solution of the poisson equation by differential quadrature the method of differential quadrature is demonstrated by solving the two dimensional poisson equation the results for three test problems are compared with the exact analytical solutions and the numerical solutions obtained by others for the galerkin the control volume and the five point finite difference methods the method of differential quadrature leads to more accurate results for comparable levels of computational effort

free vibration analysis of circular plates with variable thickness by the generalized differential quadrature rule the generalized differential quadrature rule proposed recently by these authors is applied here to the free vibration analyses of solid circular plates with radially varying thickness and elastic restraints the thickness can vary radially in any given continuous form and it varies exponentially and linearly in this work two regularity conditions corresponding to the plate center are expressed in explicit formulae since they have been inexactly or even wrongly expressed in the literature such errors are pointed out in the application of the differential quadrature techniques and in the expression of the regularity conditions at the plate center numerical results are presented for a number of plates illustrating the versatility and accuracy of the approach

vibration analysis of non circular curved panels by the differential quadrature method free vibration characteristics of cantilever non circular curved panels are analyzed by using the differential quadrature method dqm in this paper the equations of motion of a curved panel are based on the love s hypothesis and are expressed in an orthogonal curvilinear co ordinate system by applying the differential quadrature formulation and the proposed modified relationships for specified boundary conditions the free vibration equations of motion of the curved panel are transformed to a set of algebraic equations natural frequencies of a cantilever flat plate and a circular curved panel are obtained for verifying the applicability of the present approach good convergent trend and accuracy are observed effects of shallowness thickness and aspect ratios on the natural frequencies of a cantilever curved panel are also investigated furthermore natural frequencies of parabolic curved panels are obtained in all cases studied the efficiency and convenience of the dqm are illustrated

an eight node curvilinear differential quadrature formulation for reissner mindlin plates a first endeavour to exploit the differential quadrature dq method as a simple accurate and efficient numerical technique for the bending of quadrilateral reissner mindlin plates with curvilinear boundaries is made the curvilinear quadrilateral is mapped onto a square domain by using the geometric coordinate transformation this geometric mapping technique is employed to transform the governing differential equations and boundary conditions of the problem from the physical domain onto the computational domain the dq procedures are then applied to discretise the transformed set of differential equations and boundary conditions into a set of linear algebraic equations the numerical solutions to the problem are obtained by solving this set of linear algebraic equations examples illustrating the accuracy and convergence of the dq method for reissner mindlin plates with curvilinear boundaries are presented the great development potential of the dq method as an alternative to other approximate techniques for solving problems related to engineering sciences and structural mechanics is exploited in the present study

a note on approximating peak congestion in m t g infinity queues with sinusoidal arrivals we study the mt g infinity queue where customers arrive according to a sinusoidal function t a sin 2 t t and the service rate is we show that the expected number of customers in the system during peak congestion can be closely approximated by a for service distributions with coefficient of variation between 0 and 1 motivated by a result derived by eick massey and whitt that the time lag of the peak congestion from the peak of the customer arrivals is 1 2 for models with deterministic service times we show that the time lag for exponential service times is closely approximated by 1 based on a cycle length of 24 hours and regardless of the values of other system parameters these approximations are of the order of 1 accuracy for 1 and the accuracy increases rapidly with increasing

a dqem for transverse vibration analysis of multiple cracked non uniform timoshenko beams with general boundary conditions in this paper a differential quadrature element method dqem for free transverse vibration analysis of multiple cracked non uniform timoshenko beams with general boundary conditions is proposed governing equations the compatibility conditions at the damaged cross sections and implementation of the external boundary conditions are derived and formulated by the differential quadrature analogue the accuracy convergence and versatility of the proposed method are confirmed by the exact solution of the uniform beam which has been presented by other authors and 2d finite element method fem numerical results for non uniform beam after the validation of the presented method the effect of quantity depth and location of the cracks on the frequency values of vibrations are investigated the achieved results show that the existence of the crack leads to a decrease in the frequencies of the vibrations through decrease in the stiffness of the beam meanwhile the compatibility conditions at the damaged section is considered as a discontinuity in slope and vertical displacement where the effect of the discontinuity in the slope is more considerable as many authors have neglected the discontinuity in the vertical displacement as it will be shown consideration of the discontinuity in the vertical displacement causes more decrease in the frequencies

stability and accuracy of differential quadrature method in solving dynamic problems in this paper the differential quadrature method is used to solve dynamic problems governed by second order ordinary differential equations in time the legendre radau chebyshev chebyshev gauss lobatto and uniformly spaced sampling grid points are considered besides two approaches using the conventional and modified differential quadrature rules to impose the initial conditions are also investigated the stability and accuracy properties are studied by evaluating the spectral radii and truncation errors of the resultant numerical amplification matrices it is found that higher order accurate solutions can be obtained at the end of a time step if the gauss and radau sampling grid points are used however the conventional approach to impose the initial conditions in general only gives conditionally stable time step integration algorithms unconditionally stable algorithms can be obtained if the modified differential quadrature rule is used unfortunately the commonly used chebyshev gauss lobatto sampling grid points would not generate unconditionally stable algorithms

solving high order partial differential equations with indirect radial basis function networks this paper reports a new numerical method based on radial basis function networks rbfns for solving high order partial differential equations pdes the variables and their derivatives in the governing equations are represented by integrated rbfns the use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high order derivatives the proposed rbfn method is verified successfully through the solution of thin plate bending and viscous flow problems which are governed by biharmonic equations for thermally driven cavity flows the solutions are obtained up to a high rayleigh number

unstable asymptotics for nonstationary queues we relate laws of large numbers and central limit theorems for nonstationary counting processes to corresponding limits for their inverse processes we apply these results to develop approximations for queues that are unstable in a nonstationary manner we obtain unstable nonstationary analogs of the queueing relation l w and associated central limit theorem versions for modeling and to obtain the first limits we can construct nonstationary point processes as random time transformations of familiar point processes such as renewal processes and stationary point processes we deduce the asymptotic behavior of the nonstationary point process from the asymptotic behavior of the familiar point process and the time transformation

a fluid limit for an overloaded x model via a stochastic averaging principle we prove a many server heavy traffic fluid limit for an overloaded markovian queueing system having two customer classes and two service pools known in the call center literature as the x model the system uses the fixed queue ratio with thresholds fqr t control which we proposed in a recent paper as a way for one service system to help another in face of an unexpected overload under fqr t customers are served by their own service pool until a threshold is exceeded then one way sharing is activated with customers from one class allowed to be served in both pools after the control is activated it aims to keep the two queues at a prespecified fixed ratio for large systems that fixed ratio is achieved approximately for the fluid limit or fwlln functional weak law of large numbers we consider a sequence of properly scaled x models in overload operating under fqr t our proof of the fwlln follows the compactness approach i e we show that the sequence of scaled processes is tight and then show that all converging subsequences have the specified limit the characterization step is complicated because the queue difference processes which determine the customer server assignments need to be considered without spatial scaling asymptotically these queue difference processes operate on a faster time scale than the fluid scaled processes in the limit because of a separation of time scales the driving processes converge to a time dependent steady state or local average of a time varying fast time scale process ftsp this averaging principle allows us to replace the driving processes with the long run average behavior of the ftsp

a simple adaptive procedure leading to correlated equilibrium we propose a new and simple adaptive procedure for playing a game regret matching in this procedure players may depart from their current play with probabilities that are proportional to measures of regret for not having used other strategies in the past it is shown that our adaptive procedure guarantees that with probability one the empirical distributions of play converge to the set of correlated equilibria of the game

three dimensional elasticity solutions to some orthotropic plate problems abstract this paper presents a three dimensional analysis of rectangular orthotropic plates by employing the differential quadrature dq method the derivation of the governing equations from the governing equations and the stress strain relationship of the three dimensional elasticity model is detailed the constrained conditions of the orthotropic plate edges are given the governing equations and boundary conditions are first normalized and discretized according to the dq procedure example problems pertaining to the bending buckling and free vibration of orthotropic plates with generic boundary conditions are selected to illustrate the simplicity and applicability of the dq method the convergence characteristics of the dq method are first obtained based on numerical studies after which the dq solutions are compared where possible with exact solutions it is found that the dq method yields accurate results for the plate problems under investigation

approximation of the non stationary m t m t c t queue using stationary queueing models the stationary backlog carryover approach this paper proposes a new approach for the time dependent analysis of stochastic and non stationary queueing systems the analysis of a series of stationary queueing models leads to a new approximation of time dependent performance measures based on a stationary backlog carryover sbc approximation of the time dependent expected utilization different approximations of the time dependent expected queue length and the number of customers in the system are discussed limiting results are given for the case of constant rates the accuracy of the sbc approach is shown for non stationary m t m t c t queueing systems with time dependent and piecewise constant arrival rates in numerical experiments we demonstrate the reliability of this approach and compare it with the lagged stationary independent period by period sipp approach in addition the approximation is applied to temporarily overloaded systems that cannot be analyzed by the variants of the sipp approach

on the accuracy of the simple peak hour approximation for markovian queues we empirically explore the accuracy of the simple stationary peak hour approximation spha for estimating peak hour performance in multiserver queuing systems with exponential service times and periodic sinusoidal poisson arrival processes we show that the spha is very good for a range of parameter values corresponding to a reasonably broad spectrum of real systems however we do find and document that there are many situations in which this approximation will be very inaccurate we postulate and then support empirically a set of hypotheses that link the accuracy of the spha and the related point wise stationary approximation psa to key parameter values and model characteristics we also present results on the time dependent behavior of these systems as a function of key parameters finally we present results which indicate that our findings developed for models with sinusoidal input streams may apply to a much broader range of markovian models with more general cyclic inputs

on the equivalence of generalized differential quadrature and highest order finite difference scheme abstract it is shown in this paper that the global method of generalized differential quadrature gdq is equivalent to the highest order finite difference scheme which is the extension of low order finite difference schemes the effect of grid point distribution on the accuracy of numerical results is studied by two numerical examples the results showed that the ability to alter the grid point distribution in the gdq method is an important feature to obtain accurate numerical solutions in certain problems

the physics of the m t g symbol queue we establish some general structural results and derive some simple formulas describing the time dependent performance of the mt g queue with a nonhomogeneous poisson arrival process we know that for appropriate initial conditions the number of busy servers at time t has a poisson distribution for each t our results show how the time dependent mean function m depends on the time dependent arrival rate function and the service time distribution for example when is quadratic the mean m t coincides with the pointwise stationary approximation t e s where s is a service time except for a time lag and a space shift it is significant that the well known insensitivity property of the stationary m g model does not hold for the nonstationary mt g model the time dependent mean function m depends on the service time distribution beyond its mean the service time stationary excess distribution plays an important role when is decreasing before time t m t is increasing in the service time

improving service by informing customers about anticipated delays this paper investigates the effect upon performance in a service system such as a telephone call center of giving waiting customers state information in particular the paper studies two m m s r queueing models with balking and reneging for simplicity it is assumed that each customer is willing to wait a fixed time before beginning service however customers differ so the delay tolerances for successive customers are random in particular it is assumed that the delay tolerance of each customer is zero with probability and is exponentially distributed with mean 1 conditional on the delay tolerance being positive let n be the number of customers found by an arrival in model 1 no state information is provided so that if n s the customer balks with probability if the customer enters the system he reneges after an exponentially distributed time with mean 1 if he has not begun service by that time in model 2 if n s k s then the customer is told the system state k and the remaining service times of all customers in the system so that he balks with probability 1 1 qk where qk p t sk t is exponentially distributed with mean 1 sk is the sum of k 1 independent exponential random variables each with mean s 1 and 1 is the mean service time in model 2 all reneging is replaced by balking the number of customers in the system for model 1 is shown to be larger than that for model 2 in the likelihood ratio stochastic ordering thus customers are more likely to be blocked in model 1 and are more likely to be served without waiting in model 2 algorithms are also developed for computing important performance measures in these and more general birth and death models

on the performance of three iterative methods for solution of gdq algebraic equations abstract in this paper the three conventional iterative methods namely j oe or jacobi gauss seidel g s and sor methods are extended to solve the gdq algebraic equations the performances of these three iterative methods for the gdq and second order finite difference fd algebraic equations are comparatively studied by their application to solve the two dimensional poisson equation it was found that sor method gives the fastest convergence rate for both gdq and fd algebraic equations however the effect of relaxation factor for gdq algebraic equations is reduced the high efficiency and accuracy of gdq over second order fd scheme are also shown in the paper furthermore the performance of j oe and sor methods was validated by their application to a complicated problem that is numerical simulation of natural convection in the annulus between two concentric cylinders it was found that for this complicated case jacobi and gauss seidel methods cannot get converged solution and the efficiency of sor method is reduced to be around three times faster than the j oe method

static analysis of frame structures by the differential quadrature element method in this paper a new version of differential quadrature method dqm is proposed and then extended to analyse frame structures the new method called the differential quadrature element method dqem retains all advantages of the earlier version of the differential quadrature method and overcomes some critical shortcomings existing in the original dqm the proposed method is however different from the quadrature element method qem proposed earlier by striz et al the methodology is explained in details herein via a differential quadrature beam element and some numerical examples are given to show the efficiency of the present method 1997 by john wiley sons ltd

regime switching stochastic approximation algorithms with application to adaptive discrete stochastic optimization this work is devoted to a class of stochastic approximation problems with regime switching modulated by a discrete time markov chain our motivation stems from using stochastic recursive algorithms for tracking markovian parameters such as those in spreading code optimization in cdma code division multiple access wireless communication the algorithm uses constant step size to update the increments of a sequence of occupation measures it is proved that least squares estimates of the tracking errors can be developed assume that the adaptation rate is of the same order of magnitude as that of the time varying parameter which is more difficult to deal with than that of slower parameter variations due to the time varying characteristics and markovian jumps the usual stochastic approximation sa techniques cannot be carried over in the analysis by a combined use of the sa method and two time scale markov chains asymptotic properties of the algorithm are obtained which are distinct from the usual sa

static analysis of mindlin plates the differential quadrature element method dqem abstract the differential quadrature method dqm is a new numerical method for fast solving linear and nonlinear differential equations based on global basis functions in the present study a discrete approximate method based on the dqm the differential quadrature element method dqem is developed for the axisymmetric static analysis of moderately thick circular and annular plates described by the mindlin shear deformable theory the basic idea of the dqem is 1 to divide the whole variable domain into several sub domains elements 2 to form discretized element governing equations by applying the dqm to each element and 3 to assemble all the discretized element governing equations into the overall characteristic equations with the consideration of displacement and stress compatibility conditions between adjacent elements the annular and circular mindlin plate elements of differential quadrature dq are established the convergence characteristics of the proposed method are carefully investigated from the view points of element refinement and element grid refinement and some general regulations and suggestions on element division and element grid selection are provided a number of numerical examples are calculated and the results are compared with the corresponding exact solutions which exhibits high accuracy simplicity and applicability of the present method

large deflection analysis of plates under thermal loading abstract a global method of generalized differential quadrature is presented to solve the problems of large deflections of thin plates under thermal loading by applying the generalized differential quadrature the drawbacks of existing analytical and numerical solution techniques for the problem of non linear bending of heated plates such as complexity lack of generality and computational inefficiency have been overcome the weighting coefficients for the approximation of derivatives required in differential quadrature formulation are calculated in a very simple way yet without any restriction on the choice of grid points numerical implementation of the method is straightforward and different boundary conditions can be easily incorporated examples are given to demonstrate the accuracy and efficiency of the proposed method to validate the method a case of large deflection of plates without thermal loading is also considered and the results compare well with existing analytical and numerical solutions

vibration characteristics and transient response of shear deformable functionally graded plates in thermal environments free and forced vibration analyses for initially stressed functionally graded plates in thermal environment are presented material properties are assumed to be temperature dependent and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents theoretical formulations are based on reddy s higher order shear deformation plate theory and include the thermal effects due to uniform temperature variation the plate is assumed to be clamped on two opposite edges with the remaining two others either free simply supported or clamped one dimensional differential quadrature technique galerkin approach and the modal superposition method are used to determine the transient response of the plate subjected to lateral dynamic loads comprehensive numerical results for silicon nitride stainless steel rectangular plates are presented in dimensionless tabular and graphical forms the roles played by the constituent volume fraction index temperature rise shape and duration of dynamic loads initial membrane stresses as well as the character of boundary conditions are studied the results reveal that when thermal effects are considered functionally graded plates with material properties intermediate to those of isotropic ones do not necessarily have intermediate natural frequencies and dynamic responses

application of differential quadrature to transport processes the methodology and numerical solution of problems concerning transport processes via the method of differential quadrature are presented application of the method is demonstrated by solving a simple one dimensional time dependent transient diffusion process involving an irreversible reaction without any flux across the end boundary in addition the same technique is used for the first time to the authors knowledge to solve a steady state problem for this purpose a convection diffusion problem involving an irreversible reaction is considered the demonstration is carried out in two ways 1 using the bellman et al technique which employs approximation formulas for higher order partial derivatives derived by iterating the linear quadrature approximation for the first order partial derivative and 2 using individual quadratures to approximate the partial derivatives of first as well as higher orders as suggested by mingle both approaches give the same results however the latter saves an appreciable amount of iterative computing effort despite the fact that it requires separate weighting coefficients for each individual quadrature since the technique of differential quadrature can produce solutions with sufficient accuracy even when using as few as three discrete points both the programming task and computational effort are alleviated considerably for these reasons the differential quadrature approach appears to be very practical in solving a variety of problems related to transport phenomena

on optimal selection of interior points for applying discretized boundary conditions in dq vibration analysis of beams and plates the direct implementation of derivative boundary conditions at edge points was demonstrated to be simple flexible and computationally efficient in the vibration analysis of beams and plates by using the differential quadrature dq method in the approach the discretized governing equations at certain interior points need to be replaced by the discretized boundary conditions naturally one may have questions as to whether the replacement points can be selected arbitrarily and how to choose the optimal replacement points this paper tries to answer these questions and its focus is to investigate systematically the effect of the replacement location on the accuracy of the numerical solution from the error distribution analysis of derivative approximation it is found that the optimal position for the replacement is the interior point just adjacent to the boundary this conclusion is confirmed by the numerical experiments of free vibration analysis for beams and plates with various boundary conditions

comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates this paper presents a comprehensive comparison study between the discrete singular convolution dsc and the well known global method of generalized differential quadrature gdq for vibration analysis so as to enhance the understanding of the dsc algorithm the dsc method is implemented through lagrange s delta sequence kernel dsc lk which utilizes local lagrange polynomials to calculate weighting coefficients whereas the gdq requires global ones moreover it is shown that the treatments of boundary conditions and the use of grid systems are different in the two methods comparison study is carried out on 21 rectangular plates of different combinations of simply supported s clamped c and transversely supported with nonuniform elastic rotational restraint e edges and five rectangular plates of mixed supporting edges some of which with a range of aspect ratios and rotational spring coefficients all the results of the dsc lk agree very well with both those in the literature and newly computed gdq results furthermore it is observed that the dsc lk performs much better for plates vibrating at higher order eigenfrequencies unlike the gdq the dsc lk is numerically stable for problems which require a large number of grid points

nonstationary analysis of the loss queue and of queueing networks of loss queues abstract we present an iterative scheme based on the fixed point approximation method for the numerical calculation of the time dependent mean number of customers and blocking probability functions in a nonstationary queueing network with multi rate loss queues we first show how the proposed method can be used to analyze a single class multi class and multi rate nonstationary loss queue subsequently the proposed method is extended to the analysis of a nonstationary queueing network of multi rate loss queues comparisons with exact and simulation results showed that the results are consistently close to the exact results and they are always within simulation confidence intervals

two new approximate methods for analyzing free vibration of structural components two approximate methods which have not previously been used for structural dynamics problems are applied to the free vibration analysis of various structural components the first method is a new version of the complementary energy method it is shown to be considerably more accurate than the conventional rayleigh and rayleigh schmidt methods when applied to spatially one dimensional free vibration problems prismatic and tapered bars prismatic beams and axisymmetric motion of circular membranes the second method is the differential quadrature method introduced by bellman and his associates it is applied successfully here to all of the problems mentioned plus square membranes and circular and square plates

a new differential quadrature methodology for beam analysis and the associated differential quadrature element method this paper presents a computationally efficient and an accurate new methodology in differential quadrature analysis of beam elements the methodology would overcome the difficulties in boundary conditions implementations of fourth order differential equations encountered in such problems the methodology benefited from defining the second order derivatives along the boundaries as independent degrees of freedom would enable the differential quadrature method to exactly satisfy some types of boundary conditions where by most other conventional algorithms have to be satisfied approximately the weighting coefficients employed are not exclusive and any accurate and efficient method such as the generalized differential quadrature method may be used to produce the methods weighting coefficients by solving some typical stability deflection and frequency analysis beam problems and by comparing the results with those of exact solutions and or those of other methodologies accuracy convergency and efficiency of the methodology is asserted in order to generalize its application to large scale beam structures and to the cases with the discontinuity in loading conditions and geometry a new one dimensional differential quadrature element method formulation is presented and implemented

non axisymmetric bending of thin annular plates due to circumferentially distributed moments abstract the non linear deformation of a thin annular plate subjected to circumferentially distributed bending moments is studied a von karman plate model is adopted to formulate the equations of motion free free boundary conditions have been applied at both inner and outer edges the matrix formulation of the differential quadrature method is used to discretise and solve the governing equations linear analysis predicts that the annular disk deforms axisymmetrically into a spherical dome however the proposed non linear analysis shows that a symmetry breaking bifurcation may occur after which the linear solution becomes unstable and the plate transitions into a non axisymmetric cylindrical deformation this is the case when at least one of two parameters reaches a critical value these parameters are the non dimensionalised ratio between applied moment and bending stiffness and the ratio between inner and outer radius furthermore it is noted that free free boundary conditions and circumferentially distributed bending moments do not break the circular symmetry of the annular disk hence the principal axes of curvature in the deformed configuration do not have a preferred orientation therefore the present work describes a shell possessing infinite identical equilibria having different yet no favoured direction and hence links to previous researches on neutrally stable structures

tracking a markov modulated stationary degree distribution of a dynamic random graph this paper considers a markov modulated duplication deletion random graph where at each time instant one node can either join or leave the network the probabilities of joining or leaving evolve according to the realization of a finite state markov chain two results are presented first motivated by social network applications the asymptotic behavior of the degree distribution is analyzed second a stochastic approximation algorithm is presented to track empirical degree distribution as it evolves over time the tracking performance of the algorithm is analyzed in terms of mean square error and a functional central limit theorem is presented for the asymptotic tracking error also a hilbert space valued stochastic approximation algorithm that tracks a markov modulated probability mass function with support on the set of nonnegative integers is analyzed

three dimensional asymptotic nonlocal elasticity theory for the free vibration analysis of embedded single walled carbon nanotubes abstract within the framework of three dimensional 3d nonlocal elasticity theory the authors develop an asymptotic theory to investigate the free vibration characteristics of simply supported single walled carbon nanotubes swcnts non embedded or embedded in an elastic medium using the multiple time scale method eringen s nonlocal constitutive relations are adopted to account for the small length scale effect in the formulation the interactions between the swcnt and its surrounding medium are modeled as a two parameter pasternak foundation model after performing a series of mathematical processes including nondimensionalization asymptotic expansion and successive integration etc the authors obtain recurrent sets of motion equations for various order problems the nonlocal classical shell theory cst is derived as a first order approximation of the current 3d nonlocal elasticity problem and the equations of motion for higher order problems retain the same differential operators as those of the nonlocal cst although with different nonhomogeneous terms the current asymptotic solutions for the natural frequency parameters of non embedded or embedded swcnts and their corresponding through thickness modal stress and displacement component distributions are obtained to assess the accuracy of various nonlocal shell and beam theories available in the literature

application of a new differential quadrature methodology for free vibration analysis of plates a new methodology is introduced in the differential quadrature dq analysis of plate problems the proposed approach is distinct from other dq methods by employing the multiple boundary conditions in a different manner for structural and plate problems the methodology employs the displacement within the domain as the only degree of freedom whereas along the boundaries the displacements as well as the second derivatives of the displacements with respect to the co ordinate variable normal to the boundary in the computational domain are considered as the degrees of freedom for the problem employing such a procedure would facilitate the boundary conditions to be implemented exactly and conveniently in order to demonstrate the capability of the new methodology all cases of free vibration analysis of rectangular isotropic plates in which the conventional dq methods have had some sort of difficulty to arrive at a converged or accurate solution are carried out excellent convergence behaviour and accuracy in comparison with exact results and or results obtained by other approximate methods were obtained the analogous dq formulation for a general rectangular plate is derived and for each individual boundary condition the general format for imposing the given conditions is devised it must be emphasized that the computational efforts of this new methodology are not more than for the conventional differential quadrature methods copyright 2002 john wiley sons ltd

an efficient differential quadrature methodology for free vibration analysis of arbitrary straight sided quadrilateral thin plates in this paper a new differential quadrature dq methodology is employed to study free vibration of irregular quadrilateral straight sided thin plates a four nodded super element is used to map the irregular physical domain into a square domain in the computational domain second order transformation schemes with relative ease and less computation are employed to transform the fourth order governing equation of thin plates between the two domains the only degree of freedom within the domain is the displacement whereas along the boundaries the displacement as well as the second order derivative of the displacement with respect to associated normal co ordinate variable in computational domain are the two degrees of freedom implementing the method the formulation for the dq method for the free vibration analysis of plates of straight sided shapes was presented together with the implementation procedure for the different boundary conditions to demonstrate the accuracy convergency and stability of the new methodology detail studies are made on isotropic plates at acute angles with different geometries boundary and loading conditions including dq free edge boundary condition implementations accurate results even with fewer degrees of freedom than for those of comparable numerical algorithms were achieved

vibration analysis of edge cracked beam on elastic foundation with axial loading using the differential quadrature method abstract the eigenvalue problems of clamped free and hinged hinged bernoulli euler beams on elastic foundation with a single edge crack axial loading and excitation force were numerically formulated using the differential quadrature method dqm appropriate boundary conditions accompanied the dqm to transform the partial differential equation of a bernoulli euler beam with a single edge crack into a discrete eigenvalue problem the dqm results for the natural frequencies of cracked beams agree well with other literature values the sampling point number effect the location of the crack effect and the depth of the crack effect on the accuracy variation of calculated natural frequencies are presented by using two elements in this work the effects of axial loading foundation stiffness opening crack and closing crack are also studied

differential quadrature domain decomposition method for a class of parabolic equations in this paper the differential quadrature method dqm and the domain decomposition method ddm are combined to form the differential quadrature domain decomposition method dqddm in which the boundary reduction technique brm is adopted the dqddm is applied to a class of parabolic equations which have discontinuity in the coefficients of the equation or weak discontinuity in the initial value condition two numerical examples belonging to this class are computed it is found that the application of this method to the above mentioned problems is seen to lead to accurate results with relatively small computational effort

asymptotic analysis of the time dependent m m 1 queue using operator analytic techniques we develop a nonstationary markovian queueing theory starting with the mt mt 1 queue we employ an asymptotic approach quite different from the usual large time analysis instead we uniformly accelerate the queue length process that is we divide the arrival and service rate by a common parameter e then for a fixed time interval we consider the asymptotics for the distribution mean and variance of the queue length process as e goes to zero the effects of e can be quite different for the given time interval this gives us a dynamic asymptotic behavior for the queue length process we can formulate a time dependent traffic intensity parameter that determines when the process is asymptotically stable and when it is asymptotically unstable

solving initial value problems by differential quadrature method part 1 first order equations in this paper the differential quadrature method is used to solve first order initial value problems the initial condition is given at the beginning of a time interval the time derivative at a sampling grid point within the time interval can be expressed as a weighted linear sum of the given initial condition and the function values at the sampling grid points within the time interval the order of accuracy and the stability property of the quadrature solutions depend on the locations of the sampling grid points it is shown that the order of accuracy of the quadrature solutions at the end of a time interval can be improved to 2n 1 or 2n if the n sampling grid points are chosen carefully in fact the approximate solutions are equivalent to the generalized pade approximations the resultant algorithms are therefore unconditionally stable with controllable numerical dissipation the corresponding sampling grid points are found to be given by the roots of the modified shifted legendre polynomials from the numerical examples the accuracy of the quadrature solutions obtained by using the proposed sampling grid points is found to be better than those obtained by the commonly used uniformly spaced or chebyshev gauss lobatto sampling grid points copyright 2001 john wiley sons ltd

on stability of randomly switched nonlinear systems this note is concerned with stability analysis and stabilization of randomly switched systems these systems may be regarded as piecewise deterministic stochastic systems the discrete switches are triggered by a stochastic process which is independent of the state of the system and between two consecutive switching instants the dynamics are deterministic our results provide sufficient conditions for almost sure stability and stability in the mean using lyapunov based methods when individual subsystems are stable and a certain ldquoslow switchingrdquo condition holds this slow switching condition takes the form of an asymptotic upper bound on the probability mass function of the number of switches that occur between the initial and current time instants this condition is shown to hold for switching signals coming from the states of finite dimensional continuous time markov chains our results therefore hold for markovian jump systems in particular for systems with control inputs we provide explicit control schemes for feedback stabilization using the universal formula for stabilization of nonlinear systems

buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method abstract in this article a combination of the finite element fe and differential quadrature dq methods is used to solve the eigenvalue buckling and free vibration equations of rectangular thick plates resting on elastic foundations the elastic foundation is described by the pasternak two parameter model the three dimensional linear and small strain theory of elasticity and energy principle are employed to derive the governing equations the in plane domain is discretized using two dimensional finite elements the spatial derivatives of equations in the thickness direction are discretized in strong form using dqm buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with pasternak elastic foundation are investigated using the proposed fe dq method the results obtained by the mixed method have been verified by the few analytical solutions in the literature it is concluded that the mixed fe dq method has good convergancy behavior and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom

upper bounds for single server queues with doubly stochastic poisson arrivals consider the following type of single server queues arrivals are according to a doubly stochastic poisson process with a stationary ergodic random intensity t service times are independent identically distributed also independent from arrivals it is proven that the mean stationary work load is not greater than e 0 where a denotes the mean stationary work load in the m gi 1 queue with arrival intensity a and the same service process similar results are given for the mean stationary queue size and the mean stationary delay

approximate analysis of non stationary loss queues and networks of loss queues with general service time distributions a fixed point approximation fpa method has recently been suggested for non stationary analysis of loss queues and networks of loss queues with exponential service times deriving exact equations relating time dependent mean numbers of busy servers to blocking probabilities we generalize the fpa method to loss systems with general service time distributions these equations are combined with associated formulae for stationary analysis of loss systems in steady state through a carried load to offered load transformation the accuracy and speed of the generalized methods are illustrated through a wide set of examples

an approximation method for a single server queue with cyclically indexed arrival and service rates in this paper we develop an approximation scheme for a single server markovian queue with cyclic arrival and or service patterns the arrival and service patterns change cyclically according to a job or customer sequence number the model has found application in production planning for multiple items on constrained resources using cyclic schedules it has also been applied to a computer simulation sensitivity analysis technique called frequency domain methodology the approximation scheme provides results to any desired degree of accuracy performance guidelines for the approximation are established analytically using stochastic ordering in addition performance results are numerically evaluated for the model with cyclic arrival patterns and independent and identically distributed exponential service patterns using a nonlinear optimization model

distributed nash equilibrium seeking this paper presents an asynchronous gossip based algorithm for finding a nash equilibrium ne of a game in a distributed multi player network the algorithm is designed in such a way that players make decisions based on estimates of the other players actions obtained from local neighbors using a set of standard assumptions on the cost functions and communication graph the paper proves almost sure convergence to a ne for diminishing step sizes for constant step sizes an error bound on expected distance from a ne is established the effectiveness of the proposed algorithm is demonstrated via simulation for both diminishing and constant step sizes

implementation of clamped and simply supported boundary conditions in the gdq free vibration analysis of beams and plates abstract in this paper a new methodology for implementing the clamped and simply supported boundary conditions is presented for the free vibration analysis of beams and plates using the generalized differential quadrature gdq method the proposed approach directly substitutes the boundary conditions into the governing equations and is referred to as sbcge approach the sbcge approach is presented to overcome the drawbacks of previous approaches in treating the boundary conditions a comparison of the sbcge approach with the method of modifying weighting coefficient matrices mwcm is made by their application to the vibration analysis of beams and plates with combinations of simply supported and clamped boundary conditions some details of the gdq method are also described in the paper

differential quadrature for multi dimensional problems the technique of differential quadrature for the solution of partial differential equations introduced by bellman et al is extended and generalized to encompass partial differential equations involving multiple space variables approximation formulae for a variety of first and second order partial derivatives and typical weighting coefficients are presented application of these formulae is demonstrated on the solution of the convection diffusion equation for the two and three dimensional space dependent cases and for both the transient and steady state dispersion of inert neutrally buoyant pollutants from continuous sources into an unbounded atmosphere

imposition of boundary conditions by modifying the weighting coefficient matrices in the differential quadrature method one of the important issues in the implementation of the differential quadrature method is the imposition of the given boundary conditions there may be multiple boundary conditions involving higher order derivatives at the boundary points the boundary conditions can be imposed by modifying the weighting coefficient matrices directly however the existing method is not robust and is known to have many limitations in this paper a systematic procedure is proposed to construct the modified weighting coefficient matrices to overcome these limitations the given boundary conditions are imposed exactly furthermore it is found that the numerical results depend only on those sampling grid points where the differential quadrature analogous equations of the governing differential equations are established the other sampling grid points with no associated boundary conditions are not essential copyright 2002 john wiley sons ltd

dynamic analysis of a spinning timoshenko beam by the differential quadrature method thedynamiccharacteristicsofaspinningtimoshenkobeamareinvestigatedbyusingthedifferentialquadrature method dqm the beam is subject to any combination of free simply supported clamped and elastically supported boundary conditions the weighting matrices in the differential quadrature formulation are modie ed to incorporate classical boundary conditions of the beam whereas boundary conditions of elastic supports are incorporated into the discretized equations numerical results of the spinning timoshenko beam obtained by the dqm are compared with the exact solutions orresults obtained by the e nite element method the results show the high accuracy and efe ciency of the differential quadrature method nomenclature a cross sectional area of beam c b damping matrix of elastic supports cxx cyy damping coefe cients e young s modulus g shear modulus g gyroscopic matrix i transverse moment of inertia jp polar mass moment of inertia k stiffness matrix k b stiffness matrix of elastic supports kxx kyy spring constants l

application of generalized differential quadrature method to structural problems summary this paper presents the first endeavour to exploit a generalized differential quadrature method as an accurate efficient and simple numerical technique for structural analysis firstly drawbacks existing in the method of differential quadrature dq are evaluated and discussed then an improved and simpler generalized differential quadrature method gdq is introduced to overcome the existing drawback and to simplify the procedure for determining the weighting coefficients subsequently the generalized differential quadrature is systematically employed to solve problems in structural analysis numerical examples have shown the superb accuracy efficiency convenience and the great potential of this method numerical approximation methods for solving partial differential equations have been widely used in various engineering fields classical techniques such as finite element and finite difference methods are well developed and well known these methods can provide very accurate results by using a large number of grid points however in a large number of cases reasonably approximate solutions are desired at only a few specific points in the physical domain in order to get results even at or around a point of interest with acceptable accuracy conventional finite element and finite difference methods still require the use of a large number of grid points consequently the requirement for computer capacity is often unnecessarily large in such cases in seeking an alternate numerical method using fewer grid points to find results with acceptable accuracy the method of differential quadrature dq was introduced by bellman et a the method of dq is a global approximate method this method is based on the ideas that the derivative of a function with respect to a co ordinate direction can be expressed as a weighted linear sum of all the function values at all mesh points along that direction and that a continuous function can be approximated by a higher order polynomial in the overall domain the dq method differs from the finite element method fem in two aspects firstly the fem uses lower order polynomials to approximate a function on a local element level while the dq method approximates a function on the global area using higher order polynomials secondly the dq method directly approximates the derivatives of a function at a point while the fem approximates a function over a local element and the derivatives can then be derived from the approximate function in this aspect the dq method is more similar to the finite difference method fdm however the fdm is also a local approximation method based on lower order polynomial approximation in fact it can be shown that the fdm is just a special case of the dq method where it is applied locally on the range xi xi l owing to the higher order

thermally induced dynamic instability of laminated composite conical shells abstract thermally induced dynamic instability of laminated composite conical shells is investigated by means of a perturbation method the laminated composite conical shells are subjected to static and periodic thermal loads the linear instability approach is adopted in the present study a set of initial membrane stresses due to the elevated temperature field is assumed to exist just before the instability occurs the formulation begins with three dimensional equations of motion in terms of incremental stresses perturbed from the state of neutral equilibrium after proper nondimensionalization asymptotic expansion and successive integration we obtain recursive sets of differential equations at various levels the method of multiple scales is used to eliminate the secular terms and make an asymptotic expansion feasible using the method of differential quadrature and bolotin s method and imposing the orthonormality and solvability conditions on the present asymptotic formulation we determine the boundary frequencies of dynamic instability regions for various orders in a consistent and hierarchical manner the principal instability regions of cross ply conical shells with simply supported simply supported boundary conditions are studied to demonstrate the performance of the present asymptotic theory

departures from a queue with many busy servers to analyze networks of queues it is important to be able to analyze departure processes from single queues for the m m s and m g models the stationary departure process is simple poisson but in general the stationary departure process is quite complicated as a basis for approximations this paper shows that the stationary departure process is approximately poisson when there are many busy slow servers in a large class of stationary g gi s congestion models having s servers infinite waiting room the first come first served discipline and mutually independent and identically distributed service times that are independent of a stationary arrival process limit theorems are proved for the departure process in a g gi s system in which the number of servers and the offered load arrival rate divided by the service rate both increase the asymptotic behavior of the departure process depends on the way the arrival rate changes if the arrival rate is held fixed so that the offered load increases by

differential quadrature and partial differential equations some numerical results abstract the differential quadrature method is used to solve partial differential equations arising in chemical reactor design the results are compared with the finite difference approach although only 9th order quadrature is used or only 9 grid points are calculated as compared to the calculation of 480 grid points in difference methods a reasonably good accuracy is obtained with differential quadrature

the pointwise stationary approximation for m 1 m 1 s green kolesar and svoronos in press and green and kolesar in press use numerical methods to investigate the behavior of multiserver markov queues with a poisson arrival process having a sinusoidal arrival rate for this model they propose an approximation for long run average performance measures called the pointwise stationary approximation psa which consists of an appropriate weighted average of the performance measure that would result at each point in time if the system were stationary with the arrival rate that applies at that point in time in this paper we verify their conjecture that psa is asymptotically correct as the service and arrival rates increase with the instantaneous traffic intensity held fixed corresponding to long arrival rate cycles we actually establish both pointwise and average versions of this result for general time dependent birth and death processes

combining differential quadrature method with simulation technique to solve non linear differential equations nowadays most of the ordinary differential equations odes can be solved by modelica based approaches such as matlab simulink dymola and labview which use simulation technique st however these kinds of approaches restrict the users in the enforcement of conditions at any instant of the time domain this limitation is one of the most important drawbacks of the st another method of solution differential quadrature method dqm leads to very accurate results using only a few grids on the domain on the other hand dqm is not flexible for the solution of non linear odes and it is not so easy to impose multiple conditions on the same location for these reasons the author aims to eliminate the mentioned disadvantages of the simulation technique st and dqm using favorable characteristics of each method in the other this work aims to show how the combining method cm works simply by solving some non linear problems and how the cm gives more accurate results compared with those of other methods copyright 2008 john wiley sons ltd

application of differential quadrature to static analysis of structural components the numerical technique of differential quadrature for the solution of linear and non linear partial differential equations first introduced by bellman and his associates is applied to the equations governing the deflection and buckling behaviour of one and two dimensional structural components separate transformations are used for higher order derivatives as suggested by mingle thus extending the method to treat fourth order equations and to include multiple boundary conditions in the respective co ordinate directions results are obtained for various boundary and loading conditions and are compared with existing exact and numerical solutions by other methods the application of differential quadrature to this class of problems is seen to lead to accurate results with relatively small computational effort

the generalized differential quadrature rule for fourth order differential equations the generalized differential quadrature rule gdqr proposed here is aimed at solving high order differential equations the improved approach is completely exempted from the use of the existing point technique by applying multiple conditions in a rigorous manner the gdqr is used here to static and dynamic analyses of bernoulli euler beams and classical rectangular plates numerical error analysis caused by the method itself is carried out in the beam analysis independent variables for the plate are first defined the explicit weighting coefficients are derived for a fourth order differential equation with two conditions at two different points it is quite evident that the gdqr expressions and weighting coefficients for two dimensional problems are not a direct application of those for one dimensional problems the gdqr are implemented through a number of examples good results are obtained in this work copyright 2001 john wiley sons ltd

distributed tracking of correlated equilibria in regime switching noncooperative games we consider a class of regime switching noncooperative repeated games where agents exchange information over a graph the parameters of the game number of agents payoffs information exchange graph evolve randomly over time according to a markov chain we present a regret based stochastic approximation algorithm with constant step size that prescribes how individual agents update their randomized strategies over time we show that if the markov chain jump changes on the same timescale as the adaptation rate of the stochastic approximation algorithm and agents independently follow this algorithm their collective behavior is agile in tracking the time varying convex polytope of correlated equilibria the analysis is carried out using weak convergence methods and lyapunov stability of switched markovian differential inclusions

stochastic approximations and differential inclusions part ii applications we apply the theoretical results on stochastic approximations and differential inclusions developed in benaim et al m benaim j hofbauer s sorin 2005 stochastic approximations and differential inclusions siam j control optim 44 328 348 to several adaptive processes used in game theory including classical and generalized approachability no regret potential procedures hart and mas colell s hart a mas colell 2003 regret based continuous time dynamics games econom behav 45 375 394 and smooth fictitious play d fudenberg d k levine 1995 consistency and cautious fictitious play j econom dynam control19 1065 1089

an invariant measure approach to the convergence of stochastic approximations with state dependent noise a new method is presented for quickly getting the ode ordinary differential equation associated with the asymptotic properties of the stochastic approximation x n 1 x n a n f x n zeta n or the projected algorithm for the constrained problem either a n to 0 or a n can be constant in which case the analysis is on the sequence obtained when a to 0 the method requires that x n zeta n 1 be markov with a feller transition function but little else the simplest result requires that if x n equiv x the corresponding noise process zeta n x n geqq 0 have a unique invariant measure but the nonunique case can also be treated no mixing condition is required nor the construction of averaged test functions and f cdot cdot need not be continuous a detailed analysis of the way that zeta n varies with x n is not required for the class of sequences treated the conditions seem easier to verify than for other methods there

on the solution of the thomas fermi equation by differential quadrature on utilise la methode de quadrature differentielle pour resoudre l equation de thomas fermi

the pointwise stationary approximation for queues with nonstationary arrivals we empirically explore the accuracy of an easily computed approximation for long run average performance measures such as expected delay and probability of delay in multiserver queueing systems with exponential service times and periodic sinusoidal poisson arrival processes the pointwise stationary approximation is computed by integrating over time that is taking the expectation of the formula for the stationary performance measure with the arrival rate that applies at each point in time this approximation which has been empirically confirmed as a tight upper bound of the true value is shown to be very accurate for a range of parameter values corresponding to a reasonably broad spectrum of real systems

stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple lyapunov functions this paper presents a general framework for analyzing stability of nonlinear switched systems by combining the method of multiple lyapunov functions with a suitably adapted comparison principle in the context of stability in terms of two measures for deterministic switched systems this leads to a unification of representative existing results and an improvement upon the current scope of the method of multiple lyapunov functions for switched systems perturbed by white noise we develop new results which may be viewed as natural stochastic counterparts of the deterministic ones in particular we study stability of deterministic and stochastic switched systems under average dwell time switching