                            
                          
                          ×
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  ×
                  
                    {
                    
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      x
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      x
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                a
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  xj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      -
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      x
                                      
                                        01
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      x
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                a
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  xj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      +
                    
                  
                
              
            
          
          
            
              
                
                  
                  ⁢
                  
                    
                      +
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      x
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    +
                                    
                                      x
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                a
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  xj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    x
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ×
              
            
          
          
            
              
                
                ⁢
                
                  ×
                  
                    {
                    
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    -
                                    
                                      y
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  
                                    y
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    +
                                    
                                      y
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  
                                    y
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                  ×
                
              
            
          
          
            
              
                
                ⁢
                
                  ×
                  
                    {
                    
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      z
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      -
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    +
                                    
                                      z
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      -
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                    ⁢
                    
                      
                        -
                        
                          
                            Θ
                            3
                            ∫
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      z
                                      -
                                      
                                        z
                                        
                                          02
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  
                                    (
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                    )
                                  
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          
                                            
                                              d
                                              
                                                j
                                                +
                                                1
                                              
                                            
                                            -
                                            
                                              d
                                              j
                                            
                                          
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    zj
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    +
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                t
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                  ⁢
                  
                    
                  
                  ⁢
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  τ
                
                +
              
            
          
        
      
    
    
      
        (
        0.10
        )
      
    
  
  
    
      
         
        
          
            
              
                
                  
                
                ⁢
                
                  
                    +
                    
                    ⁢
                    
                      a
                      
                        2
                        ⁢
                        
                          
                            (
                            
                              φc
                              t
                            
                            )
                          
                          j
                        
                        ⁢
                        
                          b
                          ⁡
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                        ⁢
                        
                          (
                          
                            
                              x
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  jj
                                
                              
                            
                          
                          )
                        
                      
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        
                          
                            L
                            l
                          
                          +
                          1
                        
                      
                      
                        M
                        l
                      
                    
                    ⁢
                    
                      
                        U
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        
                          ∫
                          0
                          
                            t
                            -
                            
                              t
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        ⁢
                        
                          
                            
                              q
                              
                                i
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  t
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                τ
                              
                              )
                            
                          
                          ×
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  ×
                  
                    {
                    
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    -
                                    
                                      y
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  yj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    +
                                    
                                      y
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  yj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                  ×
                
              
            
          
          
            
              
                
                ⁢
                
                  ×
                  
                    {
                    
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    +
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                t
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                  ×
                
              
            
          
          
            
              
                
                ⁢
                
                  ×
                  
                    {
                    
                      
                        
                          Θ
                          3
                          
                            ∫
                            
                              
                            
                            ∫
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      x
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      x
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                a
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  
                                    x
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      -
                      
                        
                          Θ
                          3
                          
                            ∫
                            
                              
                            
                            ∫
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      x
                                      
                                        01
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      x
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                a
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  
                                    x
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      -
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    -
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    x
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    -
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    x
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                    
                    ⁢
                    
                      
                        +
                        
                          
                            Θ
                            3
                            
                              ∫
                              
                                
                              
                              ∫
                            
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      
                                        x
                                        
                                          02
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                      
                                      +
                                      
                                        x
                                        
                                          02
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    xj
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                      
                      -
                      
                        
                          Θ
                          3
                          
                            ∫
                            
                              
                            
                            ∫
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      x
                                      
                                        01
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    +
                                    
                                      x
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                a
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  xj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                  ⁢
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  τ
                
                +
              
            
          
        
      
    
    
      
        
      
    
  










  
    
  
  ⁢
  
     
    
      
        
          
            
              
                
              
              ⁢
              
                
                  +
                  
                  ⁢
                  
                    1
                    
                      2
                      ⁢
                      
                        
                          (
                          
                            φc
                            t
                          
                          )
                        
                        j
                      
                      ⁢
                      
                        (
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          
                            x
                            
                              02
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          -
                          
                            x
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      
                        
                          M
                          ι
                        
                        +
                        1
                      
                    
                    
                      N
                      ι
                    
                  
                  ⁢
                  
                    
                      U
                      (
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        ∫
                        0
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                      
                      ⁢
                      
                        
                          
                            q
                            
                              i
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                t
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              τ
                            
                            )
                          
                        
                        ×
                      
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    x
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    x
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    x
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  x
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ×
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  -
                                  
                                    y
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  +
                                  
                                    y
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    
                      -
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    y
                                    -
                                    
                                      y
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                b
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        b
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  yj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  +
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                yj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ⁢
                d
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
       
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                    
                    j
                  
                  ⁢
                  
                    b
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      ι
                    
                    +
                    1
                  
                
                
                  N
                  d
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  θ
                  
                    0
                    ⁢
                    
                      i
                      ⁢
                      j
                    
                  
                
                ×
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                
                  t
                  -
                  
                    t
                    
                      0
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                  
                
              
              ⁢
              
                
                  
                    q
                    
                      i
                      ⁢
                      j
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                      -
                      τ
                    
                    )
                  
                
                ⁢
                
                  
                    ∫
                    
                      z
                      ⁢
                      
                        
                      
                      ⁢
                      01
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                    
                      z
                      
                        02
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  ⁢
                  
                    [
                    
                      {
                      
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  
                                    2
                                    ⁢
                                    b
                                  
                                
                                ⁢
                                
                                  {
                                  
                                    y
                                    -
                                    
                                      
                                        (
                                        
                                          
                                            z
                                            
                                              0
                                              ⁢
                                              
                                                i
                                                ⁢
                                                j
                                              
                                            
                                          
                                          -
                                          
                                            γ
                                            
                                              0
                                              ⁢
                                              
                                                i
                                                ⁢
                                                j
                                              
                                            
                                          
                                        
                                        )
                                      
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      cot
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                      ⁢
                                      cos
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                  
                                  }
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          b
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    ⁢
                                    yj
                                    ⁢
                                    τ
                                  
                                
                              
                            
                            )
                          
                        
                        +
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              +
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        {
                        
                          y
                          -
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                
                                  
                                
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          ⁢
                          yj
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            a
                          
                        
                        ⁢
                        
                          {
                          
                            
                              x
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              
                                (
                                
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              ⁢
                              
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                          
                          }
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            ⁢
                            xj
                            ⁢
                            τ
                          
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          a
                        
                      
                      ⁢
                      
                        {
                        
                          
                            x
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          -
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          ⁢
                          xj
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            +
          
        
      
    
    
      
        
          
          ⁢
          
            
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          a
                        
                      
                      ⁢
                      
                        {
                        
                          
                            x
                            
                              02
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          ⁢
                          xj
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            -
          
        
      
    
    
      
        
          
            
            ⁢
            
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          a
                        
                      
                      ⁢
                      
                        {
                        
                          
                            x
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          ⁢
                          xj
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              
                                2
                                ⁢
                                d
                              
                            
                            ⁢
                            
                              (
                              
                                z
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      d
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                ⁢
                                yj
                                ⁢
                                τ
                              
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              
                                2
                                ⁢
                                d
                              
                            
                            ⁢
                            
                              (
                              
                                z
                                +
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      d
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                ⁢
                                yj
                                ⁢
                                τ
                              
                            
                          
                        
                        )
                      
                    
                  
                  }
                
              
              ]
            
            ⁢
            
              
            
            ⁢
            
              dz
              
                0
                ⁢
                
                  i
                  ⁢
                  j
                
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   










  
    
  
  ⁢
  
    
      
        
          
            
              
            
            ⁢
            
              
                +
                
                ⁢
                
                  a
                  
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      j
                    
                    ⁢
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          x
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          x
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
              ⁢
              
                
                  ∑
                  
                    i
                    =
                    
                      
                        N
                        d
                      
                      +
                      1
                    
                  
                  
                    L
                    r
                  
                
                ⁢
                
                  
                    U
                    (
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      ∫
                      0
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                    
                    ⁢
                    
                      
                        
                          q
                          
                            i
                            ⁢
                            j
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              t
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            τ
                          
                          )
                        
                      
                      ×
                    
                  
                
              
            
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
        
      
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              x
                              
                                01
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              -
            
          
        
      
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              +
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  +
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                -
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                +
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                -
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ⁢
                
                  
                
                ⁢
                d
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
    
     
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      r
                    
                    +
                    1
                  
                
                
                  M
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              x
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            y
                            +
                            
                              y
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          yj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                a
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    b
                    ⁡
                    
                      (
                      
                        
                          x
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          x
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      r
                    
                    +
                    1
                  
                
                
                  N
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              x
                              
                                01
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
            
              
                Θ
                3
                
                  ∫
                  
                    
                  
                  ∫
                
              
              ⁡
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            x
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            x
                            
                              01
                              ⁢
                              ij
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      a
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        xj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            -
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
            
              
                Θ
                3
                
                  ∫
                  
                    
                  
                  ∫
                
              
              ⁡
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            x
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            x
                            
                              02
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      a
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        xj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
  
  ⁢
  
    
      
        
                
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  









  
    
      
    
    ⁢
    
      
        
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    b
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                    ∫
                    0
                    b
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            ψ
                            j
                          
                          ⁡
                          
                            (
                            
                              u
                              ,
                              v
                              ,
                              τ
                            
                            )
                          
                        
                        ⁢
                        
                          Θ
                          3
                        
                        ⁢
                        
                          {
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  
                                    η
                                    zj
                                  
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      ψ
                      
                        j
                        +
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        u
                        ,
                        v
                        ,
                        τ
                      
                      )
                    
                  
                
                ⁢
                
                  Θ
                  4
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            z
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            zj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              ]
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                [
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                u
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                u
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  
                    +
                    
                      Θ
                      3
                      ∫
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              ]
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              
                ×
                
                  [
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  -
                                  v
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    y
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  y
                                  +
                                  v
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    y
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                dudvdτ
              
              +
            
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    b
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            j
                            ⁢
                            
                              
                            
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                
                  ∫
                  0
                  t
                
                ⁢
                
                  
                    ∫
                    0
                    b
                  
                  ⁢
                  
                    
                      ∫
                      0
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            
                              ψ
                              
                                0
                                ⁢
                                
                                  yz
                                  ⁢
                                  j
                                
                              
                            
                            (
                            
                              v
                              ,
                              w
                              ,
                              τ
                            
                            )
                          
                          ⁢
                          
                            {
                            
                              
                                
                                  Θ
                                  3
                                  ∫
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          π
                                          ⁢
                                          x
                                        
                                        
                                          02
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                      
                                      
                                        2
                                        ⁢
                                        a
                                      
                                    
                                    ,
                                    
                                      e
                                      
                                        
                                          -
                                          
                                            
                                              (
                                              
                                                π
                                                a
                                              
                                              )
                                            
                                            2
                                          
                                        
                                        ⁢
                                        
                                          
                                            η
                                            xj
                                          
                                          (
                                          
                                            t
                                            -
                                            τ
                                          
                                          )
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              -
                              
                                
                                  Θ
                                  3
                                  ∫
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          π
                                          ⁢
                                          x
                                        
                                        
                                          01
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                      
                                      
                                        2
                                        ⁢
                                        a
                                      
                                    
                                    ,
                                    
                                      e
                                      
                                        
                                          -
                                          
                                            
                                              (
                                              
                                                π
                                                a
                                              
                                              )
                                            
                                            2
                                          
                                        
                                        ⁢
                                        
                                          
                                            η
                                            xj
                                          
                                          (
                                          
                                            t
                                            -
                                            τ
                                          
                                          )
                                        
                                      
                                    
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                        -
                      
                    
                  
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    ψ
                    ayzj
                  
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    v
                    ,
                    w
                    ,
                    τ
                  
                  )
                
                ⁢
                
                  {
                  
                    
                      
                        Θ
                        4
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                π
                                ⁢
                                x
                              
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  xj
                                
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        4
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                π
                                ⁢
                                x
                              
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  xj
                                
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                  
                  }
                
              
              }
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                [
                
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                -
                                v
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                y
                                +
                                v
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                
                ]
              
              ⁢
              
                
              
              ×
            
          
        
      
      
        
          
            
            ⁢
            
              
                ×
                
                  [
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    d
                                    
                                      j
                                      ⁢
                                      
                                        
                                      
                                    
                                  
                                  -
                                  w
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  zj
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    d
                                    
                                      j
                                      ⁢
                                      
                                        
                                      
                                    
                                  
                                  +
                                  w
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  zj
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                dudwd
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
    
  









  
    
  
  ⁢
  
    
      
        
          
            
              
            
            ⁢
            
              
                +
                
                ⁢
                
                  1
                  
                    2
                    ⁢
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      j
                    
                    ⁢
                    
                      b
                      ⁡
                      
                        (
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                    
                    ⁢
                    
                      (
                      
                        
                          x
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          x
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
              ×
            
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                
                  ∫
                  0
                  t
                
                ⁢
                
                  
                    ∫
                    0
                    a
                  
                  ⁢
                  
                    
                      ∫
                      0
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            
                              ψ
                              
                                x
                                ⁢
                                
                                  
                                
                                ⁢
                                0
                                ⁢
                                zj
                              
                            
                            (
                            
                              u
                              ,
                              w
                              ,
                              τ
                            
                            )
                          
                          ⁢
                          
                            
                              Θ
                              3
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁢
                                    y
                                  
                                  
                                    2
                                    ⁢
                                    b
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            b
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        yj
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                        
                        -
                        
                          
                            
                              ψ
                              xbzj
                            
                            (
                            
                              u
                              ,
                              w
                              ,
                              τ
                            
                            )
                          
                          ⁢
                          
                            
                              Θ
                              4
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁢
                                    y
                                  
                                  
                                    2
                                    ⁢
                                    b
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            b
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        yj
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                        
                      
                      }
                    
                  
                
              
            
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                [
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                u
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                u
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  
                    +
                    
                      Θ
                      3
                      ∫
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              ]
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              
                ×
                
                  [
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    d
                                    
                                      j
                                      ⁢
                                      
                                        
                                      
                                    
                                  
                                  -
                                  w
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  zj
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    d
                                    
                                      j
                                      ⁢
                                      
                                        
                                      
                                    
                                  
                                  +
                                  w
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  zj
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                dudwd
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
    
     
  










  
    
  
  ⁢
  
    
      
        
          
            +
            
            ⁢
            
              1
              
                4
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      d
                      
                        j
                        +
                        1
                      
                    
                    -
                    
                      d
                      j
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      x
                      
                        02
                        ⁢
                        
                          ◇
                          ⁢
                          j
                        
                      
                    
                    -
                    
                      x
                      
                        01
                        ⁢
                        
                          ◇
                          ⁢
                          jj
                        
                      
                    
                  
                  )
                
              
            
          
          ×
          
            
              ∫
              0
              
                
                  d
                  
                    j
                    +
                    1
                  
                
                -
                
                  d
                  j
                
              
            
            ⁢
            
              
                ∫
                0
                b
              
              ⁢
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                    
                      φ
                      j
                    
                    (
                    
                      u
                      ,
                      v
                      ,
                      
                        w
                        +
                        
                          d
                          j
                        
                      
                    
                    )
                  
                  ×
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                
                  +
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            u
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            u
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
            
            ]
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              -
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              y
                              +
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                d
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          t
                        
                      
                    
                    }
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                d
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          t
                        
                      
                    
                    }
                  
                
              
              }
            
            ⁢
            dudvdw
          
        
      
    
  




















TABLE 5





 






  
    The
    ⁢
    
      
    
    ⁢
    spatial
    ⁢
    
      
    
    ⁢
    average
    ⁢
    
      
    
    ⁢
    pressure
    ⁢
    
      
    
    ⁢
    response
    ⁢
    
      
    
    ⁢
    of
    ⁢
    
      
    
    ⁢
    the
    ⁢
    
      
    
    ⁢
    inclined
    ⁢
    
      
    
    ⁢
    
      line
      ⁢
      
        
      
      [
      
        
          (
          
            
              z
              
                02
                ⁢
                
                  ◇
                  ⁢
                  j
                
              
            
            -
            
              z
              
                01
                ⁢
                
                  ◇
                  ⁢
                  j
                
              
            
          
          )
        
        ⁢
        
          
        
        ⁢
        sin
        ⁢
        
          
        
        ⁢
        
          θ
          
            0
            ⁢
            
              ◇
              ⁢
              j
            
          
        
      
      ]
    
  
  ,
  
    i
    =
    ◇j
  
  ,
  
    

  
  ⁢
  
    
      N
      
        ι
        +
        1
      
    
    ≤
    ◇
    ≤
    
      
        N
        d
      
      ⁢
      
        
      
      ⁢
      is
      ⁢
      
        
      
      ⁢
      given
      ⁢
      
        
      
      ⁢
      
        by
        .
      
    
  















  
    
      
        
          
            
              p
              =
              
              ⁢
              
                
                  1
                  
                    4
                    ⁢
                    
                      πa
                      ⁢
                      b
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      j
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  
                    ∑
                    
                      i
                      =
                      1
                    
                    
                      L
                      ι
                    
                  
                  ⁢
                  
                    
                      U
                      (
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        ∫
                        0
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                      
                      ⁢
                      
                        [
                        
                          
                            
                              q
                              ij
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  t
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            
                              ∫
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            ⁢
                            
                              {
                              
                                
                                  
                                    Θ
                                    3
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          π
                                          
                                            2
                                            ⁢
                                            a
                                          
                                        
                                        ⁢
                                        
                                          (
                                          
                                            
                                              
                                                (
                                                
                                                  z
                                                  -
                                                  
                                                    γ
                                                    
                                                      0
                                                      ⁢
                                                      
                                                        ◇
                                                        ⁢
                                                        j
                                                      
                                                    
                                                  
                                                
                                                )
                                              
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              cot
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              
                                                θ
                                                
                                                  
                                                    0
                                                    ⁢
                                                    
                                                      ◇
                                                      ⁢
                                                      j
                                                    
                                                  
                                                  ⁢
                                                  
                                                    
                                                  
                                                
                                              
                                              ⁢
                                              cos
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              
                                                θ
                                                
                                                  0
                                                  ⁢
                                                  
                                                    ◇
                                                    ⁢
                                                    j
                                                  
                                                
                                              
                                            
                                            -
                                            
                                              x
                                              
                                                0
                                                ⁢
                                                
                                                  i
                                                  ⁢
                                                  j
                                                
                                              
                                            
                                          
                                          )
                                        
                                      
                                      ,
                                      
                                        e
                                        
                                          
                                            -
                                            
                                              
                                                (
                                                
                                                  π
                                                  a
                                                
                                                )
                                              
                                              2
                                            
                                          
                                          ⁢
                                          
                                            η
                                            
                                              x
                                              ⁢
                                              j
                                            
                                          
                                          ⁢
                                          τ
                                        
                                      
                                    
                                    )
                                  
                                
                                +
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      Θ
                      3
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            a
                          
                        
                        ⁢
                        
                          (
                          
                            
                              
                                (
                                
                                  z
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            +
                            
                              x
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                }
              
              ×
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              π
                              
                                2
                                ⁢
                                b
                              
                            
                            ⁢
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      z
                                      -
                                      
                                        γ
                                        
                                          0
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  cot
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      
                                        0
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                      ⁢
                                      
                                        
                                      
                                    
                                  
                                  ⁢
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                -
                                
                                  y
                                  
                                    0
                                    ⁢
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    j
                                  
                                
                              
                              )
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                yj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      Θ
                      3
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            b
                          
                        
                        ⁢
                        
                          (
                          
                            
                              
                                (
                                
                                  z
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            +
                            
                              y
                              
                                0
                                ⁢
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                j
                              
                            
                          
                          )
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                }
              
              ×
            
          
        
      
    
    
      
        (
        0.11
        )
      
    
  
  
    
      
        
          
            
                       
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  z
                                  
                                    01
                                    ⁢
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                z
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      
                      ⁢
                      
                        
                          +
                          
                            
                              Θ
                              3
                              ∫
                            
                            (
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      z
                                      -
                                      
                                        z
                                        
                                          02
                                          ⁢
                                          i
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          j
                                        
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  
                                    (
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                    )
                                  
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          
                                            
                                              d
                                              
                                                j
                                                +
                                                1
                                              
                                            
                                            -
                                            
                                              d
                                              j
                                            
                                          
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    
                                      z
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                        +
                        
                          
                            Θ
                            3
                            ∫
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      z
                                      +
                                      
                                        z
                                        
                                          02
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                      -
                                      
                                        2
                                        ⁢
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  
                                    (
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                    )
                                  
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          
                                            
                                              d
                                              
                                                j
                                                +
                                                1
                                              
                                            
                                            -
                                            
                                              d
                                              j
                                            
                                          
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    zj
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                      
                      }
                    
                    ⁢
                    
                      
                    
                    ⁢
                    dz
                  
                  ]
                
                ⁢
                d
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
    
    
      
        
      
    
  









  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  πb
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      ι
                    
                    +
                    1
                  
                
                
                  M
                  ι
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    [
                    
                      
                        
                          q
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            j
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              t
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            τ
                          
                          )
                        
                      
                      ⁢
                      
                        
                          ∫
                          
                            z
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          
                            z
                            
                              02
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        ⁢
                        
                          {
                          
                            
                              
                                Θ
                                3
                              
                              ⁡
                              
                                (
                                
                                  
                                    
                                      π
                                      ⁡
                                      
                                        (
                                        
                                          z
                                          -
                                          
                                            z
                                            
                                              0
                                              ⁢
                                              i
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              j
                                            
                                          
                                        
                                        )
                                      
                                    
                                    
                                      2
                                      ⁢
                                      
                                        (
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ,
                                  
                                    e
                                    
                                      
                                        -
                                        
                                          
                                            (
                                            
                                              π
                                              
                                                
                                                  d
                                                  
                                                    j
                                                    +
                                                    1
                                                  
                                                
                                                -
                                                
                                                  d
                                                  j
                                                
                                              
                                            
                                            )
                                          
                                          2
                                        
                                      
                                      ⁢
                                      
                                        η
                                        
                                          z
                                          ⁢
                                          j
                                        
                                      
                                      ⁢
                                      τ
                                    
                                  
                                
                                )
                              
                            
                            +
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  Θ
                  3
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          z
                          +
                          
                            z
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          
                            2
                            ⁢
                            
                              d
                              j
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      
                        (
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        
                          z
                          ⁢
                          j
                        
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            }
          
          ×
        
      
    
  




  
    
      
        
          
                        
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            π
                            
                              2
                              ⁢
                              b
                            
                          
                          ⁢
                          
                            (
                            
                              
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      γ
                                      
                                        0
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                cot
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                    ⁢
                                    
                                      
                                    
                                  
                                
                                ⁢
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  j
                                
                              
                            
                            )
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              yj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  +
                  
                    Θ
                    3
                  
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              (
                              
                                z
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                
                                  
                                
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                          
                          +
                          
                            y
                            
                              0
                              ⁢
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              j
                            
                          
                        
                        )
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          yj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              }
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            π
                            
                              2
                              ⁢
                              a
                            
                          
                          ⁢
                          
                            (
                            
                              
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      γ
                                      
                                        0
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                cot
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                    ⁢
                                    
                                      
                                    
                                  
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
            ⁢
            
              
                
                  +
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          a
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              (
                              
                                z
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                
                                  
                                
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                          
                          +
                          
                            x
                            
                              01
                              ⁢
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              j
                            
                          
                        
                        )
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              +
            
          
        
      
      
        
          
            
            ⁢
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        a
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        -
                        
                          x
                          
                            02
                            ⁢
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            j
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        xj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
          
        
      
      
        
          
            
              
                
                  
                    
                    ⁢
                    
                      
                        +
                        
                          Θ
                          3
                          ∫
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              π
                              
                                2
                                ⁢
                                a
                              
                            
                            ⁢
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      z
                                      -
                                      
                                        γ
                                        
                                          0
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  cot
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      
                                        0
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                      ⁢
                                      
                                        
                                      
                                    
                                  
                                  ⁢
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                +
                                
                                  x
                                  
                                    02
                                    ⁢
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    j
                                  
                                
                              
                              )
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    }
                  
                  ⁢
                  
                    
                  
                  ⁢
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  z
                
                ]
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
     
  









  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  4
                  ⁢
                  
                    π
                    ⁢
                    a
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      ι
                    
                    +
                    1
                  
                
                
                  N
                  ι
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ×
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                
                  t
                  -
                  
                    t
                    
                      0
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                  
                
              
              ⁢
              
                [
                
                  
                    
                      q
                      ij
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                        -
                        τ
                      
                      )
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        (
                                        
                                          z
                                          -
                                          
                                            γ
                                            
                                              0
                                              ⁢
                                              
                                                ◇
                                                ⁢
                                                j
                                              
                                            
                                          
                                        
                                        )
                                      
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      cot
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          
                                            0
                                            ⁢
                                            
                                              ◇
                                              ⁢
                                              j
                                            
                                          
                                          ⁢
                                          
                                            
                                          
                                        
                                      
                                      ⁢
                                      cos
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          0
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    -
                                    
                                      x
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    
                                      x
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                        +
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  Θ
                  3
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        a
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          x
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        xj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            z
                            -
                            
                              z
                              
                                0
                                ⁢
                                
                                  
                                
                                ⁢
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                j
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            z
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
                   
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            π
                            
                              2
                              ⁢
                              b
                            
                          
                          ⁢
                          
                            (
                            
                              
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      γ
                                      
                                        0
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                cot
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                    ⁢
                                    
                                      
                                    
                                  
                                
                                ⁢
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  j
                                
                              
                            
                            )
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              yj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
            ⁢
            
              
                
                  +
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              (
                              
                                z
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                
                                  
                                
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                          
                          +
                          
                            y
                            
                              0
                              ⁢
                              
                                
                              
                              ⁢
                              1
                              ⁢
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              j
                            
                          
                        
                        )
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          yj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              +
            
          
        
      
      
        
          
            
            ⁢
            
              
                
                  +
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              (
                              
                                z
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                
                                  
                                
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                          
                          -
                          
                            y
                            
                              02
                              ⁢
                              
                                
                              
                              ⁢
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              j
                            
                          
                        
                        )
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          yj
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              +
            
          
        
      
      
        
          
            
              
                
                  
                    
                    ⁢
                    
                      
                        +
                        
                          Θ
                          3
                          ∫
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              π
                              
                                2
                                ⁢
                                b
                              
                            
                            ⁢
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      z
                                      -
                                      
                                        γ
                                        
                                          0
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  cot
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      
                                        0
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                      ⁢
                                      
                                        
                                      
                                    
                                  
                                  ⁢
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                +
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    j
                                  
                                
                              
                              )
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                yj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    }
                  
                  ⁢
                  
                    
                  
                  ⁢
                  dz
                
                ]
              
              ⁢
              
                
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  8
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    ab
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
           
          ⁢
          
            ×
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      ι
                    
                    +
                    1
                  
                
                
                  N
                  d
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  θ
                  
                    0
                    ⁢
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    j
                  
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      q
                      ij
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                        -
                        τ
                      
                      )
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                
                  z
                  
                    01
                    ⁢
                    
                      i
                      ⁢
                      j
                    
                  
                
                
                  z
                  
                    02
                    ⁢
                    
                      i
                      ⁢
                      j
                    
                  
                
              
              ⁢
              
                
                  ∫
                  
                    z
                    
                      01
                      ⁢
                      
                        ◇
                        ⁢
                        j
                      
                    
                  
                  
                    z
                    
                      02
                      ⁢
                      
                        ◇
                        ⁢
                        j
                      
                    
                  
                
                ⁢
                
                  [
                  
                    {
                    
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      (
                                      
                                        z
                                        -
                                        
                                          γ
                                          
                                            0
                                            ⁢
                                            
                                              ◇
                                              ⁢
                                              j
                                            
                                          
                                        
                                      
                                      )
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    cot
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      θ
                                      
                                        
                                          0
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                        ⁢
                                        
                                          
                                        
                                      
                                    
                                    ⁢
                                    cos
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      θ
                                      
                                        0
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  -
                                  
                                    
                                      (
                                      
                                        
                                          z
                                          
                                            0
                                            ⁢
                                            
                                              i
                                              ⁢
                                              j
                                            
                                          
                                        
                                        -
                                        
                                          γ
                                          
                                            0
                                            ⁢
                                            i
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    cot
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      θ
                                      
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                        ⁢
                                        
                                          
                                        
                                      
                                    
                                    ⁢
                                    cos
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      θ
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                
                                }
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  xj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      +
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  Θ
                  3
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        a
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          
                            (
                            
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  j
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                      
                      }
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        xj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            b
                          
                        
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  z
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            -
                            
                              
                                (
                                
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      ij
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                          
                          }
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  Θ
                  3
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        b
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          
                            (
                            
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  j
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                      
                      }
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        yj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            }
          
          ×
        
      
    
    
      
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                z
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  z
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
              
              ]
            
            ⁢
            
              
            
            ⁢
            
              dzdz
              
                0
                ⁢
                i
                ⁢
                
                  
                
                ⁢
                j
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   









  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      π
                      2
                    
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                  ⁢
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      d
                    
                    +
                    1
                  
                
                
                  L
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ×
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                
                  t
                  -
                  
                    t
                    
                      0
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                  
                
              
              ⁢
              
                [
                
                  
                    
                      q
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        j
                      
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                        -
                        τ
                      
                      )
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            Θ
                            3
                          
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                        +
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  Θ
                  3
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          z
                          +
                          
                            z
                            
                              0
                              ⁢
                              j
                            
                          
                          -
                          
                            2
                            ⁢
                            
                              d
                              j
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      
                        (
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        
                          z
                          ⁢
                          j
                        
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            }
          
          ×
        
      
    
  




  
    
      
        
                   
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            b
                          
                        
                        ⁢
                        
                          (
                          
                            
                              
                                (
                                
                                  z
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            -
                            
                              y
                              
                                01
                                ⁢
                                
                                  
                                
                                ⁢
                                i
                              
                            
                          
                          )
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        b
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          y
                          
                            01
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        yj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        b
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        -
                        
                          y
                          
                            02
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        yj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        b
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          y
                          
                            02
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        yj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            a
                          
                        
                        ⁢
                        
                          (
                          
                            
                              
                                (
                                
                                  z
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            -
                            
                              x
                              
                                01
                                ⁢
                                
                                  
                                
                                ⁢
                                i
                              
                            
                          
                          )
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        a
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          x
                          
                            01
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        xj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        a
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        -
                        
                          x
                          
                            02
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        xj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
            
              
                
                  
                  ⁢
                  
                    
                      +
                      
                        Θ
                        3
                        ∫
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            π
                            
                              2
                              ⁢
                              a
                            
                          
                          ⁢
                          
                            (
                            
                              
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      γ
                                      
                                        
                                          0
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                        ⁢
                                        
                                          
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                cot
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                    ⁢
                                    
                                      
                                    
                                  
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              +
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  i
                                
                              
                            
                            )
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  }
                
                ⁢
                
                  
                
                ⁢
                dz
              
              ]
            
            ⁢
            
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  









  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    π
                    2
                  
                  ⁢
                  
                    
                      a
                      ⁡
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                    
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      r
                    
                    +
                    1
                  
                
                
                  M
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ×
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                
                  t
                  -
                  
                    t
                    
                      0
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                  
                
              
              ⁢
              
                [
                
                  
                    
                      q
                      
                        i
                        ⁢
                        j
                      
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                        -
                        τ
                      
                      )
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        (
                                        
                                          z
                                          -
                                          
                                            γ
                                            
                                              0
                                              ⁢
                                              
                                                ◇
                                                ⁢
                                                j
                                              
                                            
                                          
                                        
                                        )
                                      
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      cot
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          
                                            0
                                            ⁢
                                            
                                              ◇
                                              ⁢
                                              j
                                            
                                          
                                          ⁢
                                          
                                            
                                          
                                        
                                      
                                      ⁢
                                      cos
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          0
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    -
                                    
                                      x
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    xj
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                        +
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  Θ
                  3
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        a
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          x
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        xj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            b
                          
                        
                        ⁢
                        
                          (
                          
                            
                              
                                (
                                
                                  z
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            -
                            
                              y
                              
                                01
                                ⁢
                                
                                  
                                
                                ⁢
                                i
                              
                            
                          
                          )
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            yj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        b
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          y
                          
                            01
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        yj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        b
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        -
                        
                          y
                          
                            02
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        yj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        b
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          y
                          
                            02
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        yj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            }
          
          ×
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
                   
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  i
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              z
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                  +
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                +
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
                
                  
                    
                    ⁢
                    
                      
                        +
                        
                          
                            Θ
                            3
                            ∫
                          
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        i
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  
                                    z
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    +
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                  ⁢
                  
                    
                  
                  ⁢
                  dz
                
                ]
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    π
                    2
                  
                  ⁢
                  
                    
                      b
                      ⁡
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                    
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      r
                    
                    +
                    1
                  
                
                
                  N
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    [
                    
                      
                        
                          q
                          i
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              t
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            τ
                          
                          )
                        
                      
                      ⁢
                      
                        
                          ∫
                          
                            z
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          
                            z
                            
                              02
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        ⁢
                        
                          {
                          
                            
                              
                                Θ
                                3
                              
                              ⁡
                              
                                (
                                
                                  
                                    
                                      π
                                      
                                        2
                                        ⁢
                                        b
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            (
                                            
                                              z
                                              -
                                              
                                                γ
                                                
                                                  0
                                                  ⁢
                                                  
                                                    ◇
                                                    ⁢
                                                    j
                                                  
                                                
                                              
                                            
                                            )
                                          
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          cot
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            θ
                                            
                                              
                                                0
                                                ⁢
                                                
                                                  ◇
                                                  ⁢
                                                  j
                                                
                                              
                                              ⁢
                                              
                                                
                                              
                                            
                                          
                                          ⁢
                                          sin
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            θ
                                            
                                              0
                                              ⁢
                                              
                                                ◇
                                                ⁢
                                                j
                                              
                                            
                                          
                                        
                                        -
                                        
                                          y
                                          
                                            0
                                            ⁢
                                            
                                              i
                                              ⁢
                                              j
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  ,
                                  
                                    e
                                    
                                      
                                        -
                                        
                                          
                                            (
                                            
                                              π
                                              b
                                            
                                            )
                                          
                                          2
                                        
                                      
                                      ⁢
                                      
                                        η
                                        
                                          y
                                          ⁢
                                          j
                                        
                                      
                                      ⁢
                                      τ
                                    
                                  
                                
                                )
                              
                            
                            +
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  Θ
                  3
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        b
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          y
                          
                            0
                            ⁢
                            
                              
                            
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        yj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            a
                          
                        
                        ⁢
                        
                          (
                          
                            
                              
                                (
                                
                                  z
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            -
                            
                              x
                              
                                01
                                ⁢
                                
                                  
                                
                                ⁢
                                i
                              
                            
                          
                          )
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        a
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          x
                          
                            01
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        xj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        a
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        -
                        
                          x
                          
                            02
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        xj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      π
                      
                        2
                        ⁢
                        a
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        +
                        
                          x
                          
                            02
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                        
                      
                      )
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        xj
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            }
          
          ×
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
                    
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  i
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              z
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                  +
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                +
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
                
                  
                    
                    ⁢
                    
                      
                        +
                        
                          
                            Θ
                            3
                            ∫
                          
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        i
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  
                                    z
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    +
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                  ⁢
                  
                    
                  
                  ⁢
                  dz
                
                ]
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  









  
    
      
        
          
            
          
          ⁢
          
            
              
                +
                
                ⁢
                
                  4
                  
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      j
                    
                    ⁢
                    
                      
                        a
                        ⁢
                        b
                      
                      ⁡
                      
                        (
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                    
                    ⁢
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    0
                  
                  ∞
                
                ⁢
                
                  ∑
                  
                    l
                    =
                    0
                  
                  ∞
                
              
            
            ⁢
            
              ∋
              m
            
            ⁢
            
              ∋
              l
            
            ⁢
            
              
                ∫
                
                  z
                  
                    01
                    ⁢
                    
                      ◇
                      ⁢
                      j
                    
                  
                
                
                  z
                  
                    02
                    ⁢
                    
                      ◇
                      ⁢
                      j
                    
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        m
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          
                            z
                            -
                            
                              γ
                              
                                0
                                ⁢
                                ◇
                                ⁢
                                
                                  
                                
                                ⁢
                                j
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                        
                        ⁢
                        cot
                        ⁢
                        
                          
                        
                        ⁢
                        
                          θ
                          
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                            ⁢
                            
                              
                            
                          
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          θ
                          
                            0
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      b
                    
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      l
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        ⁡
                        
                          (
                          
                            z
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                  )
                
                ×
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              ×
              
                
                  ∫
                  0
                  t
                
                ⁢
                
                  
                
                ⁢
                
                  {
                  
                    
                      
                        ψ
                        =
                      
                      
                        0
                        ⁢
                        yz
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        m
                        ,
                        l
                        ,
                        τ
                      
                      )
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Θ
                      3
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          
                            (
                            
                              z
                              -
                              
                                γ
                                
                                  0
                                  ⁢
                                  ◇
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  j
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          cot
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      )
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            xj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  )
                
              
            
            -
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      ψ
                      =
                    
                    ayz
                  
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    m
                    ,
                    l
                    ,
                    τ
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                
                  Θ
                  4
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          
                            z
                            -
                            
                              γ
                              
                                0
                                ⁢
                                ◇
                                ⁢
                                
                                  
                                
                                ⁢
                                j
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                        
                        ⁢
                        cot
                        ⁢
                        
                          
                        
                        ⁢
                        
                          θ
                          
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                            ⁢
                            
                              
                            
                          
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          θ
                          
                            0
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            xj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  )
                
              
              }
            
            ⁢
            
              
            
            ⁢
            
              e
              
                
                  -
                  
                    {
                    
                      
                        
                          
                            (
                            
                              
                                m
                                ⁢
                                
                                  
                                
                                ⁢
                                π
                              
                              b
                            
                            )
                          
                          2
                        
                        ⁢
                        
                          η
                          yj
                        
                      
                      +
                      
                        
                          
                            (
                            
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                π
                              
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                          2
                        
                        ⁢
                        
                          η
                          zj
                        
                      
                    
                    }
                  
                
                ⁢
                
                  (
                  
                    t
                    -
                    τ
                  
                  )
                
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τdz
          
          +
        
      
    
  




  
    
      
        
          
            
          
          ⁢
          
            
              
                +
                
                ⁢
                
                  4
                  
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      j
                    
                    ⁢
                    
                      
                        a
                        ⁢
                        b
                      
                      ⁡
                      
                        (
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                    
                    ⁢
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    0
                  
                  ∞
                
                ⁢
                
                  ∑
                  
                    l
                    =
                    0
                  
                  ∞
                
              
            
            ⁢
            
              ∋
              n
            
            ⁢
            
              ∋
              l
            
            ⁢
            
              
                ∫
                
                  z
                  
                    01
                    ⁢
                    
                      ◇
                      ⁢
                      j
                    
                  
                
                
                  z
                  
                    02
                    ⁢
                    
                      ◇
                      ⁢
                      j
                    
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          
                            z
                            -
                            
                              γ
                              
                                0
                                ⁢
                                ◇
                                ⁢
                                
                                  
                                
                                ⁢
                                j
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                        
                        ⁢
                        cot
                        ⁢
                        
                          
                        
                        ⁢
                        
                          θ
                          
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                            ⁢
                            
                              
                            
                          
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          θ
                          
                            0
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      a
                    
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      l
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        ⁡
                        
                          (
                          
                            z
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                  
                  )
                
                ×
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              ×
              
                
                  ∫
                  0
                  t
                
                ⁢
                
                  
                
                ⁢
                
                  {
                  
                    
                      
                        ψ
                        =
                      
                      
                        x
                        ⁢
                        
                          
                        
                        ⁢
                        0
                        ⁢
                        z
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        n
                        ,
                        l
                        ,
                        τ
                      
                      )
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Θ
                      3
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    jj
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            θ
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      )
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            yj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  )
                
              
            
            -
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      ψ
                      =
                    
                    xbz
                  
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    n
                    ,
                    l
                    ,
                    τ
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                
                  Θ
                  4
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          
                            
                              z
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              z
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  jj
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          θ
                          
                            
                              0
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                            ⁢
                            
                              
                            
                          
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            yj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  )
                
              
              }
            
            ⁢
            
              
            
            ⁢
            
              e
              
                
                  -
                  
                    {
                    
                      
                        
                          
                            (
                            
                              
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                π
                              
                              a
                            
                            )
                          
                          2
                        
                        ⁢
                        
                          η
                          xj
                        
                      
                      +
                      
                        
                          
                            (
                            
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                π
                              
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                          2
                        
                        ⁢
                        
                          η
                          zj
                        
                      
                    
                    }
                  
                
                ⁢
                
                  (
                  
                    t
                    -
                    τ
                  
                  )
                
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τdz
          
          +
        
      
    
  









  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                4
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    
                      a
                      ⁢
                      b
                    
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            
          
          ⁢
          
            
              ×
              
                
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    0
                  
                  ∞
                
                ⁢
                
                  ∑
                  
                    m
                    =
                    0
                  
                  ∞
                
              
            
            ⁢
            
              ∋
              n
            
            ⁢
            
              ∋
              m
            
            ⁢
            
              
                ∫
                
                  z
                  
                    01
                    ⁢
                    
                      ◇
                      ⁢
                      j
                    
                  
                
                
                  z
                  
                    02
                    ⁢
                    
                      ◇
                      ⁢
                      j
                    
                  
                
              
              ⁢
              
                
              
              ⁢
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          z
                          -
                          
                            γ
                            
                              0
                              ⁢
                              ◇
                              ⁢
                              
                                
                              
                              ⁢
                              j
                            
                          
                        
                        )
                      
                      ⁢
                      
                        
                      
                      ⁢
                      cot
                      ⁢
                      
                        
                      
                      ⁢
                      
                        θ
                        
                          
                            0
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                          ⁢
                          
                            
                          
                        
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        θ
                        
                          0
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    a
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      m
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          z
                          -
                          
                            γ
                            
                              0
                              ⁢
                              ◇
                              ⁢
                              
                                
                              
                              ⁢
                              j
                            
                          
                        
                        )
                      
                      ⁢
                      
                        
                      
                      ⁢
                      cot
                      ⁢
                      
                        
                      
                      ⁢
                      
                        θ
                        
                          
                            0
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                          ⁢
                          
                            
                          
                        
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        θ
                        
                          0
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    b
                  
                  )
                
                ×
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
              
              ⁢
              
                {
                
                  
                    
                      
                        ψ
                        =
                      
                      
                        xy
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        n
                        ,
                        m
                        ,
                        τ
                      
                      )
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Θ
                      3
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  d
                                  jj
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      ψ
                      =
                    
                    xyd
                  
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    n
                    ,
                    m
                    ,
                    τ
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                
                  Θ
                  4
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            z
                            -
                            
                              d
                              jj
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            zj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  )
                
              
              }
            
            ⁢
            
              
            
            ⁢
            
              e
              
                
                  -
                  
                    {
                    
                      
                        
                          
                            (
                            
                              
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                π
                              
                              a
                            
                            )
                          
                          2
                        
                        ⁢
                        
                          η
                          xj
                        
                      
                      +
                      
                        
                          
                            (
                            
                              
                                m
                                ⁢
                                
                                  
                                
                                ⁢
                                π
                              
                              b
                            
                            )
                          
                          2
                        
                        ⁢
                        
                          η
                          yj
                        
                      
                    
                    }
                  
                
                ⁢
                
                  (
                  
                    t
                    -
                    τ
                  
                  )
                
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τdz
          
          +
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
            
              
            
            ⁢
            
              
                +
                
                ⁢
                
                  1
                  
                    8
                    ⁢
                    
                      ab
                      ⁡
                      
                        (
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                    
                    ⁢
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  ∫
                  0
                  d
                
                ⁢
                
                  
                    ∫
                    0
                    b
                  
                  ⁢
                  
                    
                      ∫
                      0
                      a
                    
                    ⁢
                    
                      
                        
                          φ
                          j
                        
                        ⁡
                        
                          (
                          
                            u
                            ,
                            v
                            ,
                            w
                            ,
                            
                              +
                              
                                d
                                j
                              
                            
                          
                          )
                        
                      
                      ×
                    
                  
                
              
            
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                
                  ∫
                  
                    z
                    
                      01
                      ⁢
                      
                        ◇
                        ⁢
                        j
                      
                    
                  
                  
                    z
                    
                      02
                      ⁢
                      
                        ◇
                        ⁢
                        j
                      
                    
                  
                
                ⁢
                
                  [
                  
                    {
                    
                      
                        
                          Θ
                          3
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          
                            
                              
                                π
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      (
                                      
                                        z
                                        -
                                        
                                          γ
                                          
                                            
                                              0
                                              ⁢
                                              
                                                ◇
                                                ⁢
                                                j
                                              
                                            
                                            ⁢
                                            
                                              
                                            
                                          
                                        
                                      
                                      )
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    cot
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      θ
                                      
                                        
                                          0
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                        ⁢
                                        
                                          
                                        
                                      
                                    
                                    ⁢
                                    cos
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      θ
                                      
                                        0
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  -
                                  u
                                
                                )
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        a
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  xj
                                
                                ⁢
                                t
                              
                            
                          
                          )
                        
                      
                      +
                    
                  
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  +
                  
                    Θ
                    3
                  
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          a
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              (
                              
                                z
                                -
                                
                                  γ
                                  
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                    ⁢
                                    
                                      
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                
                                  
                                
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                          
                          +
                          u
                        
                        )
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          xj
                        
                        ⁢
                        t
                      
                    
                  
                  )
                
              
              }
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            π
                            
                              2
                              ⁢
                              b
                            
                          
                          ⁢
                          
                            (
                            
                              
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      γ
                                      
                                        
                                          0
                                          ⁢
                                          
                                            ◇
                                            ⁢
                                            j
                                          
                                        
                                        ⁢
                                        
                                          
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                cot
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                    ⁢
                                    
                                      
                                    
                                  
                                
                                ⁢
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    0
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              -
                              v
                            
                            )
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              yj
                            
                            ⁢
                            t
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  +
                  
                    Θ
                    3
                  
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              (
                              
                                z
                                -
                                
                                  γ
                                  
                                    
                                      0
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                    ⁢
                                    
                                      
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                
                                  
                                
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                          
                          +
                          v
                        
                        )
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          yj
                        
                        ⁢
                        t
                      
                    
                  
                  )
                
              
              }
            
            ×
          
        
      
      
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    d
                                    
                                      j
                                      ⁢
                                      
                                        
                                      
                                    
                                  
                                  -
                                  w
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  z
                                  ⁢
                                  j
                                
                              
                              ⁢
                              t
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    d
                                    
                                      j
                                      ⁢
                                      
                                        
                                      
                                    
                                  
                                  +
                                  w
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              t
                            
                          
                        
                        )
                      
                    
                  
                  }
                
              
              ]
            
            ⁢
            
              
            
            ⁢
            dudvdwdz
          
        
      
    
  




















TABLE 6





 






  
    The
    ⁢
    
      
    
    ⁢
    spatial
    ⁢
    
      
    
    ⁢
    average
    ⁢
    
      
    
    ⁢
    pressure
    ⁢
    
      
    
    ⁢
    response
    ⁢
    
      
    
    ⁢
    of
    ⁢
    
      
    
    ⁢
    the
    ⁢
    
      
    
    ⁢
    
      rectangle
      ⁢
      
        
      
      [
      
        
          (
          
            
              x
              
                02
                ⁢
                
                  ◇
                  ⁢
                  j
                
              
            
            -
            
              x
              
                01
                ⁢
                
                  ◇
                  ⁢
                  j
                
              
            
          
          )
        
        ⁢
        
          
        
        ⁢
        
          (
          
            
              y
              
                02
                ⁢
                
                  ◇
                  ⁢
                  j
                
              
            
            -
            
              y
              
                01
                ⁢
                
                  ◇
                  ⁢
                  j
                
              
            
          
          )
        
      
      ]
    
  
  ,
  
    i
    =
    ◇
  
  ,
  
    

  
  ⁢
  
    
      
        N
        ι
      
      +
      1
    
    ≤
    ◇
    ≤
    
      
        L
        r
      
      ⁢
      
        
      
      ⁢
      is
      ⁢
      
        
      
      ⁢
      given
      ⁢
      
        
      
      ⁢
      
        by
        .
      
    
  















  
    
      
        
          
            
              
                p
                j
              
              =
              
              ⁢
              
                
                  1
                  
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      j
                    
                    ⁢
                    
                      (
                      
                        
                          x
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          x
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          y
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          y
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      1
                    
                    
                      L
                      ι
                    
                  
                  ⁢
                  
                    
                      U
                      (
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        ∫
                        0
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                      
                      ⁢
                      
                        
                          
                            q
                            
                              i
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                t
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              τ
                            
                            )
                          
                        
                        ×
                      
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    x
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    x
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    x
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  x
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ×
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    y
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    y
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    y
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  y
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ×
            
          
        
      
    
    
      
        (
        0.12
        )
      
    
  
  
    
      
        
          
            
                        
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    
                      -
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ⁢
                
                  
                
                ⁢
                d
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
    
    
      
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                a
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      ι
                    
                    +
                    1
                  
                
                
                  M
                  ι
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              y
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
            
                      
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    x
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    x
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    x
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              +
            
          
        
      
       
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                b
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      ι
                    
                    +
                    1
                  
                
                
                  N
                  ι
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              x
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              +
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
                      
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
        
      
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              y
                              
                                01
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              -
            
          
        
      
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              y
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              +
            
          
        
      
      
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ⁢
              
                
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      
                        π
                        2
                      
                      ⁡
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      ι
                    
                    +
                    1
                  
                
                
                  N
                  d
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  θ
                  
                    0
                    ⁢
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    j
                  
                
                ×
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                
                  t
                  -
                  
                    t
                    
                      0
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                  
                
              
              ⁢
              
                
                  
                    q
                    
                      i
                      ⁢
                      j
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                      -
                      τ
                    
                    )
                  
                
                ⁢
                
                  
                    ∫
                    
                      z
                      
                        01
                        ⁢
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        j
                      
                    
                    
                      z
                      
                        02
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  ⁢
                  
                    [
                    
                      {
                      
                        
                          
                            Θ
                            3
                            f
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  
                                    2
                                    ⁢
                                    b
                                  
                                
                                ⁢
                                
                                  {
                                  
                                    
                                      y
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      
                                        (
                                        
                                          
                                            z
                                            
                                              0
                                              ⁢
                                              
                                                i
                                                ⁢
                                                j
                                              
                                            
                                          
                                          -
                                          
                                            γ
                                            
                                              0
                                              ⁢
                                              
                                                i
                                                ⁢
                                                j
                                              
                                            
                                          
                                        
                                        )
                                      
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      cot
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          0
                                          ⁢
                                          ij
                                        
                                      
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      sin
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          0
                                          ⁢
                                          ij
                                        
                                      
                                    
                                  
                                  }
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          b
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    
                                      y
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                        -
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  f
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        {
                        
                          
                            y
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          -
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                ij
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                ij
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            
                              
                            
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
          
        
      
    
    
      
        
          
          ⁢
          
            
              +
              
                
                  Θ
                  3
                  f
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        {
                        
                          
                            y
                            
                              02
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                ij
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                ij
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            
                              
                            
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            -
          
        
      
    
    
      
        
          
            
            ⁢
            
              -
              
                
                  Θ
                  3
                  f
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        {
                        
                          
                            y
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                j
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                j
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            
                              
                            
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
            
                      
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        f
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              
                                2
                                ⁢
                                a
                              
                            
                            ⁢
                            
                              {
                              
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  
                                    (
                                    
                                      
                                        z
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                      -
                                      
                                        γ
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  cot
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      0
                                      ⁢
                                      i
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    θ
                                    
                                      0
                                      ⁢
                                      i
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              }
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  x
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      f
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            
                              2
                              ⁢
                              a
                            
                          
                          ⁢
                          
                            {
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                
                                  (
                                  
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      γ
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                cot
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    0
                                    ⁢
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    0
                                    ⁢
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    j
                                  
                                
                              
                            
                            }
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                x
                                ⁢
                                
                                  
                                
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  +
                  
                    
                      Θ
                      3
                      f
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            
                              2
                              ⁢
                              a
                            
                          
                          ⁢
                          
                            {
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                
                                  (
                                  
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      γ
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                cot
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    0
                                    ⁢
                                    ij
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    0
                                    ⁢
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    j
                                  
                                
                              
                            
                            }
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                x
                                ⁢
                                
                                  
                                
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                -
              
            
          
        
        
          
            
              
                
                ⁢
                
                  -
                  
                    
                      Θ
                      3
                      f
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            
                              2
                              ⁢
                              a
                            
                          
                          ⁢
                          
                            {
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                
                                  (
                                  
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      γ
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                cot
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    0
                                    ⁢
                                    ij
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  θ
                                  
                                    0
                                    ⁢
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    j
                                  
                                
                              
                            
                            }
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                x
                                ⁢
                                
                                  
                                
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ×
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    ×
                    
                      {
                      
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  
                                    2
                                    ⁢
                                    d
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          d
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    
                                      y
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                        +
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  
                                    2
                                    ⁢
                                    d
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    z
                                    +
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          d
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    
                                      y
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      j
                                    
                                  
                                  ⁢
                                  τ
                                
                              
                            
                            )
                          
                        
                      
                      }
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                
                  dz
                  
                    0
                    ⁢
                    ij
                  
                
                ⁢
                d
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
       
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                
                  2
                  ⁢
                  
                    a
                    ⁢
                    b
                  
                
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      d
                    
                    +
                    1
                  
                
                
                  L
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              x
                              
                                01
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
            
              
                Θ
                3
                
                  ∫
                  
                    
                  
                  ∫
                
              
              ⁡
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            x
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            x
                            
                              01
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      a
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        
                          x
                          ⁢
                          j
                        
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            -
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
            
              
                Θ
                3
                
                  ∫
                  
                    
                  
                  ∫
                
              
              ⁡
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            x
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          -
                          
                            x
                            
                              02
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      a
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        
                          x
                          ⁢
                          j
                        
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          +
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
            
                        
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    y
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    y
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ×
            
          
        
        
          
            
              
              ⁢
              
                
                  ×
                  
                    {
                    
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          Θ
                          3
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    +
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                  ⁢
                  
                    
                  
                  ⁢
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  τ
                
                +
              
            
          
        
      
       
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                
                  2
                  ⁢
                  b
                
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      r
                    
                    +
                    1
                  
                
                
                  M
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              x
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
            
                       
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ×
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    
                      -
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    z
                                    -
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  +
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ⁢
                
                  
                
                ⁢
                d
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
       
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                
                  2
                  ⁢
                  a
                
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      r
                    
                    +
                    1
                  
                
                
                  N
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              x
                              
                                01
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
            
              
                Θ
                3
                
                  ∫
                  
                    
                  
                  ∫
                
              
              ⁡
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            x
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            x
                            
                              01
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      a
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        
                          x
                          ⁢
                          j
                        
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            -
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
            
              
                Θ
                3
                
                  ∫
                  
                    
                  
                  ∫
                
              
              ⁡
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            x
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          -
                          
                            x
                            
                              02
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      a
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        
                          x
                          ⁢
                          j
                        
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          +
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
                      
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  y
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  y
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  +
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  y
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                +
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
        
      
      
        
          
            
              
                
                ⁢
                
                  
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  z
                                  -
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                +
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ⁢
              
                
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  










   
  
    
      
        
          
                     
            ⁢
            
              
                +
                
                ⁢
                
                  1
                  
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      j
                    
                    ⁢
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          x
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          x
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              jj
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          y
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          y
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                
              
              ×
            
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                
                  ∫
                  0
                  t
                
                ⁢
                
                  
                    ∫
                    0
                    a
                  
                  ⁢
                  
                    
                      ∫
                      0
                      b
                    
                    ⁢
                    
                      [
                      
                        
                          
                            
                              ψ
                              j
                            
                            ⁡
                            
                              (
                              
                                u
                                ,
                                v
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            Θ
                            3
                          
                          ⁢
                          
                            {
                            
                              
                                
                                  π
                                  ⁡
                                  
                                    (
                                    
                                      z
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                    )
                                  
                                
                                
                                  2
                                  ⁢
                                  
                                    (
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                    )
                                  
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          
                                            
                                              d
                                              
                                                j
                                                +
                                                1
                                              
                                            
                                            -
                                            
                                              d
                                              j
                                            
                                          
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    
                                      η
                                      zj
                                    
                                    (
                                    
                                      t
                                      -
                                      τ
                                    
                                    )
                                  
                                
                              
                            
                            }
                          
                        
                        -
                      
                    
                  
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      ψ
                      
                        j
                        +
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        u
                        ,
                        v
                        ,
                        τ
                      
                      )
                    
                  
                
                ⁢
                
                  Θ
                  4
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            z
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            zj
                          
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                    
                  
                  }
                
              
              ]
            
            ×
          
        
      
    
    ⁢
    
      

    
    ⁢
    
      
        
          
                      
            ⁢
            
              ×
              
                [
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                u
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                u
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  
                    +
                    
                      Θ
                      3
                      ∫
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              ]
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                [
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                v
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                v
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
                
                ⁢
                
                  
                    
                      +
                      
                        Θ
                        3
                        ∫
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                v
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                v
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                
                ]
              
              ⁢
              
                
              
              ⁢
              dudvd
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  b
                
                ⁢
                
                  
                    ∫
                    0
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            ψ
                            
                              0
                              ⁢
                              yzj
                            
                          
                          ⁡
                          
                            (
                            
                              v
                              ,
                              w
                              ,
                              τ
                            
                            )
                          
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          {
                          
                            
                              
                                Θ
                                3
                                ∫
                              
                              (
                              
                                
                                  
                                    πx
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            a
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        zj
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                            -
                            
                              
                                Θ
                                3
                                ∫
                              
                              (
                              
                                
                                  
                                    πx
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            a
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        
                                          z
                                          ⁢
                                          j
                                        
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    ψ
                    ayzj
                  
                  ⁡
                  
                    (
                    
                      v
                      ,
                      w
                      ,
                      τ
                    
                    )
                  
                
              
              ⁢
              
                {
                
                  
                    
                      Θ
                      4
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            πx
                            
                              02
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  x
                                  ⁢
                                  j
                                
                              
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      4
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            πx
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                xj
                              
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                }
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                
                  +
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            v
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            v
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
            
            ]
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            
              ×
              
                [
                
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  d
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                  
                                
                                -
                                w
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  d
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                  
                                
                                +
                                w
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                
                ]
              
              ⁢
              
                
              
              ⁢
              dvdwd
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  
   










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        x
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            jj
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                    ∫
                    0
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            ψ
                            
                              x
                              ⁢
                              
                                
                              
                              ⁢
                              0
                              ⁢
                              zj
                            
                          
                          (
                          
                            u
                            ,
                            w
                            ,
                            τ
                          
                          )
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          {
                          
                            
                              
                                Θ
                                3
                                ∫
                              
                              (
                              
                                
                                  
                                    πy
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    b
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            b
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        
                                          y
                                          ⁢
                                          j
                                        
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                            -
                            
                              
                                Θ
                                3
                                ∫
                              
                              (
                              
                                
                                  
                                    
                                      π
                                      ⁢
                                      y
                                    
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    b
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            b
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        
                                          y
                                          ⁢
                                          j
                                        
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    ψ
                    xbzj
                  
                  ⁡
                  
                    (
                    
                      u
                      ,
                      w
                      ,
                      τ
                    
                    )
                  
                
              
              ⁢
              
                {
                
                  
                    
                      Θ
                      4
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            πy
                            
                              02
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      4
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            πy
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                yj
                              
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                }
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                z
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                z
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                
                  +
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            u
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              x
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            u
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
            
            ]
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            
              ×
              
                [
                
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  d
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                  
                                
                                -
                                w
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                  
                    
                      Θ
                      3
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                z
                                -
                                
                                  d
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                  
                                
                                +
                                w
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                
                ]
              
              ⁢
              
                
              
              ⁢
              dudwd
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  
   










  
    
      
        
          
        
        ⁢
        
          
            +
            
            ⁢
            
              1
              
                2
                ⁢
                
                  (
                  
                    
                      d
                      
                        j
                        +
                        1
                      
                    
                    -
                    
                      d
                      j
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      x
                      
                        02
                        ⁢
                        
                          ◇
                          ⁢
                          j
                        
                      
                    
                    -
                    
                      x
                      
                        01
                        ⁢
                        
                          ◇
                          ⁢
                          jj
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      y
                      
                        02
                        ⁢
                        
                          ◇
                          ⁢
                          j
                        
                      
                    
                    -
                    
                      y
                      
                        01
                        ⁢
                        
                          ◇
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
              
            
          
          ×
        
      
    
  
  
    
      
        
        ⁢
        
          ×
          
            
              ∫
              0
              
                
                  d
                  
                    j
                    +
                    1
                  
                
                -
                
                  d
                  j
                
              
            
            ⁢
            
              
                ∫
                0
                b
              
              ⁢
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                    
                      φ
                      j
                    
                    (
                    
                      u
                      ,
                      v
                      ,
                      
                        w
                        +
                        
                          d
                          j
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                  
                  [
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              (
                              
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                u
                              
                              )
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    x
                                    ⁢
                                    j
                                  
                                
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                        }
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  u
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    x
                                    ⁢
                                    j
                                  
                                
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                        }
                      
                    
                    +
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
          ⁢
          
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                {
                
                  
                    
                      π
                      (
                      
                        
                          x
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        +
                        u
                      
                      )
                    
                    
                      2
                      ⁢
                      a
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        (
                        
                          t
                          -
                          τ
                        
                        )
                      
                    
                  
                
                }
              
            
            -
            
              
                Θ
                3
                ∫
              
              ⁢
              
                {
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            x
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          u
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      a
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              a
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        (
                        
                          t
                          -
                          τ
                        
                        )
                      
                    
                  
                
                }
              
            
          
          ]
        
        ×
      
    
  
  
    
      
        
        ⁢
        
          ×
          
            [
            
              
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            v
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            v
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              +
            
          
        
      
    
  
  
    
      
        
          
          ⁢
          
            
              
                +
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                {
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            y
                            
                              02
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          v
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      b
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                    
                  
                
                }
              
            
            -
            
              
                Θ
                3
                ∫
              
              ⁢
              
                {
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            y
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          v
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      b
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                    
                  
                
                }
              
            
          
          ]
        
        ×
      
    
  
  
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                d
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          t
                        
                      
                    
                    }
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              z
                              -
                              
                                d
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          t
                        
                      
                    
                    }
                  
                
              
              }
            
          
          ]
        
        ⁢
        
          
        
        ⁢
        dudvdw
      
    
  




















TABLE 7





 






  
    The
    ⁢
    
      
    
    ⁢
    spatial
    ⁢
    
      
    
    ⁢
    average
    ⁢
    
      
    
    ⁢
    pressure
    ⁢
    
      
    
    ⁢
    response
    ⁢
    
      
    
    ⁢
    of
    ⁢
    
      
    
    ⁢
    the
    ⁢
    
      
    
    ⁢
    
      rectangle
      ⁢
      
        
      
      [
      
        
          (
          
            
              y
              
                02
                ⁢
                
                  ◇
                  ⁢
                  j
                
              
            
            -
            
              y
              
                01
                ⁢
                
                  ◇
                  ⁢
                  j
                
              
            
          
          )
        
        ⁢
        
          (
          
            
              z
              
                02
                ⁢
                
                  ◇
                  ⁢
                  j
                
              
            
            -
            
              z
              
                01
                ⁢
                
                  ◇
                  ⁢
                  j
                
              
            
          
          )
        
      
      ]
    
  
  ,
  
    

  
  ⁢
  
    i
    =
    ◇
  
  ,
  
    
      
        L
        r
      
      +
      1
    
    ≤
    ◇
    ≤
    
      
        M
        r
      
      ⁢
      
        
      
      ⁢
      is
      ⁢
      
        
      
      ⁢
      given
      ⁢
      
        
      
      ⁢
      
        by
        .
      
    
  















  
    
      
        
          
            
              
                p
                j
              
              =
              
              ⁢
              
                
                  
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          d
                          
                            j
                            +
                            1
                          
                        
                        -
                        
                          d
                          j
                        
                      
                      )
                    
                  
                  
                    
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                      j
                    
                    ⁢
                    
                      a
                      ⁡
                      
                        (
                        
                          
                            y
                            
                              02
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          -
                          
                            y
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      (
                      
                        
                          z
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          z
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      1
                    
                    
                      L
                      ι
                    
                  
                  ⁢
                  
                    
                      U
                      (
                      
                        t
                        -
                        
                          t
                          
                            0
                            ⁢
                            
                              i
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        ∫
                        0
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                      
                      ⁢
                      
                        
                          
                            q
                            
                              i
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                t
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              τ
                            
                            )
                          
                        
                        ×
                      
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  -
                                  
                                    x
                                    
                                      0
                                      ⁢
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  +
                                  
                                    x
                                    
                                      0
                                      ⁢
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                xj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    y
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    y
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    +
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    y
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  y
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                }
              
              ×
            
          
        
      
    
    
      
        (
        0.13
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              
                
                ⁢
                
                  ×
                  
                    {
                    
                      
                        
                          Θ
                          3
                          
                            ∫
                            
                              
                            
                            ∫
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      z
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      -
                      
                        
                          Θ
                          3
                          
                            ∫
                            
                              
                            
                            ∫
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      z
                                      
                                        01
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      z
                                      
                                        01
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                      -
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    -
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    -
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                  ⁢
                  
                    
                      +
                      
                        
                          Θ
                          3
                          
                            ∫
                            
                              
                            
                            ∫
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    +
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    -
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                +
              
            
          
        
      
    
    
      
        
      
    
  









  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      ι
                    
                    +
                    1
                  
                
                
                  M
                  ι
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              y
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
  




  
    
      
        
          
                         
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                      }
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  
                    +
                    
                      Θ
                      3
                      ∫
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                    }
                  
                
              
              }
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                x
                                -
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                x
                                +
                                
                                  x
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              xy
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                x
                                -
                                
                                  x
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            a
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    a
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
        
      
    
     
  









  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                b
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        
                          y
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          y
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      ι
                    
                    +
                    1
                  
                
                
                  N
                  ι
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  0
                                  ⁢
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                
                  +
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              z
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                d
                                j
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            zj
                          
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                    
                  
                  }
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              z
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                d
                                j
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            zj
                          
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                    
                  
                  }
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              y
                              
                                01
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
            
              
                Θ
                3
                
                  ∫
                  
                    
                  
                  ∫
                
              
              ⁡
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            y
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            y
                            
                              01
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      b
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        
                          y
                          ⁢
                          j
                        
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            -
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              y
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
            
              
                Θ
                3
                
                  ∫
                  
                    
                  
                  ∫
                
              
              ⁡
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            y
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          -
                          
                            y
                            
                              02
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      b
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        
                          y
                          ⁢
                          j
                        
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  +
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ⁢
            
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    
                      
                        π
                        2
                      
                      ⁡
                      
                        (
                        
                          φc
                          t
                        
                        )
                      
                    
                    j
                  
                  ⁢
                  a
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      ι
                    
                    +
                    1
                  
                
                
                  N
                  d
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  θ
                  
                    0
                    ⁢
                    ij
                  
                
                ×
              
            
          
        
      
    
    
      
        
           
          ⁢
          
            ×
            
              
                ∫
                0
                
                  t
                  -
                  
                    t
                    
                      0
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                  
                
              
              ⁢
              
                
                  q
                  j
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                    -
                    τ
                  
                  )
                
                ⁢
                
                  
                    ∫
                    
                      z
                      ⁢
                      
                        
                      
                      ⁢
                      01
                      ⁢
                      
                        i
                        ⁢
                        j
                      
                    
                    
                      z
                      
                        02
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  ⁢
                  
                    [
                    
                      {
                      
                        
                          
                            Θ
                            3
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  π
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ⁢
                                
                                  {
                                  
                                    x
                                    -
                                    
                                      
                                        (
                                        
                                          
                                            z
                                            
                                              0
                                              ⁢
                                              
                                                i
                                                ⁢
                                                j
                                              
                                            
                                          
                                          -
                                          
                                            γ
                                            
                                              0
                                              ⁢
                                              
                                                i
                                                ⁢
                                                j
                                              
                                            
                                          
                                        
                                        )
                                      
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      cot
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                      ⁢
                                      cos
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        θ
                                        
                                          0
                                          ⁢
                                          
                                            i
                                            ⁢
                                            j
                                          
                                        
                                      
                                    
                                  
                                  }
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    η
                                    ⁢
                                    xj
                                    ⁢
                                    τ
                                  
                                
                              
                            
                            )
                          
                        
                        +
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              +
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          a
                        
                      
                      ⁢
                      
                        {
                        
                          x
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                                ⁢
                                
                                  
                                
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          ⁢
                          xj
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          
                            2
                            ⁢
                            b
                          
                        
                        ⁢
                        
                          {
                          
                            
                              y
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              
                                (
                                
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    γ
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              cot
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              ⁢
                              
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                θ
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                          
                          }
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          η
                          ⁢
                          
                            
                          
                          ⁢
                          
                            y
                            ⁢
                            j
                            ⁢
                            τ
                          
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        {
                        
                          
                            y
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          -
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        η
                        ⁢
                        
                          
                        
                        ⁢
                        
                          y
                          ⁢
                          j
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            +
          
        
      
    
    
      
        
          
          ⁢
          
            
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        {
                        
                          
                            y
                            
                              02
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        η
                        ⁢
                        
                          
                        
                        ⁢
                        
                          y
                          ⁢
                          j
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            -
          
        
      
    
    
      
        
          
            
            ⁢
            
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        
                          2
                          ⁢
                          b
                        
                      
                      ⁢
                      
                        {
                        
                          
                            y
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            
                              (
                              
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  γ
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cot
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              θ
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                        
                        }
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        η
                        ⁢
                        
                          
                        
                        ⁢
                        
                          y
                          ⁢
                          j
                          ⁢
                          τ
                        
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
                       
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  
                                    η
                                    zj
                                  
                                  ⁡
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                
                                2
                              
                            
                            ⁢
                            t
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  
                                    η
                                    zj
                                  
                                  ⁡
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                
                                2
                              
                            
                            ⁢
                            t
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
                
                  
                  ⁢
                  
                    
                      +
                      
                        
                          Θ
                          3
                          ∫
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    +
                                    
                                      z
                                      
                                        0
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      
                                        η
                                        zj
                                      
                                      ⁡
                                      
                                        (
                                        
                                          π
                                          
                                            
                                              d
                                              
                                                j
                                                +
                                                1
                                              
                                            
                                            -
                                            
                                              d
                                              j
                                            
                                          
                                        
                                        )
                                      
                                    
                                    2
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          )
                        
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    z
                                    
                                      0
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    
                                      η
                                      zj
                                    
                                    ⁡
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                  
                                  2
                                
                              
                              ⁢
                              t
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ]
              
              ⁢
              
                
              
              ⁢
              
                dz
                
                  0
                  ⁢
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  j
                
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              τ
            
            +
          
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                
                  2
                  ⁢
                  b
                
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      N
                      d
                    
                    +
                    1
                  
                
                
                  L
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            x
                            +
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              y
                              
                                01
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
            
              
                Θ
                3
                
                  ∫
                  
                    
                  
                  ∫
                
              
              ⁡
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            y
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            y
                            
                              01
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      b
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        
                          y
                          ⁢
                          j
                        
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            -
          
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
    
    ⁢
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              y
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
              +
            
          
        
      
      
        
          
            
              
              ⁢
              
                
                  +
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  y
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
      
        
          
            
            ⁢
            
              ×
              
                {
                
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                      }
                    
                  
                  -
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                
              
            
          
        
      
      
        
          
            
              
                
                ⁢
                
                  
                    
                      +
                      
                        Θ
                        3
                        ∫
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                  +
                  
                    
                      Θ
                      3
                      ∫
                    
                    ⁢
                    
                      {
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    01
                                    ⁢
                                    j
                                  
                                
                                +
                                
                                  z
                                  
                                    0
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                zj
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  τ
                                
                                )
                              
                            
                          
                        
                      
                      }
                    
                  
                
                }
              
              ⁢
              
                
              
              ⁢
              dτ
            
            +
          
        
      
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                
                  2
                  ⁢
                  b
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        
                          y
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          y
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      L
                      r
                    
                    +
                    1
                  
                
                
                  M
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  0
                                  ⁢
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              }
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              y
                              
                                01
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
            
              
                Θ
                3
                
                  ∫
                  
                    
                  
                  ∫
                
              
              ⁡
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            y
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            y
                            
                              01
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      b
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        
                          y
                          ⁢
                          j
                        
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            -
          
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              y
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            +
            
              
                Θ
                3
                
                  ∫
                  
                    
                  
                  ∫
                
              
              ⁡
              
                (
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            y
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          -
                          
                            y
                            
                              02
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      b
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              b
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        η
                        
                          y
                          ⁢
                          j
                        
                      
                      ⁢
                      τ
                    
                  
                
                )
              
            
            +
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  Θ
                  3
                  
                    ∫
                    
                      
                    
                    ∫
                  
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              y
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
    
    ⁢
    
      
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    
                      +
                      
                        
                          Θ
                          3
                          
                            ∫
                            
                              
                            
                            ∫
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    +
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    -
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ⁢
                
                  
                
                ⁢
                dτ
              
              +
            
          
        
      
       
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                    )
                  
                
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    
                      M
                      r
                    
                    +
                    1
                  
                
                
                  N
                  r
                
              
              ⁢
              
                
                  U
                  (
                  
                    t
                    -
                    
                      t
                      
                        0
                        ⁢
                        
                          i
                          ⁢
                          j
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ∫
                    0
                    
                      t
                      -
                      
                        t
                        
                          0
                          ⁢
                          
                            i
                            ⁢
                            j
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        q
                        
                          i
                          ⁢
                          j
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            t
                            
                              0
                              ⁢
                              
                                i
                                ⁢
                                j
                              
                            
                          
                          -
                          τ
                        
                        )
                      
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              
                                x
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            xj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              
                                x
                                
                                  02
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              x
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            x
                            +
                            
                              x
                              
                                02
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        a
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                a
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            x
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                +
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                y
                                
                                  0
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁡
                
                  (
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              y
                              
                                0
                                ⁢
                                
                                  i
                                  ⁢
                                  j
                                
                              
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          
                            y
                            ⁢
                            j
                          
                        
                        ⁢
                        τ
                      
                    
                  
                  )
                
              
            
            }
          
          ×
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
    
    ⁢
    
      
        
          
            
              
              ⁢
              
                ×
                
                  {
                  
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                +
                                
                                  z
                                  
                                    01
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              +
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  -
                  
                    
                      Θ
                      3
                      
                        ∫
                        
                          
                        
                        ∫
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            π
                            ⁡
                            
                              (
                              
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      ◇
                                      ⁢
                                      j
                                    
                                  
                                
                                -
                                
                                  z
                                  
                                    02
                                    ⁢
                                    
                                      i
                                      ⁢
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                          
                            2
                            ⁢
                            
                              (
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                              )
                            
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    
                                      
                                        d
                                        
                                          j
                                          +
                                          1
                                        
                                      
                                      -
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              η
                              zj
                            
                            ⁢
                            τ
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    Θ
                    3
                    
                      ∫
                      
                        
                      
                      ∫
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    i
                                    ⁢
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          τ
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    
                      +
                      
                        
                          Θ
                          3
                          
                            ∫
                            
                              
                            
                            ∫
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                π
                                ⁡
                                
                                  (
                                  
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    +
                                    
                                      z
                                      
                                        02
                                        ⁢
                                        
                                          i
                                          ⁢
                                          j
                                        
                                      
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        d
                                        j
                                      
                                    
                                  
                                  )
                                
                              
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            ,
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      (
                                      
                                        π
                                        
                                          
                                            d
                                            
                                              j
                                              +
                                              1
                                            
                                          
                                          -
                                          
                                            d
                                            j
                                          
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  η
                                  zj
                                
                                ⁢
                                τ
                              
                            
                          
                          )
                        
                      
                    
                    -
                    
                      
                        Θ
                        3
                        
                          ∫
                          
                            
                          
                          ∫
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    z
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  
                                    z
                                    
                                      02
                                      ⁢
                                      
                                        i
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              
                                (
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                                )
                              
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                η
                                zj
                              
                              ⁢
                              τ
                            
                          
                        
                        )
                      
                    
                  
                  }
                
                ⁢
                
                  
                
                ⁢
                dτ
              
              +
            
          
        
      
       
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        
                          y
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          y
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                    ∫
                    0
                    b
                  
                  ⁢
                  
                    {
                    
                      
                        
                          ψ
                          j
                        
                        ⁡
                        
                          (
                          
                            u
                            ,
                            v
                            ,
                            τ
                          
                          )
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              Θ
                              3
                              ∫
                            
                            ⁢
                            
                              {
                              
                                
                                  
                                    π
                                    ⁡
                                    
                                      (
                                      
                                        
                                          z
                                          
                                            02
                                            ⁢
                                            
                                              i
                                              ⁢
                                              j
                                            
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                      )
                                    
                                  
                                  
                                    2
                                    ⁢
                                    
                                      (
                                      
                                        
                                          d
                                          
                                            j
                                            +
                                            1
                                          
                                        
                                        -
                                        
                                          d
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            
                                              
                                                d
                                                
                                                  j
                                                  +
                                                  1
                                                
                                              
                                              -
                                              
                                                d
                                                j
                                              
                                            
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        η
                                        zj
                                      
                                      (
                                      
                                        t
                                        -
                                        τ
                                      
                                      )
                                    
                                  
                                
                              
                              }
                            
                          
                          -
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  Θ
                  3
                  ∫
                
              
              ⁢
              
                {
                
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            z
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      
                        (
                        
                          
                            d
                            
                              j
                              +
                              1
                            
                          
                          -
                          
                            d
                            j
                          
                        
                        )
                      
                    
                  
                  ,
                  
                    e
                    
                      
                        -
                        
                          
                            (
                            
                              π
                              
                                
                                  d
                                  
                                    j
                                    +
                                    1
                                  
                                
                                -
                                
                                  d
                                  j
                                
                              
                            
                            )
                          
                          2
                        
                      
                      ⁢
                      
                        
                          η
                          zj
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                    
                  
                
                }
              
            
            }
          
          -
        
      
    
    
      
        
          
          ⁢
          
            
              -
              
                
                  ψ
                  
                    j
                    +
                    1
                  
                
                ⁡
                
                  (
                  
                    u
                    ,
                    v
                    ,
                    τ
                  
                  )
                
              
            
            ⁢
            
              {
              
                
                  
                    Θ
                    4
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                    }
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  -
                  
                    Θ
                    4
                    ∫
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            zj
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              }
            
            ]
          
          ×
        
      
    
  
  ⁢
  
     
    
      

    
    ⁢
    
      
        
          
            
                           
              ⁢
              
                ×
                
                  [
                  
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  -
                                  u
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    x
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    +
                    
                      
                        Θ
                        3
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  x
                                  +
                                  u
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              a
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    x
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                  
                  ]
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                ×
                
                  [
                  
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  v
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    y
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  -
                                  v
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    y
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    
                      
                        +
                        
                          Θ
                          3
                          ∫
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      02
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  v
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    y
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                    -
                    
                      
                        Θ
                        3
                        ∫
                      
                      ⁢
                      
                        {
                        
                          
                            
                              π
                              ⁡
                              
                                (
                                
                                  
                                    y
                                    
                                      01
                                      ⁢
                                      
                                        ◇
                                        ⁢
                                        j
                                      
                                    
                                  
                                  +
                                  v
                                
                                )
                              
                            
                            
                              2
                              ⁢
                              b
                            
                          
                          ,
                          
                            e
                            
                              
                                -
                                
                                  
                                    (
                                    
                                      π
                                      b
                                    
                                    )
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  η
                                  
                                    y
                                    ⁢
                                    j
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    τ
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                dudvd
                ⁢
                
                  
                
                ⁢
                τ
              
              +
            
          
        
      
       
    
  










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        
                          y
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          y
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  b
                
                ⁢
                
                  
                    ∫
                    0
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                  
                  ⁢
                  
                    
                      {
                      
                        
                          
                            
                              ψ
                              
                                0
                                ⁢
                                yzj
                              
                            
                            ⁡
                            
                              (
                              
                                v
                                ,
                                w
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              Θ
                              3
                            
                            (
                            
                              
                                πx
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    
                                      η
                                      
                                        x
                                        ⁢
                                        j
                                      
                                    
                                    (
                                    
                                      t
                                      -
                                      τ
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        -
                        
                          
                            
                              ψ
                              ayzj
                            
                            ⁡
                            
                              (
                              
                                v
                                ,
                                w
                                ,
                                τ
                              
                              )
                            
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              Θ
                              4
                            
                            (
                            
                              
                                πx
                                
                                  2
                                  ⁢
                                  a
                                
                              
                              ,
                              
                                e
                                
                                  
                                    -
                                    
                                      
                                        (
                                        
                                          π
                                          a
                                        
                                        )
                                      
                                      2
                                    
                                  
                                  ⁢
                                  
                                    
                                      η
                                      
                                        x
                                        ⁢
                                        j
                                      
                                    
                                    (
                                    
                                      t
                                      -
                                      τ
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                      }
                    
                    ×
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                
                  +
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            v
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            v
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
            
            ]
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  ×
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              }
            
            ⁢
            dvdwdτ
          
          +
        
      
    
  
   










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  
                    
                      (
                      
                        φc
                        t
                      
                      )
                    
                    j
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        
                          y
                          
                            02
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                        -
                        
                          y
                          
                            01
                            ⁢
                            
                              ◇
                              ⁢
                              j
                            
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                t
              
              ⁢
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                    ∫
                    0
                    
                      
                        d
                        
                          j
                          +
                          1
                        
                      
                      -
                      
                        d
                        j
                      
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            ψ
                            
                              x
                              ⁢
                              
                                
                              
                              ⁢
                              0
                              ⁢
                              zj
                            
                          
                          (
                          
                            v
                            ,
                            w
                            ,
                            τ
                          
                          )
                        
                        ⁢
                        
                          
                            
                          
                          ⁢
                          
                            
                          
                        
                        ⁢
                        
                          {
                          
                            
                              
                                Θ
                                3
                                ∫
                              
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        π
                                        ⁢
                                        y
                                      
                                      
                                        02
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    
                                      2
                                      ⁢
                                      b
                                    
                                  
                                  ,
                                  
                                    e
                                    
                                      
                                        -
                                        
                                          
                                            (
                                            
                                              π
                                              b
                                            
                                            )
                                          
                                          2
                                        
                                      
                                      ⁢
                                      
                                        
                                          η
                                          
                                            y
                                            ⁢
                                            j
                                          
                                        
                                        ⁡
                                        
                                          (
                                          
                                            t
                                            -
                                            τ
                                          
                                          )
                                        
                                      
                                    
                                  
                                
                                )
                              
                            
                            -
                            
                              
                                Θ
                                3
                                ∫
                              
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        π
                                        ⁢
                                        y
                                      
                                      
                                        01
                                        ⁢
                                        
                                          ◇
                                          ⁢
                                          j
                                        
                                      
                                    
                                    
                                      2
                                      ⁢
                                      b
                                    
                                  
                                  ,
                                  
                                    e
                                    
                                      
                                        -
                                        
                                          
                                            (
                                            
                                              π
                                              b
                                            
                                            )
                                          
                                          2
                                        
                                      
                                      ⁢
                                      
                                        
                                          η
                                          
                                            y
                                            ⁢
                                            j
                                          
                                        
                                        ⁡
                                        
                                          (
                                          
                                            t
                                            -
                                            τ
                                          
                                          )
                                        
                                      
                                    
                                  
                                
                                )
                              
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                -
                
                  
                    ψ
                    xbzj
                  
                  ⁡
                  
                    (
                    
                      v
                      ,
                      w
                      ,
                      τ
                    
                    )
                  
                
              
              ⁢
              
                {
                
                  
                    
                      Θ
                      4
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            πy
                            
                              02
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                
                                  y
                                  ⁢
                                  j
                                
                              
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      Θ
                      4
                      ∫
                    
                    ⁡
                    
                      (
                      
                        
                          
                            πy
                            
                              01
                              ⁢
                              
                                ◇
                                ⁢
                                j
                              
                            
                          
                          
                            2
                            ⁢
                            b
                          
                        
                        ,
                        
                          e
                          
                            
                              -
                              
                                
                                  (
                                  
                                    π
                                    b
                                  
                                  )
                                
                                2
                              
                            
                            ⁢
                            
                              
                                η
                                yj
                              
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                }
              
            
            }
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              -
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
                
                  
                    Θ
                    3
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              x
                              +
                              u
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          a
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  a
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                x
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              ]
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  ×
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              +
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              zj
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
              
              }
            
            ⁢
            
              
            
            ⁢
            dvdwd
            ⁢
            
              
            
            ⁢
            τ
          
          +
        
      
    
  
   










  
    
      
        
          
            
          
          ⁢
          
            
              +
              
              ⁢
              
                1
                
                  2
                  ⁢
                  a
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        y
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        y
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        
                          02
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                      -
                      
                        z
                        
                          01
                          ⁢
                          
                            ◇
                            ⁢
                            j
                          
                        
                      
                    
                    )
                  
                
              
            
            ×
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              
                ∫
                0
                
                  
                    d
                    
                      j
                      +
                      1
                    
                  
                  -
                  
                    d
                    j
                  
                
              
              ⁢
              
                
                  ∫
                  0
                  b
                
                ⁢
                
                  
                    ∫
                    0
                    a
                  
                  ⁢
                  
                    
                      
                        φ
                        j
                      
                      ⁡
                      
                        (
                        
                          u
                          ,
                          v
                          ,
                          
                            w
                            +
                            
                              d
                              j
                            
                          
                        
                        )
                      
                    
                    [
                    
                      
                        {
                        
                          
                            
                              Θ
                              3
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        -
                                        u
                                      
                                      )
                                    
                                  
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            a
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      η
                                      
                                        x
                                        ⁢
                                        j
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              )
                            
                          
                          +
                          
                            
                              Θ
                              3
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    π
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        +
                                        u
                                      
                                      )
                                    
                                  
                                  
                                    2
                                    ⁢
                                    a
                                  
                                
                                ,
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          (
                                          
                                            π
                                            a
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      η
                                      
                                        x
                                        ⁢
                                        j
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              )
                            
                          
                        
                        }
                      
                      ×
                    
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              [
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                y
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              v
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          b
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  b
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            
                              η
                              
                                y
                                ⁢
                                j
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                τ
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                
                  +
                  
                    Θ
                    3
                    ∫
                  
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            v
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              y
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            +
                            v
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        b
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                b
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          
                            η
                            
                              y
                              ⁢
                              j
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              τ
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
            
            ]
          
          ×
        
      
    
    
      
        
          
          ⁢
          
            ×
            
              {
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  02
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          t
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁢
                  
                    {
                    
                      
                        
                          π
                          ⁡
                          
                            (
                            
                              
                                z
                                
                                  01
                                  ⁢
                                  
                                    ◇
                                    ⁢
                                    j
                                  
                                
                              
                              -
                              
                                d
                                j
                              
                              -
                              w
                            
                            )
                          
                        
                        
                          2
                          ⁢
                          
                            (
                            
                              
                                d
                                
                                  j
                                  +
                                  1
                                
                              
                              -
                              
                                d
                                j
                              
                            
                            )
                          
                        
                      
                      ,
                      
                        e
                        
                          
                            -
                            
                              
                                (
                                
                                  π
                                  
                                    
                                      d
                                      
                                        j
                                        +
                                        1
                                      
                                    
                                    -
                                    
                                      d
                                      j
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          ⁢
                          
                            η
                            zj
                          
                          ⁢
                          t
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                ×
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                02
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              d
                              j
                            
                            +
                            w
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          zj
                        
                        ⁢
                        t
                      
                    
                  
                  }
                
              
              -
              
                
                  Θ
                  3
                  ∫
                
                ⁢
                
                  {
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              z
                              
                                01
                                ⁢
                                
                                  ◇
                                  ⁢
                                  j
                                
                              
                            
                            -
                            
                              d
                              j
                            
                            +
                            w
                          
                          )
                        
                      
                      
                        2
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                j
                                +
                                1
                              
                            
                            -
                            
                              d
                              j
                            
                          
                          )
                        
                      
                    
                    ,
                    
                      e
                      
                        
                          -
                          
                            
                              (
                              
                                π
                                
                                  
                                    d
                                    
                                      j
                                      +
                                      1
                                    
                                  
                                  -
                                  
                                    d
                                    j
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ⁢
                        
                          η
                          zj
                        
                        ⁢
                        t
                      
                    
                  
                  }
                
              
            
            }
          
          ⁢
          dudvdw
        
      
    
  
   













The nomenclatures used in TABLES 3 through 7 are listed in TABLE 8 below.







TABLE 8





Nomenclature















a Width of the layer, m.


b Bredth of the layer, m.






  
    
      c
      t
    
    ⁢
    
      
    
    ⁢
    Compressibility
  
  ,
  
    
      P
      a
      
        -
        1
      
    
    .
  






φ Porosity, fraction


dj+1 − dj Layer thickness, m.


kx, ky, kz Permeability in the x, y and z direction, m2


μ Viscosity, Pa · s.


 






  
    
      η
      xj
    
    =
    
      
        (
        
          
            k
            x
          
          
            
              φc
              t
            
            ⁢
            μ
          
        
        )
      
      j
    
  
  ,
  
    
      η
      yj
    
    =
    
      
        
          
            (
            
              
                k
                y
              
              
                
                  φc
                  t
                
                ⁢
                μ
              
            
            )
          
          j
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        
          η
          zj
        
      
      =
      
        
          
            (
            
              
                k
                z
              
              
                
                  φc
                  t
                
                ⁢
                μ
              
            
            )
          
          j
        
        ⁢
        
          
        
        ⁢
        Diffusion
        ⁢
        
          
        
        ⁢
        coefficients
      
    
  






 


pj Pressure in the jth layer, Pa.


qij Production rate of the ith well or fracture in the jth layer, m3/s.


t Time, s.


t0ij Stat time of production of the ith well or fracture in the jth layer, s.


θ0ij The inclination to the x − y plane of the ith well or fracture in the jth layer


γ0j The intercept to the z axis of the ith well or fracture in the jth layer


 






  
    U
    ⁡
    
      (
      
        t
        -
        
          t
          0
        
      
      )
    
  
  =
  
    {
    
      
        
          
            
              
                0
              
              
                
                  t
                  <
                  
                    t
                    0
                  
                
              
            
            
              
                1
              
              
                
                  t
                  >
                  
                    t
                    0
                  
                
              
            
          
          ⁢
          Heaviside
          ⁢
          
            '
          
          ⁢
          s
          ⁢
          
            
          
          ⁢
          Unit
          ⁢
          
            
          
          ⁢
          step
          ⁢
          
            
          
          ⁢
          function
          ⁢
          
            
          
          ⁢
          s
          ⁢
          
            
          
          ⁢
          Laplace
          ⁢
          
            
          
          ⁢
          variable
        
        ⁢
        
          

        
        ⁢
        
          ∋
          m
        
      
      =
      
        {
        
          
            
              
                1
                2
              
            
            
              
                m
                =
                0
              
            
          
          
            
              1
            
            
              
                
                  m
                  =
                  1
                
                ,
                2
                ,
                3
                ,
                ...
              
            
          
        
      
    
  






 






  
    
      
        
          
            Θ
            3
          
          ⁡
          
            (
            
              
                π
                ⁢
                
                  
                
                ⁢
                x
              
              ,
              
                e
                
                  
                    -
                    
                      π
                      2
                    
                  
                  ⁢
                  t
                
              
            
            )
          
        
        =
        
          
            {
            
              
                
                  
                    1
                    +
                    
                      2
                      ⁢
                      
                        
                          
                            
                          
                          ∑
                        
                        
                          n
                          =
                          1
                        
                        ∞
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              n
                              2
                            
                          
                          ⁢
                          
                            π
                            2
                          
                          ⁢
                          t
                        
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          πx
                        
                        )
                      
                    
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          
                            π
                            2
                          
                        
                        ⁢
                        t
                      
                    
                    >
                    
                      1
                      π
                    
                  
                
              
              
                
                  
                    
                      1
                      
                        πt
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          n
                          =
                          
                            -
                            ∞
                          
                        
                        ∞
                      
                      ⁢
                      
                        e
                        
                          -
                          
                            
                              
                                (
                                
                                  x
                                  +
                                  n
                                
                                )
                              
                              2
                            
                            t
                          
                        
                      
                    
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          
                            π
                            2
                          
                        
                        ⁢
                        t
                      
                    
                    ≤
                    
                      1
                      π
                    
                  
                
              
            
            }
          
          ⁢
          
            
          
          ⁢
          Eliptic
          ⁢
          
            
          
          ⁢
          theta
          ⁢
          
            
          
          ⁢
          function
          ⁢
          
            
          
          ⁢
          of
          ⁢
          
            
          
          ⁢
          the
          ⁢
          
            
          
          ⁢
          third
          ⁢
          
            
          
          ⁢
          kind
        
      
    
  
  
    
      
        
          
            Θ
            3
            ∫
          
          ⁡
          
            (
            
              
                π
                ⁢
                
                  
                
                ⁢
                x
              
              ,
              
                e
                
                  
                    -
                    
                      π
                      2
                    
                  
                  ⁢
                  t
                
              
            
            )
          
        
        =
        
          
            
              ∫
              0
              x
            
            ⁢
            
              
                
                  Θ
                  3
                
                ⁡
                
                  (
                  
                    πu
                    ,
                    
                      e
                      
                        
                          -
                          
                            π
                            2
                          
                        
                        ⁢
                        t
                      
                    
                  
                  )
                
              
              ⁢
              
                
              
              ⁢
              du
            
          
          =
          
            
              {
              
                
                  
                    
                      x
                      +
                      
                        
                          1
                          π
                        
                        ⁢
                        
                          
                            
                              
                            
                            ∑
                          
                          
                            n
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                            e
                            
                              
                                -
                                
                                  n
                                  2
                                
                              
                              ⁢
                              
                                π
                                2
                              
                              ⁢
                              t
                            
                          
                          n
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            πx
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        e
                        
                          
                            -
                            
                              π
                              2
                            
                          
                          ⁢
                          t
                        
                      
                      >
                      
                        1
                        π
                      
                    
                  
                
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            n
                            =
                            
                              -
                              ∞
                            
                          
                          ∞
                        
                        ⁢
                        
                          {
                          
                            
                              erf
                              ⁢
                              
                                
                              
                              ⁢
                              
                                (
                                
                                  
                                    x
                                    +
                                    n
                                  
                                  
                                    t
                                  
                                
                                )
                              
                            
                            -
                            
                              erf
                              ⁢
                              
                                
                              
                              ⁢
                              
                                (
                                
                                  n
                                  
                                    t
                                  
                                
                                )
                              
                            
                          
                          }
                        
                      
                    
                  
                  
                    
                      
                        e
                        
                          
                            -
                            
                              π
                              2
                            
                          
                          ⁢
                          t
                        
                      
                      ≤
                      
                        1
                        π
                      
                    
                  
                
              
              }
            
            ⁢
            
              
            
            ⁢
            Integral
            ⁢
            
              
            
            ⁢
            of
            ⁢
            
              
            
            ⁢
            Eliptic
          
        
      
    
  






 


theta function of the third kind


 






   
  
    
      
        
          
            
              Θ
              3
              
                ∫
                
                  
                
                ∫
              
            
            ⁡
            
              (
              
                πx
                ,
                
                  e
                  
                    
                      -
                      
                        π
                        2
                      
                    
                    ⁢
                    t
                  
                
              
              )
            
          
          =
          
            
              
                ∫
                0
                x
              
              ⁢
              
                
                  
                    Θ
                    3
                    ∫
                  
                  ⁡
                  
                    (
                    
                      πu
                      ,
                      
                        e
                        
                          
                            -
                            
                              π
                              2
                            
                          
                          ⁢
                          t
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                du
              
            
            =
          
        
      
    
    
      
        
          =
          
            {
            
              
                
                  
                    
                      
                        x
                        2
                      
                      2
                    
                    +
                    
                      
                        1
                        
                          2
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            n
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                            
                              e
                              
                                
                                  -
                                  
                                    n
                                    2
                                  
                                
                                ⁢
                                
                                  π
                                  2
                                
                                ⁢
                                t
                              
                            
                            
                              n
                              2
                            
                          
                          ⁢
                          
                            {
                            
                              1
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    2
                                    ⁢
                                    n
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    πx
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                      
                    
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          
                            π
                            2
                          
                        
                        ⁢
                        t
                      
                    
                    >
                    
                      1
                      π
                    
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          n
                          =
                          
                            -
                            ∞
                          
                        
                        ∞
                      
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                x
                                +
                                n
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            erf
                            ⁢
                            
                              
                            
                            ⁢
                            
                              (
                              
                                
                                  x
                                  +
                                  n
                                
                                
                                  t
                                
                              
                              )
                            
                          
                          +
                          
                            
                              
                                
                                  t
                                  π
                                
                                ⁢
                                
                                  
                                
                              
                            
                            ⁢
                            
                              (
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          (
                                          
                                            x
                                            +
                                            n
                                          
                                          )
                                        
                                        2
                                      
                                      t
                                    
                                  
                                
                                -
                                
                                  e
                                  
                                    -
                                    
                                      
                                        n
                                        2
                                      
                                      t
                                    
                                  
                                
                              
                              )
                            
                          
                          -
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            erf
                            ⁢
                            
                              
                            
                            ⁢
                            
                              (
                              
                                n
                                
                                  t
                                
                              
                              )
                            
                          
                        
                        }
                      
                    
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          
                            π
                            2
                          
                        
                        ⁢
                        t
                      
                    
                    ≤
                    
                      1
                      π
                    
                  
                
              
            
            }
          
        
      
    
  






 


Second integral of Elliptic theta function of the third kind.









FIG. 9 is a flow chart of a method to perform an oilfield operation using the real-time analytical simulator. The oilfield operation is performed in an oilfield, such as the oilfield (300) depicted in FIG. 3 above. This method involves using the gridless analytical simulator, such as described with respect to FIG. 8, to generate real-time simulation results for performing the oilfield operation.
In Step 901, multiple real-time parameters are obtained from sensors disposed about the oilfield (e.g., oilfield (300)). The oilfield may include multiple wellsites, such as depicted in FIG. 3 above. The multiple real-time parameters include, at least, real-time flow rate data, real-time pressure data, or real-time temperature data of the wellbore (e.g., wellbore (436) of FIG. 5). These real-time data may be monitored by a user (e.g., the surveillance engineer). In some examples, there may be missing periods of the real-time measurement, which may be supplemented with data re-construction, for example based on tubing head or bottom hole pressure measurement (Step 902). The real-time pressure data and/or the real-time flow rate data (if available) are filtered, for example by using a wavelet decomposition technique to remove outliers, noise, and identify transients (Step 902). The large amount of real-time raw data may be sampled to reduce to the filtered data to a manageable amount, while retaining all the relevant characteristics of the original larger data set.
A set of alarm conditions is calculated based on the real-time data after filtering (Step 903). The alarms may include, for example, a drawdown alarm, a downtime alarm, etc. If the alarm is triggered, detailed diagnostics are performed thereafter. For example, drawdown pressure may be selected as the alarm parameter where the running maximum and running minimum values for pressure are calculated for each hour. These running averages are reset at the end of each hour. The running maximum, minimum, and average of the pressure data are also calculated for the day. The running averages are reset at 24:00:00 each day. Static reservoir pressure (Pr) in the vicinity of the well bore is estimated and entered at predefined intervals, typically every 48 to 72 hours.
Occasionally, previously estimated Pr values are re-estimated, in which case other previously estimated values must be updated. Drawdown pressures are calculated by subtracting the gauge pressure (Pwg) from the static reservoir pressure (Pr). Limiting values for gauge pressure are calculated or estimated and entered at predefined intervals, typically 48 to 72 hours.
The sources are bubblepoint limits, sand management limits and drawdown limits. Bubblepoint limits are absolute limits for the bottomhole pressure; sand management limits are functions of the static reservoir pressure; drawdown limits are a fixed offset from the static reservoir pressure. Occasionally, these limits are recomputed, and the previous values must be updated.
Drawdown surveillance is performed each hour by comparing the hourly average, running maxima, running minima, and running averages to the appropriate limiting values for gauge pressure. Automatic alerts (e.g., indicated in color yellow) are generated whenever the gauge pressure is within a defined variance from the limit value.
A surveillance engineer analyzes automatic alerts and sets a validation condition for each alert (e.g., Green: “No action;” Yellow: “Monitor closely:” Red: “Action recommended”) with an optional comment. Green measurements indicate that a component or system is performing within specified bounds and requires no action. Essentially, green-light data can be ignored. Yellow is a low level alarm (or alert), meaning the sensor measurement is approaching upper or lower bounds. Red is an alarm (or critical level alert), which indicates that the component has been shut down because sensor measurements fall outside of specified ranges. The yellow alert is one key to asset management, helping operators avoid deferred production. Operators take proactive measures on yellow alerts, and are reactive to red alarms. Alternatively, other colors may also be used in lieu of the Green/Yellow/Red system.
While the drawdown pressure can be directly calculated from the measured real-time data in the above example, wellbore skin may be selected as the alarm parameter in another example where the running maximum and running minimum values for wellbore skin are calculated on a regular basis using the gridless simulator. Within the gridless analytical simulator, many parameters may be used to configure an appropriate model for simulating the oilfield (e.g., the oilfield (300)) (Step 904). For example, static parameters obtained through geological surveys (e.g. as depicted in FIG. 1 and FIG. 3 above) may be used to set up the initial and boundary conditions described in TABLE 1 above.
Based on the configurations of the wellsites (e.g., vertical well, horizontal well, deviated well, fractured well, etc.), the gridless analytical simulator is configured using equations shown in TABLES 3 through 7 above. For example, the coefficients in equation (0.13) are appropriately determined for each well configuration. Preferably, the model is further identified by using a neural network method based on, for example rate of change of the real-time pressure data. Additionally, a history matching method of key parameters, such as the historic value of the reservoir pressure, well skin, effective permeability, and well productivity may be used to update the model further.
Once the model is identified and the simulator is configured, real-time simulation results are then generated, for example based on equations described in TABLES 3-7 above (Step 905). The real-time simulation results include a prediction of the production rates and reservoir pressure over time. The real-time simulation results can be delivered in an automatic workflow with real-time plotting of the key parameters (e.g., the reservoir pressure, well skin, effective permeability, well productivity, etc.) and alarm setting based on pre-determined criteria. The model is automatically updated when the predicted performance diverges from the actual performance by more than a pre-determined limit (Step 906).
In Step 907, the oilfield operation is performed based on the real-time simulation results. The gridless analytical simulator may provide information indicating problems at the wellsites that require action. The simulators may also indicate that adjustments in the oilfield operation may be made to improve efficiency, or correct problems. Well management strategy may be adjusted to define different development scenarios to be included in the integrated simulation run.
The steps of portions or all of the process may be repeated as desired. Repeated steps may be selectively performed until satisfactory results achieved. For example, steps may be repeated after adjustments are made. This may be done to update the simulator and/or to determine the impact of changes made.
The data input, coupling, layout, and constraints defined in the simulation provide flexibility to the simulation process. These factors of the various simulators are selected to meet the requirements of the oilfield operation. Any combination of simulators may be selectively linked to create the overall oilfield simulation. The process of linking the simulators may be re-arranged and simulations repeated using different configurations. Depending on the type of coupling and/or the arrangement of simulators, the oilfield simulation may be selected to provide the desired results. Various combinations may be tried and compared to determine the best outcome. Adjustments to the oilfield simulation may be made based on the oilfield, the simulators, the arrangement, and other factors. The process may be repeated as desired.
It will be understood from the foregoing description that various modifications and changes may be made in the preferred and alternative embodiments of the present invention without departing from its true spirit. For example, the boundary conditions of the multi-layer model of FIG. 8 and TABLE 1 may be varied, the specific formulation of the analytic solutions of TABLES 2-7 and other equations/formulas described throughout this paper may be adjusted or otherwise modified, the simulators, couplings, and arrangement of the system may be selected to achieve the desired simulation. The simulations may be repeated according to the various configurations, and the results compared and/or analyzed.
This description is intended for purposes of illustration only and should not be construed in a limiting sense. The scope of this invention should be determined only by the language of the claims that follow. The term “comprising” within the claims is intended to mean “including at least” such that the recited listing of elements in a claim are an open group. “A,” “an” and other singular terms are intended to include the plural forms thereof unless specifically excluded.

## Claims

1. A method of performing an oilfield operation of an oilfield having at least one wellsite, each wellsite having a wellbore penetrating a subterranean formation for extracting fluid from an underground reservoir therein, the method comprising:
obtaining a plurality of real-time parameters from a plurality of sensors disposed about the oilfield, the oilfield receiving command signals from a surface unit to perform the oilfield operation, wherein the plurality of real-time parameters comprise real-time flow rate data and real-time pressure data of the wellbore;
identifying a simulation session marked by identified transients in the plurality of real-time parameters, wherein the identified transients are identified using wavelet decomposition;
identifying, within the simulation session, a first time period during which the real-time flow rate data is not available and a second time period during which the real-time pressure data is not available;
obtaining offline flow rate data for the first time period by performing flow rate re-construction based on at least one selected from a group consisting of tubing head pressure measurement and bottom hole pressure measurement;
obtaining offline pressure data for the second time period based on historical data and spot measurement;
configuring, in the surface unit, a gridless analytical simulator for the simulation session to simulate the underground reservoir based on the plurality of real-time parameters supplemented by the offline flow rate data during the first time period and the offline pressure data during the second time period;
generating real-time simulation results of the underground reservoir and the at least one wellsite in real-time using the gridless analytical simulator; and
sending, from the surface unit, the command signals to the oilfield to manage the underground reservoir in real time by performing the oilfield operation, wherein the command signals are based on the real-time simulation results.

2. The method of claim 1,
wherein at least a portion of the oilfield is modeled as a vertically stacked system of a plurality of layers using a plurality of analytic solutions corresponding to the plurality of layers, and
wherein the gridless analytical simulator is based on coupling the plurality of analytic solutions to account for crossflow among the plurality of layers.

3. The method of claim 2,
wherein a flux field at an interface of the plurality of layers is obtained by solving a Fredholm integral equation, and
wherein a time evolution of the flux field is governed by a Volterra integral equation.

4. The method of claim 1,
wherein the oilfield comprises a plurality of wellsites, and
wherein the gridless analytical model is configured to simulate an interference effect from the plurality of wellsites.

5. The method of claim 1, wherein the real-time simulation results are generated using at least one selected from a group consisting of a no-flow boundary condition, and a constant pressure boundary condition.

6. The method of claim 1, wherein configuring the gridless analytical simulator comprises identifying a reservoir model based on at least one selected from a group consisting of a neural network method, a rate of change of the real-time pressure data, and a geological parameter.

7. The method of claim 1,
wherein the at least one wellsite comprises at least one selected from a group consisting of a horizontal well, a vertical well, and a deviated well, and
wherein the underground reservoir comprises a plurality of heterogeneous layers.

8. The method of claim 1, wherein the underground reservoir is a naturally fractured reservoir.

9. The method of claim 1, wherein hydraulic fracturing is performed for the at least one wellsite.

10. The method of claim 9, wherein the wellbore comprises at least one selected from a group consisting of a finite conductivity hydraulic fracture and an infinite conductivity hydraulic fracture.

11. The method of claim 1, wherein the wellbore is modeled as a line source in the gridless analytical simulator.

12. The method of claim 11, further comprising:
simulating at least one selected from a group consisting of a wellbore storage effect and a finite wellbore radius by applying corrections to the gridless analytical simulator.

13. The method of claim 1, wherein the real-time simulation results comprises at least one selected from a group consisting of reservoir pressure, flow rate, well skin, effective permeability, fracturing performance, well drainage area, compartmentalization, and well productivity.

14. The method of claim 1, wherein performing the oilfield operation comprises at least one selected from a group consisting of anticipating an event, identifying an event, performing real-time diagnostics, performing real-time interpretation, performing real-time decision making, performing real-time corrective action, and forecasting performance of the wellsite and the reservoir in real-time.

15. The method of claim 1, further comprising:
generating an alert based on comparing at least one of the plurality of real-time parameters to a pre-determined limit; and
classifying the alert according to a plurality of pre-determined alert levels,
wherein an alert level of the plurality of pre-determined alert levels dictates at least one selected from a group consisting of a proactive action and a reactive action.

16. A non-transitory computer readable medium, embodying instructions executable by a computer to perform method steps for an oilfield operation, the oilfield having at least one wellsite, each of the at least one wellsite having a wellbore penetrating a subterranean formation for extracting fluid from an underground reservoir therein, the instructions comprising functionality to:
obtain a plurality of real-time parameters from a plurality of sensors disposed about the oilfield, the oilfield receiving command signals from a surface unit to perform the oilfield operation, wherein the plurality of real-time parameters comprise at least one selected from a group consisting of real-time flow rate data and real-time pressure data of the wellbore;
identify a simulation session marked by identified transients in the plurality of real-time parameters, wherein the identified transients are identified using wavelet decomposition;
identify a first time period during which the real-time flow rate data is not available and a second time period during which the real-time pressure data is not available;
obtain offline flow rate data for the first time period by performing flow rate re-construction based on at least one selected from a group consisting of tubing head pressure measurement and bottom hole pressure measurement;
obtain offline pressure data for the second time period based on historical data and spot measurement;
configure, in the surface unit, a gridless analytical simulator for the simulation session to simulate the underground reservoir based on the plurality of real-time parameters supplemented by the offline flow rate data and the offline pressure data during the first time period and the second time period, respectively; and
generate real-time simulation results of the reservoir and the at least one wellsite in real-time using the gridless analytical simulator,
wherein the oilfield operation is performed to manage the underground reservoir in real time by receiving the command signals that are generated based on the real-time simulation results.

17. The non-transitory computer readable medium of claim 16,
wherein at least a portion of the oilfield is modeled as a vertically stacked system of a plurality of layers using a plurality of analytic solutions corresponding to the plurality of layers, and
wherein the gridless analytical simulator is based on coupling the plurality of analytic solutions to account for crossflow among the plurality of layers.

18. The non-transitory computer readable medium of claim 16,
wherein a flux field at an interface of the plurality of layers is obtained by solving a Fredholm integral equation, and
wherein a time evolution of the flux field is governed by a Volterra integral equation.

19. The non-transitory computer readable medium of claim 16,
wherein the oilfield comprises a plurality of wellsites, and
wherein the gridless analytical model is configured to simulate an interference effect from the plurality of wellsites.

20. The non-transitory computer readable medium of claim 16, wherein the real-time simulation results are generated using at least one selected from a group consisting of a no-flow boundary condition, and a constant pressure boundary condition.

21. The non-transitory computer readable medium of claim 16, wherein configuring the gridless analytical simulator comprises identifying a reservoir model based on at least one selected from a group consisting of a neural network method, a rate of change of the real-time pressure data, and a geological parameter.

22. The non-transitory computer readable medium of claim 16,
wherein the at least one wellsite comprises at least one selected from a group consisting of a horizontal well, a vertical well, and a deviated well, and
wherein the underground reservoir comprises a plurality of heterogeneous layers.

23. The non-transitory computer readable medium of claim 16, wherein the underground reservoir is a naturally fractured reservoir.

24. The non-transitory computer readable medium of claim 16, wherein hydraulic fracturing is performed for the at least one wellsite.

25. The non-transitory computer readable medium of claim 24, wherein the wellbore comprises at least one selected from a group consisting of a finite conductivity hydraulic fracture and an infinite conductivity hydraulic fracture.

26. The non-transitory computer readable medium of claim 16, wherein the wellbore is modeled as a line source in the gridless analytical simulator.

27. The non-transitory computer readable medium of claim 26, the instructions further comprising functionality to:
simulating at least one selected from a group consisting of a wellbore storage effect and a finite wellbore radius by applying corrections to the gridless analytical simulator.

28. The non-transitory computer readable medium of claim 16, wherein the real-time simulation results comprises at least one selected from a group consisting of reservoir pressure, flow rate, well skin, effective permeability, fracturing performance, well drainage area, compartmentalization, and well productivity.

29. The non-transitory computer readable medium of claim 16, wherein performing the oilfield operation comprises at least one selected from a group consisting of anticipating an event, identifying an event, performing real-time diagnostics, performing real-time interpretation, performing real-time decision making, performing real-time corrective action, and forecasting performance of the wellsite and the reservoir in real-time.

30. The non-transitory computer readable medium of claim 16, the instructions further comprising functionality to:
generating an alert based on comparing at least one of the plurality of real-time parameters to a pre-determined limit; and
classifying the alert according to a plurality of pre-determined alert levels,
wherein an alert level of the plurality of pre-determined alert levels dictates at least one selected from a group consisting of a proactive action and a reactive action.