n soliton solutions for a 2 1 dimensional breaking soliton equation with self consistent sources abstract a 2 1 dimensional breaking soliton equation with self consistent sources 2 1 dbsescs is derived the bilinear form of it is obtained and the n soliton solutions are worked out by means of the hirota method and wronskian technique respectively by making use of the laplace expansion formula a simple method is presented to treat the nonlinear term in the bilinear equation and finish the wronskian verification the dynamics and source effects are investigated

a convergent numerical scheme for integrodifferential kinetic models of angiogenesis abstract we study a robust finite difference scheme for integrodifferential kinetic systems of fokker planck type modeling tumor driven blood vessel growth the scheme is of order one and enjoys positivity features we analyze stability and convergence properties and show that soliton like asymptotic solutions are correctly captured we also find good agreement with the solution of the original stochastic model from which the deterministic kinetic equations are derived working with ensemble averages a numerical study clarifies the influence of velocity cut offs on the solutions for exponentially decaying data

towards an ultra efficient kinetic scheme part i basics on the bgk equation in this paper we present a new ultra efficient numerical method for solving kinetic equations in this preliminary work we present the scheme in the case of the bgk relaxation operator the scheme being based on a splitting technique between transport and collision can be easily extended to other collisional operators as the boltzmann collision integral or to other kinetic equations such as the vlasov equation the key idea on which the method relies is to solve the collision part on a grid and then to solve exactly the transport linear part by following the characteristics backward in time the main difference between the method proposed and semi lagrangian methods is that here we do not need to reconstruct the distribution function at each time step this allows to tremendously reduce the computational cost of the method and it permits for the first time to the author s knowledge to compute solutions of full six dimensional kinetic equations on a single processor laptop machine numerical examples up to the full three dimensional case are presented which validate the method and assess its efficiency in 1d 2d and 3d

semi lagrangian schemes for the vlasov equation on an unstructured mesh of phase space a new scheme for solving the vlasov equation using an unstructured mesh for the phase space is proposed the algorithm is based on the semi lagrangian method which exploits the fact that the distribution function is constant along the characteristic curves we use different local interpolation operators to reconstruct the distribution function f some of which need the knowledge of the gradient of f we can use limiter coefficients to maintain the positivity and the l bound of f and optimize these coefficients to ensure the conservation of the l1 norm that is to say the mass by solving a linear programming problem several numerical results are presented in two and three axisymmetric case dimensional phase space the local interpolation technique is well suited for parallel computation

stability analysis of the green s function method gfm used as an abc for arbitrarily shaped boundaries the time domain discrete green s function of the external region beyond a given boundary has been recently introduced as a discretized version of the impedance condition it is incorporated within the framework of the finite difference time domain fdtd as a quasi local single layer boundary condition termed the green s function method gfm the stability characteristics of this method are provided the analysis is based on the general representation of the method in matrix form whose eigenvalues are investigated this formulation helps detect and remove possible instabilities of the algorithm a demonstration of the ability of the gfm absorbing boundary condition abc to deal with re entrant corner problems is given

conservative numerical schemes for the vlasov equation abstract a new scheme for solving the vlasov equation using a phase space grid is proposed the algorithm is based on the conservation of the flux of particles and the distribution function is reconstructed using various techniques that allow control of spurious oscillations or preservation of the positivity several numerical results are presented in two and four dimensional phase space and the scheme is compared with the semilagrangian method this method is almost as accurate as the semilagrangian one and the local reconstruction technique is well suited for parallel computation

darboux transformation and exact solutions for a hierarchy of nonlinear evolution equations starting from a spectral problem we derive a hierarchy of nonlinear evolution equations an explicit and universal darboux transformation for the whole hierarchy is constructed the exact solutions for the hierarchy are obtained by applying the darboux transformation

discrete green s function formulation of the fdtd method and its application in antenna modeling a discrete green s function formulation of the finite difference time domain dgf fdtd method based on both discrete system theory and the fdtd method has been developed which expresses the field response as a convolution of current sources and the impulse response of the fdtd equation system the dgf fdtd method presents the fdtd equations in a different perspective from the conventional yee algorithm it avoids the computational difficulties such as the need for computation of free space nodes and absorbing boundary conditions of the classic fdtd method the ability of the dgf fdtd method to model on antenna is demonstrated by the modeling of a yagi uda array antenna with considerable saving in memory usage

a numerical method for solving the vlasov poisson equation based on the conservative ido scheme we have applied the conservative form of the interpolated differential operator ido cf scheme in order to solve the vlasov poisson equation which is one of the multi moment schemes through numerical tests of the nonlinear landau damping and two stream instability we compared the present scheme with other schemes such as the spline and cip ones we mainly investigated the conservation property of the l1 norm energy entropy and phase space area for each scheme and demonstrated that the ido cf scheme is capable of performing stable long time scale simulation while maintaining high accuracy the scheme is based on an eulerian approach and it can thus be directly used for fokker planck high dimensional vlasov poisson and also guiding center drift simulations aiming at particular problems of plasma physics the benchmark tests for such simulations have shown that the ido cf scheme is superior in keeping the conservation properties without causing serious phase error

lyapunov stability of vlasov equilibria using fourier hermite modes we propose an efficient method to compute lyapunov exponents and lyapunov eigenvectors of long range interacting many particle systems whose dynamics is described by the vlasov equation we show that an expansion of a distribution function using hermite modes in velocity variable and fourier modes in configuration variable converges fast if an appropriate scaling parameter is introduced and identified with the inverse of the temperature as a consequence dynamics and linear stability properties of many particle states both in the close to and in the far from equilibrium regimes can be predicted using a small number of expansion coefficients as an example of a long range interacting system we investigate stability properties of stationary states the hamiltonian mean field model

vlasov simulations using velocity scaled hermite representations the efficiency accuracy and stability of two different pseudo spectral methods using scaled hermite basis and weight functions applied to the nonlinear vlasov poisson equations in one dimension 1d 1v are explored and compared a variable velocity scaleuis introduced into the hermite basis and is shown to yield orders of magnitude reduction in errors as compared to linear kinetic theory with no increase in workload a set of fourier hermite coefficients representing a periodic gaussian distribution function are advanced through time with ano t2 accurate splitting method within this splitting scheme the advection and acceleration terms of the vlasov equation are solved separately using ano t4 accurate runge kutta method the asymmetrically weighted aw hermite basis which has been used previously by many authors conserves particles and momentum exactly and total energy too t3 however the aw hermite method doesnotconserve the square integral of the distribution and is in fact numerically unstable the symmetrically weighted sw hermite algorithm applied here to the vlasov system for the first time can either conserve particles and energy fornueven or momentum fornuodd as t 0 wherenuis the largest hermite mode number the sw hermite method conserves the square integral of the distribution and therefore remains numerically stable in addition careful velocity scaling improves the conservation properties of the sw hermite method damping and growth rates oscillation frequencies e field saturation levels and phase space evolution are seen to be qualitatively correct during simulations relative errors with respect to linear landau damping and linear bump on tail instability are shown to be less than 1 using only 64 velocity scaled hermite functions comparisons to particle in cell pic simulations show that as the number of particles increases to more than 106 the pic solutions converge to scaled sw hermite solutions that were found in only 1 10 of the run time the sw hermite method with velocity scaling is well suited to kinetic simulations of warm plasmas

multilayered discrete green s functions based on mixed potential finite difference formulation an approach to calculating discrete green s functions dgfs in multilayered media is proposed based on the finite difference scheme of time domainmixed potential equations the calculated dgfs have very high accuracy in comparison to the direct fdtd solution furthermore the steady state values of the dgfs are properly estimated in terms of the scalar potential ensuring stability out of the truncating window these dgfs are applicable to the analysis of antennas with multilayered planar structures in such analysis the proposed method has the advantage of the method of moments in that the computations are performed only on the antenna regardless of the white space around it moreover the broadband frequency characteristic of the antenna is achieved with a single simulation run such as in the fdtdmethod the theoretical results have been verified by the experimental results on a wideband coplanar waveguide cpw fed monopole antenna

arbitrarily high order convected scheme solution of the vlasov poisson system abstract the convected scheme cs is a forward trajectory semi lagrangian method for solution of transport equations which has been most often applied to the kinetic description of plasmas and rarefied neutral gases in its simplest form the cs propagates the solution forward in time by advecting the so called moving cells along their characteristic trajectories and by remapping them on the mesh at the end of the time step the cs is conservative positivity preserving simple to implement and it is not subject to time step restriction to maintain stability recently guclu and hitchon 2012 1 a new methodology was introduced for reducing numerical diffusion based on a modified equation analysis the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping while the spatial accuracy was increased from 2nd to 4th order the new scheme retained the important properties of the original method and was shown to be extremely simple and efficient for constant advection problems here the cs is applied to the solution of the vlasov poisson system which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal coulomb interactions the vlasov equation is split into two constant advection equations one in configuration space and one in velocity space and high order time accuracy is achieved by proper composition of the operators the splitting procedure enables us to use the constant advection solver which we extend to arbitrarily high order of accuracy in time and space a new improved procedure is given which makes the calculation of the corrections straightforward focusing on periodic domains we describe a spectrally accurate scheme based on the fast fourier transform the proposed implementation is strictly conservative and positivity preserving the ability to correctly reproduce the system dynamics as well as resolving small scale features in the solution is shown in classical 1d 1v test cases both in the linear and the non linear regimes

stability of nonlinear vlasov poisson equilibria through spectral deformation and fourier hermite expansion we study the stability of spatially periodic nonlinear vlasov poisson equilibria as an eigenproblem in a fourier hermite basis in the space and velocity variables respectively of finite dimension n when the advection term in the vlasov equation is dominant the convergence with n of the eigenvalues is rather slow limiting the applicability of the method we use the method of spectral deformation introduced by crawford and hislop ann phys ny 189 265 1989 to selectively damp the continuum of neutral modes associated with the advection term thus accelerating convergence we validate and benchmark the performance of our method by reproducing the kinetic dispersion relation results for linear spatially homogeneous equilibria finally we study the stability of a periodic bernstein greene kruskal mode with multiple phase space vortices compare our results with numerical simulations of the vlasov poisson system and show that the initial unstable equilibrium may evolve to different asymptotic states depending on the way it was perturbed

darboux transformation and explicit solutions for the integrable sixth order kdv equation for nonlinear waves abstract korteweg de vries kdv type equations can describe some physical phenomena in fluids nonlinear optics quantum mechanics plasmas etc in this paper with the aid of symbolic computation the integrable sixth order kdv equation is investigated darboux transformation dt with an arbitrary parameter is presented explicit solutions are derived with the dt relevant properties are graphically illustrated which might be helpful to understand some physical processes in fluids plasmas optics and quantum mechanics

application of spectral deformation to the vlasov poisson system ii mathematical results this paper presents a mathematical description of the linearized vlasov poisson operator lk acting on a family of banach spaces xp related to l p r and the application of the method of spectral deformation to this model it is shown that a type a analytic family of operators lk c lk 0 lk can be associated with lk by means of this family the landau damped modes of the plasma are identified as the spectral resonances of lk existence and uniqueness of solutions to the initial value problem for the evolution equation g lk g is proven an expansion of any solution to the initial value problem with sufficiently smooth initial data is obtained in terms of the eigenfunctions of lk and a spectral integral over the essential spectrum this is applied to derive an expansion for solutions to the vlasov equation in which the landau damped portions of the distribution function are manifestly exhibited a self contained discussion of the spectral deformation method and an extension of it to ce

a finite element code for the simulation of one dimensional vlasov plasmas i theory abstract the galerkin method is used to obtain a finite element solution to the vlasov poisson equations over the two dimensional x v phase plane using bilinear element shape functions a set of linear equations for updated values of the distribution function is obtained in which time dependent and time independent coefficients are separated to reduce the computation involved the noise levels are low and the energy conservation good the finite element approach offers certain advantages over alternative methods in applications to problems in plasma physics in particular the flexibility in using elements of different size allows problems in which regions of the plasma are characterized by often widely differing scale lengths to be treated efficiently

darboux transformation and solutions for an equation in 2 1 dimensions painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2 1 dimensions this procedure has allowed us to obtain the lax pair darboux transformation and functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively

asymptotically stable particle in cell methods for the vlasov poisson system with a strong external magnetic field this paper deals with the numerical resolution of the vlasov poisson system with a strong external magnetic field by particle in cell pic methods in this regime classical pic methods are subject to stability constraints on the time and space steps related to the small larmor radius and plasma frequency here we propose an asymptotic preserving pic scheme which is not subjected to these limitations our approach is based on first and higher order semi implicit numerical schemes already validated on dissipative systems s boscarino f filbet and g russo j sci comput 2016 doi 10 1007 s10915 016 0168 y additionally when the magnitude of the external magnetic field becomes large this method provides a consistent pic discretization of the guiding center equation that is an incompressible euler equation in vorticity form we propose several numerical experiments which provide a solid validation of the method and its underlying concepts

n fold darboux transformation and soliton solutions for a nonlinear dirac system from a spectral problem and corresponding lenard operator pairs we derive a dirac soliton hierarchy associated with a nonlinear dirac system a systematic method is proposed for constructing the n fold darboux transformation of the dirac system based on its lax pair as an application of darboux transformation explicit soliton solutions of the dirac system are given

a high order semi lagrangian discontinuous galerkin method for vlasov poisson simulations without operator splitting abstract in this paper we develop a high order semi lagrangian sl discontinuous galerkin dg method for nonlinear vlasov poisson vp simulations without operator splitting in particular we combine two recently developed novel techniques one is the high order non splitting sldg transport method cai et al 2017 4 and the other is the high order characteristics tracing technique proposed in qiu and russo 2017 29 the proposed method with up to third order accuracy in both space and time is locally mass conservative free of splitting error positivity preserving stable and robust for large time stepping size the sldg vp solver is applied to classic benchmark test problems such as landau damping and two stream instabilities for vp simulations efficiency and effectiveness of the proposed scheme is extensively tested tremendous cpu savings are shown by comparisons between the proposed sl dg scheme and the classical runge kutta dg method

conservative semi lagrangian schemes for vlasov equations conservative methods for the numerical solution of the vlasov equation are developed in the context of the one dimensional splitting in the case of constant advection these methods and the traditional semi lagrangian ones are proven to be equivalent but the conservative methods offer the possibility to add adequate filters in order to ensure the positivity in the non constant advection case they present an alternative to the traditional semi lagrangian schemes which can suffer from bad mass conservation in this time splitting setting

source decomposition as a diakoptic boundary condition in fdtd with reflecting external regions the diakoptic approach is utilized for the creation of boundary conditions for the truncation of the fdtd computational grid in the presence of reflecting external media the interfaces between the regions can be of arbitrary shape and are in close proximity to the scatterers in both regions surface green s functions over these interfaces serve to characterize the external domain from a viewpoint of the computational domain previously used field definitions of such green s functions that can be viewed as analogs of parameters in microwave networks have involved effective hard boundaries at the interfaces these boundaries however can cause artificial interactions with physical reflecting bodies in the external domain to eliminate this phenomenon the green s functions are now defined in terms of outgoing and incoming wave constituents rather than fields this distinction between the aforementioned constituents is made by viewing them as originating from sources within and outside the computational domain the green s function based boundary conditions are demonstrated to operate successfully in the two dimensional case resolving the aforementioned resonance issues efficiently and accurately

accuracy of the discrete green s function formulation of the fdtd method this paper reports an evaluation of the accuracy of the discrete green s function dgf formulation of the finite difference time domain fdtd method recently the closed form expression for the dgf and its efficient numerical implementation were presented which facilitates applications of the dgf in fdtd simulations of radiation and scattering problems so far the accuracy of the dgf formulation of the fdtd method has been rather marginally treated in the literature moreover although windowing has been reported as an efficient method of dgf waveform truncation and a remedy for stability issues the accuracy and usability of this technique have not yet been fully evaluated in this paper previously unrevealed accuracy limitations of the dgf formulation of the fdtd method are demonstrated in several numerical tests specifically the truncation errors are compared for the most frequently applied windowing functions with the best performance shown for the hann s window

an analytical expression for 3 d dyadic fdtd compatible green s function in infinite free space via z transform and partial difference operators fdtd solutions have their own properties distinct from the discrete samples of corresponding continuous wave solutions thus the discrete equivalent to the green s function is needed for applications like the one using a hybrid absorbing boundary condition which couples the fdtd algorithm with integral operators for nonconvex scatterers in this paper we propose a new closed form expression for the 3 d dyadic fdtd compatible green s function in infinite free space via a novel approach with the ordinary z transform along with the spatial partial difference operators the final expression involves a summation of standing wave modes with time varying coefficients the propagation of waves in the yee s grid can be interpreted by the selective property of the time varying coefficients which is very different from the conventional concept of a traveling wave the traditional dispersion analysis using plane waves for the fdtd algorithm in a source free region may not be applicable to explain the wave propagation phenomenon through our analytic expression because the corresponding z transform diverges for z on the unit circle

darboux transformation and soliton solutions for the boiti pempinelli tu bpt hierarchy starting from a spectral problem we derive the well known boiti pempinelli tu bpt hierarchy an explicit and universal darboux transformation for the whole hierarchy is constructed the soliton solutions for the bpt hierarchy are obtained by applying the darboux transformation

a parallel deterministic solver for the schrodinger poisson boltzmann system in ultra short dg mosfets comparison with monte carlo abstract we present a parallel deterministic solver for the boltzmann schrodinger poisson system for partially confined dg mosfets our 2d model uses a dimensional coupling the electrons behave as particles along the longitudinal dimension and as waves along the transverse dimension in ben abdallah et al 2009 the authors validated in a simplified setting the solver for the eigenproblems in this paper we have focused on obtaining physically accurate results by introducing a realistic si conduction band and electron phonon interactions we have switched from cartesian to elliptic variables for the wave vector and consequently radically rebuilt the time integrator the code has been parallelized following a domain decomposition approach the results have been compared to those given by a monte carlo solver

a supertrace identity and its applications to superintegrable systems a supertrace identity on lie superalgebras is established it provides a tool for constructing super hamiltonian structures of zero curvature equations associated with lie superalgebras applications in the case of the lie superalgebra b 0 1 present super hamiltonian structures of a super akns soliton hierarchy and a super dirac soliton hierarchy

a splitting algorithm for vlasov simulation with filamentation filtration a fourier fourier transformed version of the splitting algorithm for simulating solutions of the vlasov poisson system of equations is introduced it is shown that with the inclusion of filamentation filtration in this transformed algorithm it is both faster and more stable than the standard splitting algorithm it is further shown that in a scalar computer environment this new algorithm is approximately equal in speed and far less noisy than its particle in cell counterpart it is conjectured that in a multiprocessor environment the filtered splitting algorithm would be faster while producing more precise results

evaluation of the green s function for the mixed potential integral equation mpie method in the time domain for layered media in the use of the time domain integral equation tdie method for the analysis of layered media it is important to have the time domain layered medium green s function computed for many source to field distances spl rho and time instants t in this paper a numerical method is used that computes the mixed potential green s functions g sub v spl rho t and g sub a spl rho t for a multilayered medium for many spl rho s and t s simultaneously the method is applicable to multilayered media and for lossless or lossy dispersive media salient features of the method are 1 the use of complex spl omega so that the surface wave poles are lifted off the real k sub spl rho axis such that pole extractions are not required 2 the use of half space extraction so that the integrand for the sommerfeld integral decays exponentially along the k sub spl rho axis to obtain fast convergence of the integral and 3 the use of the fast hankel transform so that the green s function is calculated for many values of spl rho simultaneously for a four layer medium we illustrate the numerical results by a three dimensional plot of spl rho g sub v spl rho t versus spl rho and t and demonstrate the space time evolution of these green s functions for a maximum frequency range of 8 ghz the method requires only a few cpu minutes to compute a table of 100 points in spl rho spl times 168 points in t uniformly spaced values of g sub v spl rho t on an 867 mhz pentium pc

solving hyperbolic elliptic problems on singular mapped disk like domains with the method of characteristics and spline finite elements abstract a common strategy in the numerical solution of partial differential equations is to define a uniform discretization of a tensor product multi dimensional logical domain which is mapped to a physical domain through a given coordinate transformation by extending this concept to a multi patch setting simple and efficient numerical algorithms can be employed on relatively complex geometries the main drawback of such an approach is the inherent difficulty in dealing with singularities of the coordinate transformation this work suggests a comprehensive numerical strategy for the common situation of disk like domains with a singularity at a unique pole where one edge of the rectangular logical domain collapses to one point of the physical domain for example a circle we present robust numerical methods for the solution of vlasov like hyperbolic equations coupled to poisson like elliptic equations in such geometries we describe a semi lagrangian advection solver that employs a novel set of coordinates named pseudo cartesian coordinates to integrate the characteristic equations in the whole domain including the pole and a finite element elliptic solver based on globally c 1 smooth splines toshniwal et al 2017 27 the two solvers are tested both independently and on a coupled model namely the 2d guiding center model for magnetized plasmas equivalent to a vorticity model for incompressible inviscid euler fluids the numerical methods presented show high order convergence in the space discretization parameters uniformly across the computational domain without effects of order reduction due to the singularity dedicated tests show that the numerical techniques described can be applied straightforwardly also in the presence of point charges equivalently point like vortices within the context of particle in cell methods

flux corrected transport this paper describes a class of explicit eulerian finite difference algorithms for solving the continuity equation which are built around a technique called flux correction these flux corrected transport fct algorithms are of indeterminate order but yield realistic accurate results in addition to the mass conserving property of most conventional algorithms the fct algorithms strictly maintain the positivity of actual mass densities so steep gradients and inviscid shocks are handled particularly well this first paper concentrates on a simple one dimensional version of fct utilizing shasta a new transport algorithm for the continuity equation which is described in detail

the time domain discrete green s function method gfm characterizing the fdtd grid boundary for a given fdtd simulation space with an arbitrarily shaped boundary and an arbitrary exterior region most existing absorbing boundary conditions become inapplicable a green s function method gfm is presented which accommodates arbitrarily shaped boundaries in close proximity to a scattering object and an arbitrary composition in the exterior of the simulation space central to this method is the numerical precomputation of a green s function tailored to each problem which represents the effects of the boundary and the external region this function becomes the kernel for a single layer absorbing boundary operator it is formulated in a manner which naturally incorporates numerically induced effects such as the numerical dispersion associated with the fdtd scheme the green s function is an exact absorber in the discretized space this property should be contrasted with other methods which are initially designed for the continuum and are subsequently discretized thereby incurring inherent errors in the discrete space which cannot be eliminated unless the continuum limit is recovered in terms of accuracy the gfm results have been shown to be of a similar quality to the pml and decidedly superior to the mur 1981 condition the properties of the gfm are substantiated by a number of numerical examples in one two and three dimensions

convergence of classes of high order semi lagrangian schemes for the vlasov poisson system abstract in this paper we present some classes of high order semi lagran gian schemes for solving the periodic one dimensional vlasov poisson system in phase space on uniform grids we prove that the distribution function f t x v and the electric field e t x converge in the l 2 norm with a rate of displaystyle mathcal o left delta t 2 h m 1 frac h m 1 delta t right where m is the degree of the polynomial reconstruction and delta t and h are respectively the time and the phase space discretization parameters

inverse scattering on the line for a dirac system the whole line version of the gelfand levitan marchenko glm equation for a dirac system is studied a new derivation of the glm equation is given under weaker hypotheses than frolov s earlier treatment sov math dokl 13 1468 1972 and the complete inversion is carried out in some explicit cases in which a spectral gap is present previous calculations of this type are restricted either to a scalar potential or degenerate gap applications are discussed in connection with optical couplers and soliton equations

symplectic realizations and action angle coordinates for the lie poisson system of dirac hierarchy abstract in this paper the relations among four finite dimensional hamiltonian systems associated with the dirac hierarchy are studied via three poisson maps from the standard symplectic structures to the lie poisson structure of lie algebra sl 2 it has shown that the canonical hamiltonian systems in the standard symplectic structures are the different symplectic realizations of the lie poisson hamiltonian system the action angle coordinates and the jacobi inversion problem for the lie poisson hamiltonian systems are also investigated in detail

a multidimensional z transform evaluation of the discrete finite difference time domain green s function analytical expressions for frequency and time domain discrete green s functions are developed below rather than discretizing the continuous green s functions directly we derive the discrete green s functions from first principles i e the difference equations this is done with the purpose of obtaining discrete integral operators that would replicate results obtained via the finite difference time domain fdtd method the main effort is directed at inversions of multidimensional frequency domain expressions in the z transform domain the green s functions obtained in this way coincide with time domain developments presented earlier both in the form of direct numerical computations and via combinatorics and the catalan triangle the z transform methodology resulting from this development provides an orderly manner for deriving expressions for multidimensional discrete integral operators that can be hybridized with fdtd based differential operators in a self consistent manner