discrete time orthogonal spline collocation methods for schrodinger equations in two space variables two discrete time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time dependent schrodinger equation in two space variables these are crank nicolson and alternating direction implicit adi schemes employing c1 piecewise polynomial spaces of arbitrary degree ge 3 in each space variable the stability of the schemes is examined and optimal order a priori l2 and h1 error estimates at each time step are derived parallel implementation of the adi scheme is discussed

spatial prediction of soil salinity using electromagnetic induction techniques 1 statistical prediction models a comparison of multiple linear regression and cokriging we describe a regression based statistical methodology suitable for predicting field scale spatial salinity ec conditions from rapidly acquired electromagnetic induction ec data this technique uses multiple linear regression mlr models to estimate soil salinity from ec survey data the mlr models incorporate multiple ec measurements and trend surface parameters to increase the prediction accuracy and can be fitted from limited amounts of ec calibration data this estimation technique is compared to some commonly recommended cokriging techniques with respect to statistical modeling assumptions calibration sample size requirements and prediction capabilities we show that mlr models are theoretically equivalent to and cost effective relative to cokriging for estimating a spatially distributed random variable when the residuals from the regression model are spatially uncorrelated mlr modeling and prediction techniques are demonstrated with data from three salinity surveys

approximations of temporal rainfall from a multidimensional model the feasibility of representing the temporal structure of a multidimensional rainfall process with simpler stochastic models and a study of the effect of parameter robustness on the time scale is investigated here via performing controlled numerical experiments a multidimensional representation for precipitation given in the theory recently proposed by e waymire et al 1984 is used for simulating rainfall in space and time the model produces moving storms with realistic mesoscale meteorological features e g clustering birth and death of cells cell intensity attenuation in time and space etc two year traces of rainfall intensities at fixed gage stations were generated at intervals of 0 1 hours for three climates these traces are then aggregated at different time scales ranging from 1 to 24 hours first and second order statistics are evaluated from the above series at each aggregation level and they are used for estimating the parameters of three one dimensional models of temporal rainfall at a point 1 poisson model with independent marks 2 rectangular pulses with independent intensity and duration and 3 neyman scott with independent marks only at very high levels of aggregation the compound poisson model was able to reproduce the statistical structure of the simulated traces the rectangular pulses model was found to have parameters which vary significantly with the time scale of aggregation and moreover it leads to large distortions in the mean depth and duration of storm events the neyman scott scheme proved to be the most stable model throughout the different time scales and it also reproduced quite well the average storm characteristics at the event level none of these three models was able to reproduce in a satisfactory manner the simulated extreme value distribution of the multidimensional model

a conservative fourier pseudo spectral method for the nonlinear schrodinger equation a fourier pseudo spectral method that conserves mass and energy is developed for a two dimensional nonlinear schrodinger equation by establishing the equivalence between the semi norm in the fourier pseudo spectral method and that in the finite difference method we are able to extend the result in ref 56 to prove that the optimal rate of convergence of the new method is in the order of o n r 2 in the discrete l 2 norm without any restrictions on the grid ratio where n is the number of modes used in the spectral method and is the time step size a fast solver is then applied to the discrete nonlinear equation system to speed up the numerical computation for the high order method numerical examples are presented to show the efficiency and accuracy of the new method

stochastic interpolation of sparsely sampled time series via multipoint fractional brownian bridges we propose and test a method to interpolate sparsely sampled signals by a stochastic process with a broad range of spatial and or temporal scales to this end we extend the notion of a fractional brownian bridge defined as fractional brownian motion with a given scaling hurst exponent h and with prescribed start and end points to a bridge process with an arbitrary number of intermediate and nonequidistant points determining the optimal value of the hurst exponent h opt appropriate to interpolate the sparse signal is a very important step of our method we demonstrate the validity of our method on a signal from fluid turbulence in a high reynolds number flow and discuss the implications of the non self similar character of the signal the method introduced here could be instrumental in several physical problems including astrophysics particle tracking and specific tailoring of surrogate data as well as in domains of natural and social sciences

a stable runge kutta methods for semilinear evolution equations we consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of hilbert spaces in this setting we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale under the same assumptions we prove that a class of implicit a stable runge kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale under the additional assumption that the linear part of the evolution equation is normal or sectorial we prove full order convergence of the semidiscretization in time for initial data on open sets our results apply in particular to the semilinear wave equation and to the nonlinear schrodinger equation 2012 elsevier inc

a space time finite element method for the nonlinear schrodinger equation the continuous galerkin method the convergence of a class of continuous galerkin methods for the nonlinear cubic schrodinger equation is analyzed in this paper these methods allow variable temporal stepsizes as well as changing of the spatial grid from one time level to the next we show the existence of the resulting approximations and prove optimal order error estimates in l infty l 2 and in l infty h 1 these estimates are valid under weak restrictions on the space time mesh these restrictions are milder if the elliptic projection is used at every time step instead of the l2 projection we also give superconvergence results at the temporal nodes tn

a deterministic geometric representation of temporal rainfall results for a storm in boston the use of a deterministic fractal multifractal fm representation to model high resolution rainfall time series via projections of fractal interpolating functions weighed by multifractal measures is reported it is shown that the intrinsic shape and variability of an 8 hour boston storm recorded every 15 s on october 25 1980 may be encoded wholistically employing the fractal geometric methodology it is illustrated that the fm methodology provides very faithful descriptions of both major trends and small noisy fluctuations for this storm resulting in preservation of not only classical statistical characteristics of the records but also multifractal and chaotic properties present in them these results and those for other storms suggest that a stochastic framework for rainfall may be bypassed in favor of a deterministic representation based on projections

multidimensional modeling of cumulative rainfall parameter estimation and model adequacy through a continuum of scales this paper presents the multidimensional representation of the cumulative rainfall process and derives a closed form expression for the mean and space time covariance functions a parameter estimation procedure is developed sensitivity analysis provides useful information for devising the parameter estimation strategy the cumulative rainfall model is applied to 8 years of digitized rain gage data from the walnut gulch experimental watershed the most useful feature of the model is its self consistency implying that even after using information from a given limited set of data it still leads to realistic parameters which reproduce historical statistics at several levels of time and space scales

an energy preserving crank nicolson galerkin spectral element method for the two dimensional nonlinear schrodinger equation abstract a crank nicolson galerkin spectral element method for solving the nonlinear schrodinger nls equation in two dimensions is proposed in this paper our key idea is twofolds first the 2d nls equation is rewritten as an infinite dimensional hamiltonian pde and the hamiltonian pde is discreted by using the galerkin spectral element gse method in space second we cast the resulted odes into a finite dimensional canonical hamiltonian system and discrete the system by using the crank nicolson cn method the relay leads to a fully discretized and energy preserved scheme without grid ratio restrictions the order of convergence of our new method is o 2 h 2 if the discrete l 2 norm is employed the fast fourier transform and the matrix diagonalization method are applied to the new method to increase computing efficiency numerical examples are given to further illustrate the conservation properties and convergence of the energy preserving scheme constructed

fully discrete methods for the nonlinear schrodinger equation abstract in this paper we construct and analyze fully discrete methods for the nonlinear schrodinger equation these methods are based on a galerkin finite element formulation in space and use time stepping techniques which are modifications of implicit runge kutta methods the resulting schemes are both efficient and circumvent the well known order reduction phenomenon affecting runge kutta methods numerical experiments are also presented

butler groups of infinite rank ii a torsion free abelian group g is called a butler group if bext g t 0 for any torsion group t we show that every butler group g of cardinality ltl is a b2 group i e g is a union of a smooth ascending chain of pure subgroups g0 where g ga b0 b 0 a butler group of finite rank assuming the validity of the continuum hypothesis ch we show that every butler group of cardinality not exceeding n is a b2 group moreover we are able to prove that the derived functor bext2 a t 0 for any torsion group t and any torsion free a with jai this implies that under ch all balanced subgroups of completely decomposable groups of cardinality nt are b2 groups

stochastic modeling of multidimensional precipitation fields considering spectral structure a multidimensional stochastic precipitation model with major emphasis on its spectral structure is proposed as a hyperbolic type of stochastic partial differential equation this model is characterized by a small set of easily estimable parameters these characteristics are similar to those of the noise forced diffusive precipitation model but the representation of the physical and statistical features of the precipitation field is similar to that of the waymire gupta rodriguez iturbe wgr precipitation model the derivation was based on the autoregressive process considering advection and diffusion the dominant statistical and physical characteristics of the precipitation field propagation the model spectrum showed a good match with the global atlantic tropical experiment spectrum this model was also compared with the wgr model and the noise forced diffusive precipitation model both analytically and through applications such as the sampling error estimation from spaceborne sensors and rain gauges the sampling error from spaceborne sensors based on the proposed model was similar to that of the noise forced diffusive precipitation model but much smaller than that of the wgr model similar results were also obtained in the estimation of the sampling error from rain gauges

on convergence and stability of the explicit difference method for solution of nonlinear schrodinger equations the first boundary value problem for a nonlinear schrodinger equation is investigated the conditional convergence and stability on the initial data of the explicit three level difference scheme of dufort frankel type in c and w 2 1 norms are proved grid analogues of energy conservation laws and grid multiplicative inequalities are used

fourth order compact and energy conservative difference schemes for the nonlinear schrodinger equation in two dimensions abstract in this paper a fourth order compact and energy conservative difference scheme is proposed for solving the two dimensional nonlinear schrodinger equation with periodic boundary condition and initial condition and the optimal convergent rate without any restriction on the grid ratio at the order of o h 4 2 in the discrete l 2 norm with time step and mesh size h is obtained besides the standard techniques of the energy method a new technique and some important lemmas are proposed to prove the high order convergence in order to avoid the outer iteration in implementation a linearized compact and energy conservative difference scheme is derived numerical examples are given to support the theoretical analysis

stochastic interpolation of rainfall data from rain gages and radar using cokriging 2 results various estimation procedures using ordinary universal and disjunctive cokriging are evaluated in merging rain gage measurements and radar rainfall data the estimation procedures and the simulation experiments were described in part 1 seo et al this issue of this two part work in this part the experiments are described in detail an objective comparison scheme devised to compare a large number of estimators is also described the results are presented focusing upon 1 the potential of radar gage estimation using cokriging over radar only estimation and gage only estimation under widely varying conditions of gage network density and the error characteristics of radar rainfall 2 the potential for using universal or disjunctive cokriging over ordinary cokriging 3 how the uncertain second order statistics affect the estimators due to lack of rain gage measurements and 4 how the statistical characteristics of ground truth rainfall affect the estimators

compatibility of continuous rainfall occurrence models with discrete rainfall observations inferences about the underlying rainfall generating mechanism made through a model fitted to the sampled rainfall series at a particular time scale are meaningful only if they are invariant under the time scale of measurement at which the fitting of the model is made following the work of i rodriguez iturbe et al 1984 this assertion is tested by examining the compatibility of the neyman scott n s model for the continuous underlying rainfall with sampled realizations taken over intervals ranging from one hour to one day the results suggest that the n s model is inconsistent at different time scales and therefore mathematical descriptions of rainfall using this model are inevitably limited to the time scale of measurement at which a description of rainfall is sought these findings are quite general in that they apply to any model with rainfall deposited instantaneously at times described by a n s process regardless of how the amounts associated with the events are distributed and of the dependence structure of the amounts

on multimodality of the likelihood in the spatial linear model summary a popular covariance scheme used for the spatial linear model in geostatistics has spherical form however the likelihood is not twice differentiable with respect to the range parameter and this raises some questions regarding the unimodality of the likelihood we compare the likelihoods of the spatial linear model for small samples under this scheme and the doubly geometric scheme also a power covariance with range parameter is proposed in view of potential multimodal likelihoods for small samples for this model a convenient profile likelihood is introduced and studied

a space time finite element method for the nonlinear schro dinger equation the discontinuous galerkin method the convergence of the discontinuous galerkin method for the nonlinear cubic schrodinger equation is analyzed in this paper we show the existence of the resulting approximations and prove optimal order error estimates in l l 2 these estimates are valid under weak restrictions on the space time mesh

convergence of runge kutta approximations for parabolic problems with neumann boundary conditions we analyze a class of algebraically stable runge kutta standard galerkin methods for inhomogeneous linear parabolic equations with time dependent coefficients under neumann boundary conditions and derive an l 2 error bound of o h r k min nu p 3 over2 provided frac k h is bounded

recovering the electrical conductivity of the soil via a linear integral model abstract electromagnetic induction techniques are often used to ascertain the presence of conductive substances in the subsoil the device we are interested in generates an electromagnetic field and measures the secondary field produced by eddy currents induced by the primary field we analyze a linear model which reproduces the device readings consisting of a system of two fredholm linear integral equations of the first kind after studying the continuous problem in various function spaces we describe three collocation methods to transform the integral equations into a system of linear equations which is solved by a regularization technique numerical experiments show the performance of the different discretizations and of a particular approach to deal with the integral truncation error we also verify the influence on the solution of regularization matrices based on the discretization of the first and second derivative

on optimal order error estimates for the nonlinear schro dinger equation implicit runge kutta methods in time are used in conjunction with the galerkin method in space to generate stable and accurate approximations to solutions of the nonlinear cubic schrodinger equation the temporal component of the discretization error is shown to decrease at the classical rates in some important special cases

order estimates in time of splitting methods for the nonlinear schrodinger equation in this paper we consider the nonlinear schrodinger equation u t i delta u f u 0 in two dimensions we show by an operator theoretic proof that the well known lie and strang formulae which are splitting methods are approximations of the exact solution of order 1 and 2 in time

a conservative spectral collocation method for the nonlinear schrdinger equation in two dimensions in this study we present a conservative fourier spectral collocation fsc method to solve the two dimensional nonlinear schrdinger nls equation we prove that the proposed method preserves the mass and energy conservation laws in semi discrete formulations using the spectral differentiation matrices the nls equation is reduced to a system of nonlinear ordinary differential equations odes the compact implicit integration factor ciif method is later developed for the nonlinear odes in this approach the storage and cpu cost are significantly reduced such that the use of ciif method becomes attractive for two dimensional nls equation numerical results are presented to demonstrate the conservation accuracy and efficiency of the method

a spectral theory of rainfall intensity at the meso β scale the available empirical descriptions of extratropical cyclonic storms are employed to formulate a physically realistic stochastic representation of the ground level rainfall intensity field in space and time the stochastic representation is based on three component stochastic point processes which possess the general features of the embedding of rain cells within small mesoscale areas within large mesoscale areas within synoptic storms certain scale idealizations and assumptions on functional forms which qualitatively reflect the physical features lead to a closed form expression for the covariance function i e the real space time spectrum of the rainfall intensity field the theoretical spectrum explains the empirical spectral features observed by zawadzki almost a decade ago of particular interest and importance in this connection is an explanation of the empirical observation that the taylorian propogation of the fine scale structure via a transformation of time to space through the storm velocity holds only for a small time lag and not throughout the results here indicate the extent of this lag in terms of the characteristic scales associated with cell durations cellular birthrates and velocities etc

continuous time versus discrete time point process models for rainfall occurrence series several authors have had apparent success in applying continuous time point process models to rainfall occurrence sequences in this paper it is shown that if rainfall occurrences are interpreted as the events of a point process and not as a censored sample the continuous time point process methodology and estimation procedures are not directly applicable since they fail to account for the time discreteness of the sample process this is demonstrated analytically by studying the effects of discretization on selected statistical properties of a poisson process a neyman scott process and a renewal cox process with markovian intensity in general the study of rainfall occurrences under the continuous time point process framework may result in misleading inferences regarding clustering dispersion and consequently incorrect interpretations of the underlying rainfall generating mechanisms for example daily rainfall occurrence structures underdispersed relative to the poisson process are usually overdispersed relative to the bernoulli process the discrete time analogue of the poisson these findings are confirmed by the statistical analysis of six daily rainfall records representative of a range of u s climates two of which are described in detail

a continuous space time finite element method for the wave equation we consider a finite element method for the nonhomogeneous second order wave equation which is formulated in terms of continuous approximation functions in both space and time thereby giving a unified treatment of the spatial and temporal discretizations our analysis uses primarily energy arguments which are quite common for spatial discretizations but not for time we present a priori nodal in time superconvergence error estimates without any special time step restrictions our method is based on tensor product spaces for the full discretization

disjunctive kriging 3 cokriging the disjunctive kriging dk method described in the first paper of this series is extended to account for more than one random function in the derivation contained herein two random functions are considered but this is easily generalized to any number an example is presented using disjunctive cokriging dck where the surface gravimetric moisture content is estimated using the bare soil temperature as an auxiliary random function the results indicate that the dck procedure produces a better estimator than ordinary cokriging in terms of reduced variance of errors and exactness of estimation also using dck an estimate of the conditional probability that the level of a property is greater than a known cutoff value can be obtained in general this conditional probability is better than the dk probability by virtue of the additional information contained in the second auxiliary random function

the mathematical structure of rainfall representations 1 a review of the stochastic rainfall models this is the first of a three part series on the mathematical structure of rainfall models several important attempts at modeling rainfall are reviewed special attention is given to the mathematical structures that arise in the rainfall descriptions a general overview of the three part series is given as preface to this part

on some high order accurate fully discrete galerkin methods for the korteweg de vries equation we construct and analyze fully discrete galerkin finite element methods of high order of accuracy for the numerical solution of the periodic initial value problem for the korteweg de vries equation the methods are based on a standard space discretization using smooth periodic splines on a uniform mesh for the time stepping we use two schemes of third resp fourth order of accuracy which are modifications of well known diagonally implicit runge kutta methods and require the solution of two resp three nonlinear systems of equations at each time step these systems are solved approximately by newton s method provided the initial iterates are chosen in a specific accurate way we show that only one newton iteration per system is needed to preserve the stability and order of accuracy of the scheme under certain mild restrictions on the space mesh length and the time step we prove l2 error estimates of optimal rate of convergence for both schemes

numerical approximation of blow up of radially symmetric solutions of the nonlinear schrodinger equation we consider the initial value problem for the radially symmetric nonlinear schrodin ger equation with cubic nonlinearity nls in d 2 and 3 space dimensions to approximate smooth solutions of this problem we construct and analyze a numerical method based on a standard galerkin finite element spatial discretization with piecewise linear continuous functions and on an implicit crank nicolson type time stepping procedure we then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the nls equation that blow up at the origin as the temporal variable t tends from below to a finite value t star for the blow up of the amplitude of the solution we recover numerically the well known rate t star t 1 2 for d 3 for d 2 our numerical evidence supports the validity of the log log law ln ln frac 1 t star t t star t 1 2 for t extremely close to t star the scheme also approximates well the details of the blow up of the phase of the solution at the origin as t to t star

daily spatiotemporal precipitation simulation using latent and transformed gaussian processes 1 a daily stochastic spatiotemporal precipitation generator that yields spatially consistent gridded quantitative precipitation realizations is described the methodology relies on a latent gaussian process to drive precipitation occurrence and a probability integral transformed gaussian process for intensity at individual locations the model reduces to a markov chain for precipitation occurrence and a gamma distribution for precipitation intensity allowing statistical parameters to be included in a generalized linear model framework statistical parameters are modeled as spatial gaussian processes which allows for interpolation to locations where there are no direct observations via kriging one advantage of such a model for the statistical parameters is that stochastic generator parameters are immediately available at any location with the ability to adapt to spatially varying precipitation characteristics a second advantage is that parameter uncertainty generally unavailable with deterministic interpolators can be immediately quantified at all locations the methodology is illustrated on two data sets the first in iowa and the second over the pampas region of argentina in both examples the method is able to capture the local and domain aggregated precipitation behavior fairly well at a wide range of time scales including daily monthly and annually

further developments of the neyman scott clustered point process for modeling rainfall this paper provides some useful results for modeling rainfall it extends work on the neyman scott cluster model for simulating rainfall time series several important properties have previously been found for the model for example the expectation and variance of the amount of rain captured in an arbitrary time interval rodriguez iturbe et al 1987a in this paper additional properties are derived such as the probability of an arbitrary interval of any chosen length being dry in applications this is a desirable property to have and is often used for fitting stochastic rainfall models to field data the model is currently being used in rainfall time series research directed toward improving sewage systems in the united kingdom to illustrate the model s performance an example is given where the model is fitted to 10 years of hourly data taken from blackpool england

torsion free modules over valuation domains whose balanced projective dimension is at most one let r denote a valuation domain torsion free balanced projective r modules possess many interesting properties including a nice structure w x theory see 7 in this paper we study the larger class of torsion free r modules whose balanced projective dimension is at most one fundamental to our investigation are the pure essential submodules and a new concept of a pseudo balanced submodule the former helps to construct w x balanced extensions as was done in 6 9 while the latter gives rise to a relative homological algebra starting from the proof of a result of the first w x author in 9 on the basic submodule of a completely decomposable torsion free module we find a noteworthy result for a torsion free r2 module b if bext b t s 0 for all torsion r modules t then b must r have balanced projective dimension f 1 our results demonstrate how recent ideas from the theory of butler groups can be successfully utilized in the study of modules over valuation domains specifically the concept

an efficient spectral collocation difference method for two dimensional schrodinger equation with neumann boundary conditions abstract based on the second order cosine spectral differentiation matrix deduced in this paper we present an efficient and high accurate numerical scheme for solving two dimensional schrodinger equation with neumann boundary conditions the crank nicolson finite difference scheme is utilized in temporal discretization and the cosine spectral collocation method is employed for the approximation of laplacian operator the new scheme conserves the mass and energy in discrete level and is implemented by a fast algorithm in terms of the relations between the cosine spectral differentiation matrix and fast cosine transform more importantly this strategy can be extended to address the problems with high even order derivatives in space numerical examples and applications are listed to confirm the validity and high accuracy of the method

stability under galerkin truncation of a stable runge kutta discretizations in time we consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of hilbert spaces we approximate their semiflow by an implicit a stable runge kutta discretization in time and a spectral galerkin truncation in space we show regularity of the galerkin truncated semiflow and its time discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value we also prove convergence of the space time discretization without any condition that couples the time step to the spatial resolution then we estimate the galerkin truncation error for the semiflow of the evolution equation its runge kutta discretization and their respective derivatives showing how the order of the galerkin truncation error depends on the smoothness of the initial data our results apply in particular to the semilinear wave equation and to the nonlinear schr odinger equation

the relationship between a continuous time renewal model and a discrete markov chain model of precipitation occurrence a relationship between a continuous wet dry renewal model with gamma distributed dry intervals and a markov chain model for daily rainfall is derived the relationship requires numerical evaluation for the general case but closed form solutions are available for the markov process case in which dry intervals are exponentially distributed the markov process equations are equivalent to those derived by j r green 1964 for first order transition probabilities but are presented in a simplified nonrecursive form the markov process relationships are examined for convergence and accuracy with simulation studies and comparison to observed rainfall records the markov process model is shown to provide a good representation in certain parts of the united states in other areas where the markov property is inappropriate due to event clustering or other phenomena the gamma model provides an improved characterization of the relationship between continuous and discrete rainfall occurrence

analysis of a leap frog pseudospectral scheme for the schrodinger equation the numerical properties of a leap frog pseudospectral scheme for the schrodinger equation are analyzed stability second order accuracy in time and spectral accuracy in space are discussed considering the linear schrodinger equation with potential in a periodic setting further issues regarding phase error gauge invariance conservation properties and commutation relations are addressed results of numerical experiments are reported to demonstrate the validity and limitations of the theoretical findings and for comparison with the well known crank nicholson finite difference scheme

maximum norm error bound of a linearized difference scheme for a coupled nonlinear schrodinger equations in this article a linearized conservative difference scheme for a coupled nonlinear schrodinger equations is studied the discrete energy method and an useful technique are used to analyze the difference scheme it is shown that the difference solution unconditionally converges to the exact solution with second order in the maximum norm numerical experiments are presented to support the theoretical results

two completely explicit and unconditionally convergent fourier pseudo spectral methods for solving the nonlinear schrodinger equation abstract this paper aims to construct and analyze two new fourier pseudo spectral fps methods for the general nonlinear schrodinger nls equation the two fps methods have two merits unconditional convergence and complete explicitness in the practical computation further more by introducing a modified mass functional and a modified energy functional the two fps methods are proved to preserve the total mass and energy in the discrete sense besides the standard energy method the key techniques used in our numerical analysis are a mathematical induction argument and a lifting technique without any restriction on the grid ratio and initial value we establish the optimal error estimate of the two fps methods for solving the general nls equation while previous work just is valid for the cubic nls equation and requires small initial value for the focusing case these two fps methods are proved to be spectrally accurate in space and second order accurate in time respectively the analysis framework can be used to prove the unconditional convergence of many other fourier pseudo spectral methods for solving the nls type equations we investigate the effect of the nonlinear term on the progression simulation of the plane wave the conservation of the invariants and the effect of initial data on the blow up solution via different parameters numerical results are reported to show the accuracy and efficiency of the proposed methods

elementary functions for constructing exact solutions to nonlinear partial differential equations with applications to nonlinear schrodinger type and mhd equations abstract in this paper we present an improvement to those methods based on the using of elementary functions like exponential trigonometric and hyperbolic functions in obtaining exact solutions to nonlinear partial differential equations npdes the improved method is applied to stable nonlinear schrodinger nls unstable nls generalized nls high order nls and derivative nls equations new solutions for these equations are obtained the obtained solutions are more general than a wide class of previous solutions solutions of a derivative nls equation that describes the large amplitude solitons propagating in an arbitrary direction in a high hall plasma are also obtained moreover the method is applied to magnetohydrodynamics mhd equations describing an ideal incompressible flow in the steady state one of the most important advantages of the solution method presented here is it deals with several types of nonlinearities associated with pdes without making a transformation of the original equation to another one

a galerkin energy preserving method for two dimensional nonlinear schrodinger equation abstract in this paper a galerkin energy preserving scheme is proposed for solving nonlinear schrodinger equation in two dimensions the nonlinear schrodinger equation is first rewritten as an infinite dimensional hamiltonian system following the method of lines the spatial derivatives of the nonlinear schrodinger equation are approximated with the aid of the galerkin methods the resulting ordinary differential equations can be cast into a canonical hamiltonian system a fully discretized scheme is then devised by considering an average vector field method in time moreover based on the fast fourier transform and the matrix diagonalization method a fast solver is developed to solving the resulting algebraic equations finally the proposed scheme is employed to capture the blow up phenomena of the nonlinear schrodinger equation

scaling limits and self similarity in precipitation fields it has been recognized in the literature on rainfall models in hydrology that the exact calculations of probabilities for models which incorporate cluster structure is typically not possible consequently one is led to considerations of limiting approximations however because of the presence of dependencies and or fat tails in several of the current models the nature of the approximations involves some important issues for both theoretical as well as experimental consideration in precipitation research the objective of the present paper is to systematically describe these issues as they pertain to rainfall models some specific large scale approximations are given for models of the type currently in the literature

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

maximum likelihood estimation of models for residual covariance in spatial regression we describe the maximum likelihood method for fitting the linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model containing unknown parameters observations are assumed to be gaussian we give conditions which ensure consistency and asymptotic normality of the estimators our main concern is with the analysis of spatial data and in this context we describe some simulation experiments to assess the small sample behaviour of estimators we also discuss an application of the spectral approximation to the likelihood for processes on a lattice

a markov renewal model for rainfall occurrences a probabilistic model for the temporal description of daily rainfall occurrences at a single location is presented by defining an event as a day with measurable precipitation the model is cast into the discrete time point process framework in the proposed model the sequence of times between events is formed by sampling from two geometric distributions according to transition probabilities specified by a markov chain the model belongs to the class of markov renewal processes and exhibits clustering relative to the independent bernoulli process as a special case it reduces to a renewal model with a mixture distribution for the interarrival times the rainfall occurrence model coupled with a mixed exponential distribution for the nonzero daily rainfall amounts was applied to the daily rainfall series for snoqualmie falls washington and was successful in preserving the short term structure of the occurrence process as well as the distributional properties of the seasonal rainfall amounts

optimal error estimates of finite difference methods for the gross pitaevskii equation with angular momentum rotation we analyze finite difference methods for the gross pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate for the conservative crank nicolson finite difference cnfd method and semi implicit finite difference sifd method at the order of o h2 2 in the l2 norm and discrete h1 norm with time step and mesh size h besides the standard techniques of the energy method the key technique in the analysis for the sifd method is to use the mathematical induction and resp for the cnfd method is to obtain a priori bound of the numerical solution in the l norm by using the inverse inequality and the l2 norm error estimate in addition for the sifd method we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts respectively which are at the same order of the convergence rate as that of the numerical solution itself finally numerical results are reported to confirm our error estimates of the numerical methods

orthogonal spline collocation methods for schr o dinger type equations in one space variable we examine the use of orthogonal spline collocation for the semi discreti za tion of the cubic schr o dinger equation and the two dimensional parabolic equation of tappert in each case an optimal order l 2 estimate of the error in the semidiscrete approximation is derived for the cubic schr o dinger equation we present the results of numerical experiments in which the integration in time is performed using a routine from a software library

statistical inference for point process models of rainfall in this paper we develop maximum likelihood procedures for parameter estimation and model selection that apply to a large class of point process models that have been used to model rainfall occurrences including cox processes neyman scott processes and renewal processes the statistical inference procedures are based on the stochastic intensity t lims 0 s 0 1 s e n t s n t n u u t the likelihood function of a point process is shown to have a simple expression in terms of the stochastic intensity the main result of this paper is a recursive procedure for computing stochastic intensities the procedure is applicable to a broad class of point process models including renewal cox process with markovian intensity processes and an important class of neyman scott processes the model selection procedure we propose which is based on likelihood ratios allows direct comparison of two classes of point processes to determine which provides a better model for a given data set the estimation and model selection procedures are applied to two data sets of simulated cox process arrivals and a data set of daily rainfall occurrences in the potomac river basin