existence of efficient and properly efficient solutions to problems of constrained vector optimization the paper is devoted to the existence of global optimal solutions for a general class of nonsmooth problems of constrained vector optimization without boundedness assumptions on constraint set the main attention is paid to the two major notions of optimality in vector problems pareto efficiency and proper efficiency in the sense of geoffrion employing adequate tools of variational analysis and generalized differentiation we first establish relationships between the notions of properness m tameness and the palais smale conditions formulated for the restriction of the vector cost mapping on the constraint set these results are instrumental to derive verifiable necessary and sufficient conditions for the existence of pareto efficient solutions in vector optimization furthermore the developed approach allows us to obtain new sufficient conditions for the existence of geoffrion properly efficient solutions to such constrained vector problems

zero stability properties of the three ordinate variable stepsize variable formula methods consider the systemy f x y y a x a b y r n wheref is continuous and lipschitzian with respect to the second argument very often linear multistep variable stepsize variable formula methods lm vsvfm s are used to computey k y xk on the points of the grid a x 0 x1 x2 xn b the general lm vsvfm is based on formulae of the following type y k sum limits i 1 s k alpha i bar h k sk y k i sum limits i 0 s k h k i beta i bar h k s k f x k i y k i whereh k xk xk 1 bar h k h k h k 1 ldots h k s k s k k k 1 1 n the coefficients i and i depend on the lasts k 1 stepsizes and on the formula used at stepk only the zero stability properties of some special classes of lm vsvfm s as for example those based on adams formulae were investigated in the literature a class of three ordinate lm vsvfm s is defined in this paper some results concerning the zero stability properties of these methods are proved it is shown that some well known results are simple corollaries of the results found for the three ordinate lm vsvfm s it is easily seen that similar results hold for the corresponding one leg vsvfm s finally the use of the theoretical results in the practical implementations is briefly discussed

clarke coderivatives of efficient point multifunctions in parametric vector optimization abstract the paper is devoted to the study of the clarke circatangent coderivatives of the efficient point multifunction of parametric vector optimization problems in banach spaces we provide inner outer estimates for evaluating the clarke circatangent coderivative of this multifunction in a broad class of conventional vector optimization problems in the presence of geometrical operator and finite and infinite functional constraints examples are given for analyzing and illustrating the obtained results

high order compact computation and nonuniform grids for streamfunction vorticity equations abstract fourth order compact difference schemes for steady streamfunction vorticity formulation of 2d incompressible navier stokes equations on nonuniform grids are derived using maple software package especially we deduce fourth order compact difference scheme of the first partial derivative terms in order to resolve boundary layers grid transformation techniques are used which maps a nonuniform grid onto a uniform one for use with the fourth order compact difference scheme a krylov subspace iterative method with an ilut preconditioning technique is employed to solve the resulting linear system the proposed high accuracy computation method is applied to two model problems computational results are compared with results computed by other schemes in the literature

unconditional stability of alternating difference schemes with intrinsic parallelism for the fourth order parabolic equation in this paper the difference method with intrinsic parallelism for the fourth order parabolic equation is studied the general alternating difference schemes are constructed and proved to be unconditionally stable the alternating group explicit scheme alternating segment explicit implicit scheme and alternating segment crank nicolson scheme are the special cases of the general schemes constructed here numerical results are presented to examine the convergence stability and accuracy of the general schemes

compact h4 finite difference approximations to operators of navier stokes type a method of obtaining compact finite difference approximations of h4 accuracy to operators of navier stokes type is considered the basic procedure is developed for operators in one space dimension and subsequently applied to problems in more space dimensions and in time four illustrative numerical examples are given which indicate clearly in various cases that excellent accuracy may be obtained using the methods comparisons with previous results and with the results of h2 accurate computations are made

a new family of 5 5 cc 4oc schemes applicable for unsteady navier stokes equations in this paper a new family of implicit compact finite difference schemes for computation of unsteady convection diffusion equation with variable convection coefficient is proposed the schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady navier stokes system the proposed schemes where transport variable and its first derivatives are carried as the unknowns combine virtues of compact discretization and pade scheme for spatial derivative these schemes which are based on a five point stencil with constant coefficients named as 5 5 constant coefficient 4th order compact 5 5 cc 4oc give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods these schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable they efficiently capture both transient and steady solutions of linear and nonlinear convection diffusion equations with dirichlet as well as neumann boundary conditions subsequently the proposed schemes are applied to problems governed by the incompressible navier stokes equations the results obtained are in excellent agreement with analytical and available numerical results in all cases establishing efficiency and accuracy of the proposed schemes

an efficient scheme for solving steady incompressible navier stokes equations abstract the steady incompressible navier stokes equations in a 2d driven cavity are solved in primitive variables by means of the multigrid method the pressure and the components of the velocity are discretized on staggered grids a block implicit relaxation technique is used to achieve a good convergence and a simplified fmg fas algorithm is proposed special focus on the finite differences scheme used to approach the convection terms is made and a large discussion with other schemes is given results in a square driven cavity are obtained for reynolds numbers as high as 15 000 on fine uniform meshes and the solution is in good agreement with other studies for re 5000 the secondary vortices are very well represented showing the robustness of the method for reynolds numbers higher than 5000 the loss of stability for the steady solution is discussed moreover some computations on a rectangular cavity of aspect ratio equal to two are presented in addition the method is very efficient as far as cpu time is concerned for instance the solution for re 1000 on a 128 128 grid is obtained within 24 s on a siemens vp 200

stability restrictions on time stepsize for numerical integration of first order partial differential equations mathematical models described by partial differential equations appear often when different phenomena related to atmospheric transport for example advection and diffusion processes are studied truncated fourier series can be used in the discretization of the space derivatives resulting in a pseudospectral method in this way the partial differential equation is transformed into a system of ordinary differential equations this system is normally solved by so called step by step integration methods the stability properties of these methods are studied some bounds for the time stepsize are derived several formulae and predictor corrector schemes with large stability intervals on the imaginary axis are constructed numerical experiments with eight time integration algorithms are carried out some recommendations concerning the choice of the time integration algorithms in three different situations are given it is emphasized that in a package for the solution of partial differential equations describing transport processes in the atmospheric environment one should have several time integration algorithms each of which can be chosen optionally finally the possibility of using variable stepsize variable formula time integration algorithms is briefly discussed

implementation of a variable stepsize variable formula method in the time integration part of a code for treatment of long range transport of air pollutants abstract a mathematical model consisting of two partial differential equations is used to study the long range transport of sulphur di oxide and sulphate over europe the discretization of the first order space derivatives the advection terms is carried out by a pseudospectral fourier algorithm a special technique is applied in the discretization of the second order space derivatives the diffusion terms two large systems of ordinary differential equations are solved at each time step it is shown that these systems can efficiently be treated by a variable stepsize variable formula method vsvfm based on the use of predictor corrector schemes the stepsize selection strategy and the formula selection strategy are discussed in detail an attempt to carry out both an accuracy control and a stability control is made at each time step the great efficiency of the vsvfm implemented in our software as well as the reliability of the results are illustrated by numerical experiments in which real meteorological data for 1979 at the grid points of a space domain covering the whole of europe were used the main ideas implemented in the time integration part might be applied in many other situations where the systems of ordinary differential equations arising after the space discretization are only moderately stiff so that the stability requirements are dominant over the accuracy requirements on a large part of the time interval but the use of implicit time integration algorithms that require solving systems of algebraic equations at each time step is not justified as an illustration only it should be mentioned that such an application has been carried out in connection with models describing long range transport of nitrogen pollutants over europe

probabilistic low rank factorization accelerates tensor network simulations of critical quantum many body ground states we provide evidence that randomized low rank factorization is a powerful tool for the determination of the ground state properties of low dimensional lattice hamiltonians through tensor network techniques in particular we show that randomized matrix factorization outperforms truncated singular value decomposition based on state of the art deterministic routines in time evolving block decimation tebd and density matrix renormalization group dmrg style simulations even when the system under study gets close to a phase transition we report linear speedups in the bond or local dimension of up to 24 times in quasi two dimensional cylindrical systems

three level fourier spectral approximations for fluid flow with low mach number fourier spectral approximation combined with a second order time differencing technique for fluid flow with low mach number is considered in this paper its generalized stability and convergence are strictly proven

a numerical method for two dimensional schrodinger equation using collocation and radial basis functions in this paper we propose a numerical scheme to solve the two dimensional 2d time dependent schrodinger equation using collocation points and approximating the solution using multiquadrics mq and the thin plate splines tps radial basis function rbf the scheme works in a similar fashion as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme

new method based on the hpm and rkhsm for solving forced duffing equations with integral boundary conditions this paper investigates the forced duffing equation with integral boundary conditions its approximate solution is developed by combining the homotopy perturbation method hpm and the reproducing kernel hilbert space method rkhsm hpm is based on the use of the traditional perturbation method and the homotopy technique the hpm can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases rkhsm is also an analytical technique which can solve powerfully linear boundary value problems therefore the forced duffing equation with integral boundary conditions can be solved using advantages of these two methods two numerical examples are presented to illustrate the strength of the method

a new paradigm for solving navier stokes equations streamfunction velocity formulation in this paper we propose a new paradigm for solving navier stokes equations the proposed methodology is based on a streamfunction velocity formulation of the two dimensional steady state navier stokes equations representing incompressible fluid flows in two dimensional domains similar formulations are also possible for three dimensional fluid flows the main advantage of our formulation is that it avoids the difficulties associated with the computation of vorticity values especially on solid boundaries encountered when solving the streamfunction vorticity formulations our formulation also avoids the difficulties associated with solving pressure equations of the conventional velocity pressure formulations of the navier stokes equations we describe the new formulation of the navier stokes equations and use this formulation to solve a couple of fluid flow problems we use a biconjugate gradient method to obtain the numerical solutions of the fluid flow problems and provide detailed comparison data for the lid driven cavity flow problem it is discovered that our new formulation successfully provides high accuracy solutions for the benchmark problem in addition we also solve a problem of flow in a rectangular cavity with aspect ratio 2 and compare our results qualitatively and quantitatively with numerical and experimental results available in the literature in all cases we obtain high accuracy solutions with little additional cost

single cell discretizations of order two and four for biharmonic problems abstract new difference formulas are derived for solving the biharmonic problem in two dimensions over a rectangular domain these methods use only the nine grid points of a single mesh cell and do not require fictitious points in order to approximate the boundary conditions derivatives of the solution are obtained as a by product of the methods second order formulas are derived for both the first and second biharmonic problems in numerical experiments the new second order formulas compare favourably with the standard second order methods extensions to fourth order formulas are given the method of deriving these formulas can be used to derive similar formulas for arbitrarily shaped regions

variants of the ekeland variational principle for a set valued map involving the clarke normal cone in this paper we present two set valued variants of the ekeland variational principle involving the clarke normal cone and establish sufficient conditions for a set valued map to have a weak minimizer or a properly positive minimizer when it satisfies palais smale type conditions 2005 elsevier inc all rights reserved

nonperturbative treatment of non markovian dynamics of open quantum systems we identify the conditions that guarantee equivalence of the reduced dynamics of an open quantum system oqs for two different types of environments one a continuous bosonic environment leading to a unitary system environment evolution and the other a discrete mode bosonic environment resulting in a system mode nonunitary lindbladian evolution assuming initial gaussian states for the environments we prove that the two oqs dynamics are equivalent if both the expectation values and two time correlation functions of the environmental interaction operators are the same at all times for the two configurations since the numerical and analytical description of a discrete mode environment undergoing a lindbladian evolution is significantly more efficient than that of a continuous bosonic environment in a unitary evolution our result represents a powerful nonperturbative tool to describe complex and possibly highly non markovian dynamics as a special application we recover and generalize the well known pseudomodes approach to open system dynamics

a detailed numerical treatment of the boundary conditions imposed by the skull on a diffusion reaction model of glioma tumor growth clinical validation aspects abstract the study of the diffusive behavior of glioma tumor growth is an active field of biomedical research with considerable therapeutic implications an important aspect of the corresponding computational problem is the mathematical handling of boundary conditions this paper aims at providing an explicit and thorough numerical formulation of the adiabatic neumann boundary conditions imposed by the skull on the diffusive growth of gliomas and in particular on glioblastoma multiforme gbm additionally a detailed exposition of the numerical solution process for a homogeneous approximation of glioma invasion using the crank nicolson technique in conjunction with the conjugate gradient system solver is provided the entire mathematical and numerical treatment is also in principle applicable to mathematically similar physical chemical and biological phenomena a comparison of the numerical solution for the special case of pure diffusion in the absence of boundary conditions or equivalently in the presence of adiabatic boundaries placed in infinity with its analytical counterpart is presented numerical simulations for various adiabatic boundary geometries and non zero net tumor growth rate support the validity of the corresponding mathematical treatment through numerical experimentation on a set of real brain imaging data a simulated tumor has shown to satisfy the expected macroscopic behavior of glioblastoma multiforme including the adiabatic behavior of the skull the paper concludes with a number of remarks pertaining to both the biological problem addressed and the more generic diffusion reaction context

improved scaling of time evolving block decimation algorithm through reduced rank randomized singular value decomposition when the amount of entanglement in a quantum system is limited the relevant dynamics of the system is restricted to a very small part of the state space when restricted to this subspace the description of the system becomes efficient in the system size a class of algorithms exemplified by the time evolving block decimation tebd algorithm make use of this observation by selecting the relevant subspace through a decimation technique relying on the singular value decomposition svd in these algorithms the complexity of each time evolution step is dominated by the svd here we show that by applying a randomized version of the svd routine rrsvd the power law governing the computational complexity of tebd is lowered by one degree resulting in a considerable speed up we exemplify the potential gains in efficiency at the hand of some real world examples to which tebd can be successfully applied and demonstrate that for those systems rrsvd delivers results as accurate as state of the art deterministic svd routines

on the existence of pareto solutions for polynomial vector optimization problems we are interested in the existence of pareto solutions to the vector optimization problem begin aligned mathrm min mathbb r m f x x in mathbb r n end aligned where f mathbb r n rightarrow mathbb r m is a polynomial map by using the tangency variety of f we first construct a semi algebraic set of dimension at most m 1 containing the set of pareto values of the problem then we establish connections between the palais smale conditions m tameness and properness for the map f based on these results we provide some sufficient conditions for the existence of pareto solutions of the problem we also introduce a generic class of polynomial vector optimization problems having at least one pareto solution

a boundary integral method for the three dimensional heat equation subject to specification of energy a boundary integral method is proposed for the numerical solution of the three dimensional heat equation subject to specification of energy a specific test problem is solved using the method

fractional step methods for parabolic equations with a non standard condition the number of physical phenomena modelled by partial differential equations involving non linear integral terms is constantly increasing these terms may appear in the boundary conditions or in the governing equation itself in the present report the integral term appears in the boundary conditions an alternating direction implicit finite difference technique which requires the solution of three tridiagonal sets of equations at each time step is derived for solving the three dimensional diffusion equation the proposed scheme is computationally efficient and easy to implement the new developed method is tested on a problem the results of numerical testing show that the numerical method based on the finite difference technique discussed in the present paper produce good results

an inverse problem of finding a parameter in a semi linear heat equation we consider the following inverse problem of finding the pair u p which satisfies the following u1 uxx p t u f x t u ux p t 0 x 1 0 t t u x 0 u0 x 0 x 1 ux 0 t f t ux 1 t g t 0 t t and 01 x t u x t dx e t 0 t t where u0 f g f and e are known functions the existence uniqueness regularity and the continuous dependence of the solution upon the data are demonstrated

the ekeland variational principle for set valued maps involving coderivatives in this paper we use the frechet clarke and mordukhovich coderivatives to obtain variants of the ekeland variational principle for a set valued map f and establish optimality conditions for set valued optimization problems our technique is based on scalarization with the help of a marginal function associated with f and estimates of subdifferentials of this function in terms of coderivatives of f

fully implicit finite differences methods for two dimensional diffusion with a non local boundary condition three new fully implicit methods which are based on the 5 5 crank nicolson method the 5 5 n h noye hayman implicit method and the 9 9 n h implicit method are developed for solving the heat equation in two dimensional space with non local boundary conditions the latter is fourth order while the others are second order while the implicit methods developed here like the scheme based on the standard implicit backward time centered space btcs method use a large amount of central processor cpu time the high accuracy of the new fourth order fully implicit scheme is significant like the btcs method the new methods are also unconditionally stable

the ekeland variational principle for henig proper minimizers and super minimizers abstract in this paper we consider for the first time approximate henig proper minimizers and approximate super minimizers of a set valued map f with values in a partially ordered vector space and formulate two versions of the ekeland variational principle for these points involving coderivatives in the sense of ioffe clarke and mordukhovich as applications we obtain sufficient conditions for f to have a henig proper minimizer or a super minimizer under the palais smale type conditions the techniques are essentially based on the characterizations of henig proper efficient points and super efficient points by mean of the henig dilating cones and the hiriart urruty signed distance function

quasi implicit and two level explicit finite difference procedures for solving the one dimensional advection equation a variety of explicit and implicit algorithms has been studied dealing with the solution of the one dimensional advection equation these schemes are based on the weighted finite difference approximations this solution approach proves to be very effective this procedure is carried out in many disciplines including hydrology oceanography and meteorology the main idea behind the finite difference methods for obtaining the solution of a given partial differential equation is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points the most usual way to generate these approximations is through the use of taylor series the basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach it is worth noting that from the truncation error of the modified equivalent equation it is possible to eliminate the dominant error terms associated with the finite difference equations that contain weights thus leading to more accurate techniques quasi implicit and two level explicit finite difference techniques are considered in this article to approximately solve the one dimensional advection problem stability of the techniques are investigated using the von neumann stability analysis the results of a numerical experiment are presented and the accuracy and central processor unit cpu time needed are discussed and compared numerical tests show that the new third order or fourth order methods are more accurate than either of the second order techniques

mathematical model for studying the sulphur pollution over europe the long range transport of sulphur pollutants in the atmosphere is an important environmental problem a mathematical model consisting of a system of partial differential equations has been developed at the danish agency of environmental protection for studying this phenomenon the model is handled numerically a pseudospectral fourier algorithm is used in the space discretization phase a variable stepsize variable formula method vsvfm based on predictor corrector schemes may be specified during the time discretization part the time integration formulae are especially constructed so that a the predictor corrector schemes have good absolute stability properties on the imaginary axis and b the fundamental properties consistency zero stability and convergence of the global vsvfm are not affected by the changes of the stepsize and or the predictor corrector scheme both the accuracy requirements and the stability requirements are taken into account in the stepsize selection strategy as well as in the formula selection strategy this double check leads to a great efficiency in the numerical treatment of problems arising in atmospheric environments this is demonstrated by many numerical results the question of the reliability of the results obtained in runs of real physical problems is discussed some results obtained in the comparison of calculated results with measurements are reported an illustration where the model is applied in studying the air pollution in the area around denmark due to danish sources is presented

a finite element approach for finding positive solutions of a logistic equation with a sign changing weight function in this article we describe the mathematical modeling and use finite element approximation to study the existence and multiplicity of numerical positive solutions of well known generalized logistic equation with sign changing weight function we investigate the range of the real parameter involved in this problem to achieve the numerical solutions and discuss the behavior of the branch of the solutions using matlab 7 0

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

integral equation formulation of an unsteady diffusionconvection equation with variable coefficient and velocity in this paper we present an integral equation formulation for the time dependent diffusionconvection equation with variable coefficient and velocity with sources the formulation is based on usage of the steady fundamental solution of the convectiondiffusion equation for a known velocity and coefficient fields which may change with location and time the formulation avoids the usage of the gradient of the unknown field function and thus avoids making the problem nonlinear two discretization approaches are proposed and compared a standard single domain boundarydomain element technique and a domain decomposition approach the validity of the formulation and comparison of discretization approaches is preformed on several challenging test cases mesh convergence is reported and the advantages and disadvantages of both approaches are examined

a high order scheme for unsteady heat conduction equations abstract in this work a high order scheme is proposed for solving the heat conduction problems in this scheme the governing equation of heat conduction is firstly written into a system of first order partial differential equations which are integrated over the control volumes around each node point and a finite time step the resulting time integrals are approximated by numerical quadrature with an undetermined weighting parameter the error analysis allows us to determine the parameter and the ratio of the time step to the square of grid spacing by making the error of the scheme as small as possible both theoretical and numerical results show that the proposed high order scheme can achieve at least sixth order accuracy at the expense of the same computing cost as the traditional finite volume method and the high order scheme demonstrates a better performance than the high order compact finite difference method and the runge kutta discontinuous galerkin method

a numerical method for finding positive solution of logistic equation abstract using a numerical methods based on sub super solution we will show the existence of positive solution for the well known logistic equation u g x u x u x 2 for x with dirichlet boundary condition

numerical solution of a parabolic equation with non local boundary specifications the parabolic partial differential equations with non local boundary specifications model various physical problems numerical schemes are developed for obtaining approximate solutions to the initial boundary value problem for one dimensional second order linear parabolic partial differential equation with non local boundary specifications replacing boundary conditions the method of lines semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the spatial derivative in the pde is approximated by a finite difference approximation the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the new algorithms are tested on two problems from the literature the central processor unit times needed are also considered

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

a fourth order compact adi method for solving two dimensional unsteady convection diffusion problems n this article an exponential high order compact ehoc alternating direction implicit adi method in which the crank nicolson scheme is used for the time discretization and an exponential fourth order compact difference formula for the steady state 1d convection diffusion problem is used for the spatial discretization is presented for the solution of the unsteady 2d convection diffusion problems the method is temporally second order accurate and spatially fourth order accurate which requires only a regular five point 2d stencil similar to that in the standard second order methods the resulting ehoc adi scheme in each adi solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal gaussian decomposition and may also be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time the unconditionally stable character of the method was verified by means of the discrete fourier or von neumann analysis numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order adi method of karaa and zhang and the spatial third order compact scheme of note and tan

the operational matrices of bernstein polynomials for solving the parabolic equation subject to specification of the mass abstract some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications in this paper a numerical method which employs the bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions the properties of bernstein polynomials and the operational matrices for integration differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the new technique

a reliable numerical algorithm for fractional advection dispersion equation arising in contaminant transport through porous media abstract this article presents a reliable numerical approach for the fractional advection dispersion equation by making use of legendre scaling functions as a basis the fractional advection dispersion equation describes the anomalous transport in surface and subsurface water using two dimensional basis formed by legendre scaling functions we get operational matrix for fractional integrations and differentiations substituting these operational matrices in the equation leads linear algebraic equations whose solutions can be derived with the aid of sylvester s approach this in turn yields approximate solutions for advection dispersion equation convergence analysis of the proposed scheme is presented the potency and accuracy of the proposed numerical algorithm are shown by plotting error figures

parallel techniques for a boundary value problem with non classic boundary conditions new parallel techniques are developed for obtaining approximate solutions to an initial boundary value problem for the three dimensional parabolic partial differential equation pde with non classic boundary conditions while sharing some common features with the one dimensional models the solution of three dimensional equations are substantially more difficult thus some considerations are taken to be able to extend some ideas of one dimensional case using a suitable transformation the solution of this problem is equivalent to the solution of two other problems the former which is a one dimensional non classic boundary value problem giving the value of m through using an unconditionally stable explicit scheme using this result the second problem will be changed to a classical three dimensional diffusion equation with neumann s boundary conditions which will be solved numerically by using the method of lines semi discretization approach the space derivatives in the pde are approximated by finite difference replacements the solution of the resulting system of first order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function the accuracy in time is controlled by choosing several subdiagonal pade approximants to replace this matrix exponential term numerical techniques are developed to compute the required solution using a splitting method leading to algorithms for sequential and parallel implementation the algorithms are tested on a model problem and the resulting numerical experiments are also presented the central processor unit times needed are also reported and are compared

a high order composite scheme for the second order elliptic problem with nonlocal boundary and its fast algorithm the elliptic problem with nonlocal boundary condition is widely applied in the field of science and engineering firstly we construct a linear finite element scheme for the nonlocal boundary problem and derive the optimal l 2 error estimate then based on the quadratic finite element and the extrapolation linear finite element methods we present a composite scheme and prove that it is convergent order three furthermore we design an upper triangular preconditioning algorithm for the linear finite element discrete system finally numerical results not only validate that the new algorithm is efficient but also show that the new scheme is convergent order three furthermore order four on uniform grids

a compact multigrid solver for convection diffusion equations we introduce a multigrid algorithm to solve the convection diffusion equations using a nine point compact difference scheme we test the efficiency of the algorithm with various smoothers and intergrid transfer operators the algorithm displays a grid independent convergence rate and produces solutions with high accuracy numerical results are presented to validate the conclusions

numerical treatment of large scale air pollution models abstract studying air pollution phenomena over a large space domain by a fairly general mathematical model is discussed the space discretization of such a model leads to huge system of odes the integration algorithm used in the solution of these systems must be efficient if a 3 d model is considered then a vector processor is to be used moreover the most time consuming parts of the code are to be vectorized the calculated results concentrations of different air pollutants must be reliable because they have to be used by other specialists therefore different checks concerningthe reliability of the results are carried out in this paper it is shown that an efficient and reliable code for studying both sulphur and nitrogen pollution has been developed at the danish agency of environmental protection some simulation processes are also described breifly

degrees of efficiency and degrees of minimality in this work we characterize different types of solutions of a vector optimization problem by means of a scalarization procedure usually different scalarizing functions are used in order to obtain the various solutions of the vector problem here we consider different kinds of solutions of the same scalarized problem our results allow us to establish a parallelism between the solutions of the scalarized problem and the various efficient frontiers stronger solution concepts of the scalar problem correspond to more restrictive notions of efficiency besides the usual notions of weakly efficient and efficient points which are characterized as global and strict global solutions of the scalarized problem we also consider some restricted notions of efficiency such as strict and proper efficiency which are characterized as tikhonov well posed minima and sharp minima for the scalarized problem

an integral constrained parabolic problem with applications in thermochronology we investigate an inverse source problem with an integral constraint for a parabolic equation the constraint is motivated by an application in thermochronology a branch of geology the existence and uniqueness of weak solutions are established by means of the rothe method and an energy method respectively the elliptic problem resulting from the time discretization is solved by homogenizing the integral constraint the implicit scheme used in the proof of existence lends itself readily for numerical studies and we present the results of numerical experiments we also report on the errors and convergence rates

short note a high order pade adi method for unsteady convection diffusion equations a high order alternating direction implicit adi method for computations of unsteady convection diffusion equations is proposed by using fourth order pade schemes for spatial derivatives the present scheme is fourth order accurate in space and second order accurate in time the solution procedure consists of a number of tridiagonal matrix operations which make the computation cost effective the method is unconditionally stable and shows higher accuracy and better phase and amplitude error characteristics than the standard second order adi method d w peaceman h h rachford jr the numerical solution of parabolic and elliptic differential equations journal of the society of industrial and applied mathematics 3 1959 28 41 and the fourth order adi scheme of karaa and zhang high order adi method for solving unsteady convection diffusion problem journal of computational physics 198 2004 1 9

a crank nicolson adi galerkin legendre spectral method for the two dimensional riesz space distributed order advection diffusion equation abstract in the paper a crank nicolson alternating direction implicit adi galerkin legendre spectral scheme is presented for the two dimensional riesz space distributed order advection diffusion equation the gauss quadrature has a higher computational accuracy than the mid point quadrature rule which is proposed to approximate the distributed order riesz space derivative so that the considered equation is transformed into a multi term fractional equation moreover the transformed equation is solved by discretizing in space by the adi galerkin legendre spectral scheme and in time using the crank nicolson difference method stability and convergence analysis are verified for the numerical approximation a lot of numerical results are demonstrated to justify the theoretical analysis

quartic and quintic b spline methods for advection diffusion equation differential quadrature methods based on b spline functions of degree four and five have been introduced to solve advection diffusion equation numerically two initial boundary value problems modeling the transportation of a concentration and distribution of an initial pulse are simulated using both methods the errors of the numerical results obtained by both methods have been computed stability analysis for both methods is also studied by the use of matrix stability

the method of approximate particular solutions for the time fractional diffusion equation with a non local boundary condition in this paper we consider the numerical solution of the time fractional diffusion equation with a non local boundary condition the method of approximate particular solutions maps using multiquadric radial basis function mq rbf is employed for this equation due to the accuracy of the mq based meshless methods is severely influenced by the shape parameter we adopt a leave one out cross validation loocv algorithm proposed by rippa 34 to enhance the performance of the maps the numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time fractional diffusion equation with a non local boundary condition

a galerkin procedure for the diffusion equation subject to the specification of mass continuous and discrete galerkin procedures are derived and analyzed for the numerical solution of the diffusion equation u 1 u xx 0 x 1 0 t leqq t u x 0 f x u x 1 t g t and the specification of the mass int 0 b u x t dx m t 0 b 1 some numerical experiments are also presented

numerical solution of the three dimensional advection diffusion equation the study of advection diffusion equation continues to be an active field of research the subject has important applications to fluid dynamics as well as many other branches of science and engineering in this paper several different numerical techniques will be developed and compared for solving the three dimensional advection diffusion equation with constant coefficient these techniques are based on the two level fully explicit and fully implicit finite difference approximations the basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyett this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes another nice feature of the modified equivalent partial differential equation approach is that a high order of accuracy can be combined with excellent stability properties the new second order accurate methods are free of numerical diffusion the results of a numerical experiment are presented and the accuracy and central processor cpu time needed are discussed and compared

numerical techniques for a parabolic equation subject to an overspecified boundary condition an inverse problem concerning diffusion equation with a control parameter is considered a new numerical scheme is developed for obtaining approximate solutions to an initial boundary value problem for the semi linear parabolic partial differential equation subject to an overspecified boundary condition the space derivative in the pde is approximated by finite difference replacements the solution of the resulting system of first order ordinary differential equations odes satisfies a recurrence relation which involves a matrix exponential function the accuracy in time can be controlled by choosing several pade approximants to replace this matrix exponential term a numerical technique is developed to compute the required solution using a splitting method leading to algorithms for sequential and parallel implementation the algorithms are tested on a model problem from the literature the results of a numerical experiment are presented and the accuracy and central processor cpu time needed for this inverse problem are discussed

comparing numerical methods for the solutions of two dimensional diffusion with an integral condition in this paper we present a numerical comparison between the adomian decomposition method adm and a fourth order explicit finite difference method for solving two dimensional diffusion problems with an integral condition the numerical results show that the adm is quite accurate and easily implemented

numerical solution of a parabolic equation subject to specification of energy in this paper special attention is devoted to one particular problem the diffusion equation subject to the specification of mass in a portion of the domain which has been studied quite extensively both analytically and numerically in recent years parabolic partial differential equations with a non local constraint in place of one of the standard boundary specifications feature in the mathematical modeling of many phenomena in this paper the application of the method of lines mol to such problems is considered the mol semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the mol is a method of solving pdes by discretizing the equation with respect to all but one variable usually time the spatial partial derivative is approximated by a finite difference method the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the matrix exponential function is approximated by a rational approximation consisting of four parameters new parallel algorithms are developed using the resulting approximation numerical experiments on two challenging examples are presented to illustrate the performance of the algorithms

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

ekeland s variational principle for set valued functions we establish several set valued function versions of ekeland s variational principle and hence provide some sufficient conditions ensuring the existence of error bounds for inequality systems defined by finitely many lower semicontinuous functions

numerical procedures for a boundary value problem with a non linear boundary condition numerical techniques based on finite difference schemes leading to parallel algorithms have been developed for obtaining approximate solutions to an initial boundary value problem for the two dimensional parabolic partial differential equation pde with a non linear boundary condition this class of parabolic pdes plays a very important role in many branches of science and engineering the non linear condition is in the form of a double integral giving the specification of mass in the solution domain not only the problem has both neumann and dirichlet boundary conditions but the dirichlet boundary condition is in a non standard form while sharing some common features with the one dimensional models the solution of two dimensional equations are substantially more difficult thus some considerations are taken to be able to extend some ideas of one dimensional case due to the structure of the boundary conditions a reduced one dimensional parabolic equation for the new unknown v y t email protected 1 0u x y t dx can be formulated the resulting problem has a non local boundary condition the new two dimensional parabolic pde with neumann s boundary conditions will be solved numerically by using the method of lines semi discretization approach the space derivatives in the pde are approximated by finite difference replacements the solution of the resulting system of first order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function the accuracy in time is controlled by choosing several subdiagonal pade approximants to replace this matrix exponential term numerical techniques are developed to compute the required solution using a splitting method leading to algorithms for sequential and parallel implementation the algorithms are tested on a model problem from the literature the article concludes with the results of some numerical experiments

a numerical method for finding positive solution of diffusive logistic equation with constant yield harvesting abstract we consider a reaction diffusion equation u x au x bu x 2 ch x 0 with dirichlet boundary condition by using a numerical method based on sub super solution we will show the existence of positive solution

new stability results for ekeland s ϵ variational principles for vector valued and set valued maps in this paper we discuss the stability of ekeland s variational principles for vector valued and set valued maps when the dominating cone is a closed pointed convex cone whose interior may be empty we provide a new approach to the study of the stability of ekeland s variational principles for vector valued and set valued maps without scalarization

numerical investigation of the parabolic mixed derivative diffusion equation via alternating direction implicit methods the two dimensional analogue of a two parameter mixed derivative equation is considered via two numerical methods the adi and compact adi method these methods are compared and the influence of the kinematic viscosity and viscoelasticity is discussed in relation to the solutions obtained the two schemes are proven to be conditionally stable and the methods compared via an error analysis

multigrid method based on the transformation free hoc scheme on nonuniform grids for 2d convection diffusion problems in this paper a multigrid method based on the high order compact hoc difference scheme on nonuniform grids which has been proposed by kalita et al j c kalita a k dass d c dalal a transformation free hoc scheme for steady convection diffusion on non uniform grids int j numer methods fluids 44 2004 33 53 is proposed to solve the two dimensional 2d convection diffusion equation the hoc scheme is not involved in any grid transformation to map the nonuniform grids to uniform grids consequently the multigrid method is brand new for solving the discrete system arising from the difference equation on nonuniform grids the corresponding multigrid projection and interpolation operators are constructed by the area ratio some boundary layer and local singularity problems are used to demonstrate the superiority of the present method numerical results show that the multigrid method with the hoc scheme on nonuniform grids almost gets as equally efficient convergence rate as on uniform grids and the computed solution on nonuniform grids retains fourth order accuracy while on uniform grids just gets very poor solution for very steep boundary layer or high local singularity problems the present method is also applied to solve the 2d incompressible navier stokes equations using the stream function vorticity formulation and the numerical solutions of the lid driven cavity flow problem are obtained and compared with solutions available in the literature

weighted finite difference techniques for the one dimensional advection diffusion equation various numerical techniques will be developed and compared for solving the one dimensional advection diffusion equation with constant coefficient these techniques are based on the two level finite difference approximations the basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyett this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes the new methods are more accurate and are more efficient than the conventional techniques these schemes are free of numerical diffusion the results of a numerical experiment are presented and the accuracy and central processor cpu time needed are discussed and compared

diffusion subject to the specification of mass abstract existence uniqueness and continuous dependence upon the data are demonstrated for the solution u u x t of the diffusion equation ut uxx s x t 0 u x 0 x 0 ux 1 t g t 0 0 b 0 x dx m 0 a numerical procedure is discussed and results of numerical experiments are presented

an efficient time splitting compact finite difference method for gross pitaevskii equation abstract we propose an efficient time splitting compact finite difference method for gross pitaevskii equation gpe in our method we solve the gpe in time with time splitting technique and in space by the compact finite difference method to find the numerical solution of the resulting discretized system in one dimension 1d two dimensions 2d and three dimensions 3d we apply the fast discrete sine transform in 1d 2d and 3d respectively and get an efficient solver for the discretized system in 1d 2d and 3d respectively our numerical algorithm at every time step does not need linear algebraic equations solver whose computation cost will be much higher when the spatial dimension increases the method also has the merit that it is unconditionally stable and conservative moreover the method can achieve spectral like accuracy in space when high order compact finite difference method is applied extensive numerical tests for the gpe in 1d 2d and 3d are presented to demonstrate the power and accuracy of the proposed numerical method

identifying a control function in two dimensional parabolic inverse problems numerical solution of the parabolic partial differential equations with an unknown parameter play a very important role in engineering applications this study presents numerical schemes for solving two two dimensional parabolic inverse problems i e for the determination of some unknown functions in the parabolic partial differential equations these schemes are developed for determining the parameter p t which satisfy u t u x x u y y p t email protected in rx 0 t u x y 0 f x y x y r 0 1 2 it is assumed that u is known on the boundary of r and subject to the integral overspecification over a portion of the spatial domain 0 email protected 0 1u x y t dxdy e t 0

advance in the theory of variable stepsize variable formula methods for ordinary differential equations the theory of the constant stepsize constant formula methods cscfm s for solving initial value problems for ordinary differential equations ode s is well developed however the integration methods are often used as variable stepsize variable formula methods vsvfm s in the codes for automatic solution of ode s this fact explains why it is necessary to develop a rigorous theory for the vsvfm s also results obtained until now in the efforts to develop a theory for the vsvfm s are presented in a systematic way and compared with the well known results for cscfm s the comparison is used to outline the directions in which further efforts in the attempt to improve the theoretical results and to obtain new results are most needed the integration methods can be separated into two large groups implicit methods and explicit methods including here the predictor corrector schemes it is discussed what kind of results are needed for each of these two groups assume that a set f of cscfm s is given from a practical point of view it is highly desirable that the cscfm s belonging to set f have the following property if every cscfm from set f is consistent zero stable and convergent then any vsvfm induced by the cscfm s of set f is also consistent zero stable and convergent for a sufficiently large class of grids a theorem concerning a class of cscfm s with this property is presented the application of this theorem in the selection of methods for solving large systems of ode s arising after the space discretization of air pollution models is discussed it is explained why the vsvfm s based on this class provide a very efficient computational process in the treatment of air pollution models

fisher equation for anisotropic diffusion simulating south american human dispersals the fisher equation is commonly used to model population dynamics this equation allows describing reaction diffusion processes considering both population growth and diffusion mechanism some results have been reported about modeling human dispersion always assuming isotropic diffusion nevertheless it is well known that dispersion depends not only on the characteristics of the habitats where individuals are but also on the properties of the places where they intend to move then isotropic approaches cannot adequately reproduce the evolution of the wave of advance of populations solutions to a fisher equation are difficult to obtain for complex geometries moreover when anisotropy has to be considered and so few studies have been conducted in this direction with this scope in mind we present in this paper a solution for a fisher equation introducing anisotropy we apply a finite difference method using the crank nicholson approximation and analyze the results as a function of the characteristic parameters finally this methodology is applied to model south american human dispersal

tensor renormalization group with randomized singular value decomposition an algorithm of the tensor renormalization group is proposed based on a randomized algorithm for singular value decomposition our algorithm is applicable to a broad range of two dimensional classical models in the case of a square lattice its computational complexity and memory usage are proportional to the fifth and the third power of the bond dimension respectively whereas those of the conventional implementation are of the sixth and the fourth power the oversampling parameter larger than the bond dimension is sufficient to reproduce the same result as full singular value decomposition even at the critical point of the two dimensional ising model

galerkin method with trigonometric basis on stable numerical differentiation abstract this paper considers the p p 1 2 3 order numerical differentiation on function y in 0 2 they are transformed into corresponding fredholm integral equation of the first kind computational schemes with analytic solution formulas are designed using galerkin method on trigonometric basis convergence and divergence are all analysed in corollaries 5 1 5 2 and a priori error estimate is uniformly obtained in theorem 6 1 7 1 7 2 therefore the algorithm achieves the optimal convergence rate o 2 2 1 1 2 or 1 with periodic sobolev source condition of order 2 p besides we indicate a noise independent a priori parameter choice when the function y possesses the form of k 0 p 1 a k t k k 1 n 1 b k cos k t k 1 n 2 c k sin k t b n 1 c n 2 0 in particular in numerical differentiations for functions above good filtering effect error approaches 0 is displayed with corresponding parameter choice in addition several numerical examples are given to show that even derivatives with discontinuity can be recovered well

locally explicit schemes for three dimensional diffusion with a non local boundary specification two finite difference schemes based on the classical three point forward time centred space ftcs method and the five point ftcs formula are used to solve the three dimensional time dependent diffusion equation with a non local boundary specification in these methods locally one dimensional lod procedures are used to extend the one dimensional techniques to solve the three dimensional problem numerical integration procedures are also employed to overcome the non local boundary specification results from some numerical experiments are presented including error estimates derived in the maximum norm the results of numerical testing show that the lod techniques give better results than the fully explicit schemes the cpu times used for the procedures developed in this article are also reported

the boundary penalty method for the diffusion equation subject to the specification of mass in this paper we concern with a new boundary penalty finite element method for the one dimensional time dependent diffusion equation subject to the specification of mass instead of solving the diffusion equation directly we will use the finite element method to solve a boundary penalty problem which is derived from the diffusion equation and analyse the error estimate of the semi discrete and fully discrete approximations when the penalty parameter e is sufficiently small the numerical experiments show that our method is very effective and supports our analysis