block monotone iterative methods for numerical solutions of nonlinear elliptic equations two block monotone iterative schemes for a nonlinear algebraic system which is a finite difference approximation of a nonlinear elliptic boundary value problem are presented and are shown to converge monotonically either from above or from below to a solution of the system this monotone convergence result yields a computational algorithm for numerical solutions as well as an existence comparison theorem of the system including a sufficient condition for the uniqueness of the solution an advantage of the block iterative schemes is that the thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one dimensional problems the block iterative schemes are compared with the point monotone iterative schemes of picard jacobi and gauss seidel and various theoretical comparison results among these monotone iterative schemes are given these comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given

numerical methods for time periodic solutions of nonlinear parabolic boundary value problems this paper is devoted to a numerical analysis of periodic solutions of a finite difference system which is a discrete version of a class of nonlinear reaction diffusion convection equations under nonlinear boundary conditions three monotone iterative schemes for the finite difference system are presented and it is shown by the method of upper and lower solutions that the sequence of iterations from each of these iterative schemes converges monotonically to either a maximal periodic solution or a minimal periodic solution depending on whether the initial iteration is an upper solution or a lower solution a comparison theorem for the various monotone sequences is given it is also shown that the maximal and minimal periodic solutions of the finite difference system converge to the corresponding maximal and minimal periodic solutions of the reaction diffusion convection equation as the mesh size decreases to zero some error estimates between the theoretical and the computed iterations for each of the three iterative schemes are obtained and a discussion on the numerical stability of these schemes is given also given are some numerical results of a logistic reaction diffusion problem

global attractor of a coupled finite difference reaction diffusion system with delays abstract in the study of asymptotic behavior of solutions for reaction diffusion systems an important concern is to determine whether and when the system has a global attractor which attracts all positive time dependent solutions the aim of this paper is to investigate the global attraction problem for a finite difference system which is a discrete approximation of a coupled system of two reaction diffusion equations with time delays sufficient conditions are obtained to ensure the existence and global attraction of a positive solution of the corresponding steady state system applications are given to three types of lotka volterra reaction diffusion models where time delays may appear in the opposing species

error and extrapolation of a compact lod method for parabolic differential equations this paper is concerned with a compact locally one dimensional lod finite difference method for solving two dimensional nonhomogeneous parabolic differential equations an explicit error estimate for the finite difference solution is given in the discrete infinity norm it is shown that the method has the accuracy of the second order in time and the fourth order in space with respect to the discrete infinity norm a richardson extrapolation algorithm is developed to make the final computed solution fourth order accurate in both time and space when the time step equals the spatial mesh size numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm

numerical methods for systems of nonlinear parabolic equations with time delays abstract the purpose of this paper is to investigate some numerical aspects of a class of coupled nonlinear parabolic systems with time delays the system of parabolic equations is discretized by the finite difference method which yields a coupled system of nonlinear algebraic equations the mathematical analysis of the nonlinear system is by the method of upper and lower solutions and its associated monotone iterations three monotone iterative schemes are presented and it is shown that the sequence of iterations from each one of these iterative schemes converges monotonically to a unique solution of the finite difference system a theoretical comparison result for the various monotone sequences and error estimates for the three monotone iterative schemes are obtained it is also shown that the finite difference solution converges to the classical solution of the parabolic system as the mesh size decreases to zero

a higher order compact adi method with monotone iterative procedure for systems of reaction diffusion equations this paper is concerned with an existing compact finite difference adi method published in the paper by liao et al 2002 3 for solving systems of two dimensional reaction diffusion equations with nonlinear reaction terms this method has an accuracy of fourth order in space and second order in time the existence and uniqueness of its solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear reaction terms the convergence of the finite difference solution to the continuous solution is proved an efficient monotone iterative algorithm is presented for solving the resulting discrete system and some techniques for the construction of upper and lower solutions are discussed an application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method

stability of numerical methods for delay differential equations abstract consider the following delay differential equation dde y t y t y t t t t 0 with the initial condition y t for t t 0 where and are such that 0 1 0 2 has a unique solution y t the author gives sufficient conditions for the asymptotic stability of the equation 0 1 for which he introduces new definitions of numerical stability the approach is reminiscent of that from the nonlinear stiff ordinary differential equation ode field he investigates stability properties of the class of one point collocation rules in particular the backward euler method turns out to be stable with respect to all the given definitions

uniqueness and reconstruction for the anisotropic helmholtz decomposition the helmholtz decomposition theorem for an anisotropic medium is explicitly stated in terms of two symmetric and positive definite dyadics which carry the directional characteristics of the medium it is shown that this general decomposition is not unique and that once the scalar invariant with respect to one of the dyadics and the vector invariant with respect to the other dyadic are given the initial field can be completely reconstructed

optimal superconvergence results for volterra functional integral equations with proportional vanishing delays abstract in this paper we develop a new technique to study the optimal convergence orders of collocation methods for volterra functional integral equations with vanishing delays on quasi geometric meshes basing on a perturbation analysis we show that for m collocation points the global convergence order of the collocation solution is only m however the collocation solution may exhibit superconvergence with order m 1 at the collocation points in particular the local convergence order may attain 2 m 1 at the nodes provided that the collocation is based on the m radau ii points finally some numerical examples are performed to verify our theoretical results

collocation methods for volterra functional integral equations with non vanishing delays abstract in this paper the existence uniqueness regularity properties and in particular the local representation of solutions for general volterra functional integral equations with non vanishing delays are investigated based on the solution representation we detailedly analyze the attainable global and local convergence order of iterated collocation solutions on invariant meshes it turns out that collocation at the m gauss legendre points neither leads to the optimal global convergence order m 1 nor yields the local convergence order 2m on the whole interval which is in sharp contrast to the case of the classical volterra delay integral equations however if the collocation is based on the m radau ii points the local superconvergence order 2 m 1 will exhibit at all mesh points finally some numerical experiments are performed to confirm our theoretical findings

positive solutions of quasilinear parabolic systems with nonlinear boundary conditions abstract the aim of this paper is to investigate the existence uniqueness and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions including a system of quasilinear parabolic and ordinary differential equations also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady state solution the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations it is shown that the time dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a porous medium type of problem a heat transfer problem and a two component competition model in ecology these applications illustrate some very interesting distinctive behavior of the time dependent solutions between density independent and density dependent diffusions

combined iterative methods for numerical solutions of parabolic problems with time delays we study monotone iterative schemes for finite difference solutions of reaction diffusion systems with time delays in this paper we derive two modified iterative schemes by combing the method of upper lower solutions and the jacobi method or the gauss seidel method the convergence and stability of the monotone iterative schemes are also discussed

accelerated monotone iterative methods for finite difference equations of reaction diffusion this paper is concerned with numerical methods for a finite difference system of reaction diffusion convection equation under nonlinear boundary condition various monotone iterative methods are presented and each of these methods leads to an existence comparison theorem as well as a computational algorithm for numerical solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function application is given to a model problem from chemical engineering and some numerical results including a test problem with known analytical solution are presented to illustrate the various rates of convergence of the iterations

a higher order compact lod method and its extrapolations for nonhomogeneous parabolic differential equations abstract a higher order compact locally one dimensional lod finite difference method for two dimensional nonhomogeneous parabolic differential equations is proposed the resulting scheme consists of two one dimensional tridiagonal systems and all computations are implemented completely in one spatial direction as for one dimensional problems the solvability and the stability of the scheme are proved almost unconditionally the error estimates are obtained in the discrete h 1 l 2 and l norms and show that the proposed compact lod method has the accuracy of the second order in time and the fourth order in space two richardson extrapolation algorithms are presented to increase the accuracy to the fourth order and the sixth order in both time and space when the time step is proportional to the spatial mesh size numerical results demonstrate the accuracy of the compact lod method and the high efficiency of its extrapolation algorithms

harmless delays for permanence in a class of population models with diffusion effects abstract this paper studies a class of time delay reaction diffusion systems modeling the dynamics of single or interacting populations in the logistic equation we prove that when the magnitude of the instantaneous term is larger than that of the delay terms the population growthuhas the same asymptotic limit as in the case of no delay for the predator prey model a condition on the interaction rates is given to ensure the permanence effect in the ecosystem regardless of the length of delay intervals a permanence condition is also obtained in then species competition system with time delays it is shown that when the natural growth rate a1 a2 an is in an unbounded parameter set the reaction diffusion system has a positive global attractor finally long term behavior of the solutions for those time delay systems is numerically demonstrated through finite difference approximations and compared with the corresponding systems without delays

general solutions of the stokes flow equations abstract the papkovich neuber solution of the stokes flow equations for a viscous incompressible flow at very low reynolds numbers is derived using three alternative approaches in each case the class of solutions which is valid varies depending on the assumptions made on the domain considered the technique is illustrated by several examples the solution is identical to that of incompressible linear elasticity upon appropriate modification

higher order compact finite difference method for systems of reaction diffusion equations this paper is concerned with a compact finite difference method for solving systems of two dimensional reaction diffusion equations this method has the accuracy of fourth order in both space and time the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term three monotone iterative algorithms are provided for solving the resulting discrete system efficiently and the sequences of iterations converge monotonically to a unique solution of the system a theoretical comparison result for the various monotone sequences is given the convergence of the finite difference solution to the continuous solution is proved and richardson extrapolation is used to achieve fourth order accuracy in time an application is given to an enzyme substrate reaction diffusion problem and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach

monotone iterates for solving nonlinear integro parabolic equations of volterra type the paper deals with numerical solving of nonlinear integro parabolic problems based on the method of upper and lower solutions a monotone iterative method is constructed existence and uniqueness of a solution to the nonlinear difference scheme are established an analysis of convergence rates of the monotone iterative method is given construction of initial upper and lower solutions is discussed numerical experiments are presented

periodic solutions of parabolic systems with time delays existence of maximal and minimal periodic solutions of a coupled system of parabolic equations with time delays and with nonlinear boundary conditions is discussed the proof of the existence theorem is based on the method of upper and lower solutions and its associated monotone iterations this method is constructive and can be used to develop a computational algorithm for numerical solutions of the periodic parabolic system an application is given to a competitor competitor mutualist model which consists of a coupled system of three reaction diffusion equations with time delays

nonlinear stability and d convergence of runge kutta methods for delay differential equations abstract this paper deals with the stability and convergence of runge kutta methods with the lagrangian interpolation rklms for nonlinear delay differential equations ddes some new concepts such as strong algebraic stability gdn stability and d convergence are introduced we show that strong algebraic stability of a rkm for odes implies gdn stability of the corresponding rklm for ddes and that a strongly algebraically stable and diagonally stable rkm with order p together with a lagrangian interpolation of order q leads a d convergent rklm of order min p q 1

an analytical solution of the navier stokes equation for internal flows this paper derives a solution to the navier stokes equation by considering vorticity generated at system boundaries the result is an explicit expression for the velocity the navier stokes equation is reformulated as a divergence and integrated giving a tensor equation that splits into a symmetric and a skew symmetric part one equation gives an algebraic system of quadratic equations involving velocity components a system of nonlinear partial differential equations is reduced to algebra the velocity is then explicitly calculated and shown to depend on boundary conditions only this removes the need to solve the navier stokes equation by a 3d numerical computation replacing it by computation of 2d surface integrals over the boundary

uniform convergent monotone iterates for semilinear singularly perturbed parabolic problems this paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems monotone sequences based on the accelerated monotone iterative method are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem this monotone convergence leads to the existence uniqueness theorem an analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and continuous problem is given numerical experiments are presented

error and stability of monotone method for numerical solutions of fourth order semilinear elliptic boundary value problems this paper is concerned with the error and stability analysis of the monotone method for numerical solutions of fourth order semilinear elliptic boundary value problems a comparison result among the various monotone sequences is given the global error is analyzed and some sufficient conditions are formulated to guarantee a geometric rate of convergence the stability of the monotone method is proved some numerical results are presented

stability and error analysis of one leg methods for nonlinear delay differential equations this paper is concerned with the numerical solution of delay dierential equations ddes we focus on the stability behaviour and error analysis of one leg methods with respect to nonlinear ddes the new concepts of gr stability gar stability and weak gar stability are introduced it is proved that a strongly a stable one leg method with linear interpolation is gar stable and that an a stable one leg method with linear interpolation is gr stable weakly gar stable and d convergent of order s if it is consistent of order s in the classical sense c 1999 elsevier science b v all rights reserved ams classication 65l06 65l20

some global uniqueness and solvability results for linear complementarity problems over symmetric cones the purpose of this paper is to give a numerical treatment for a class of quasi linear elliptic equations under nonlinear boundary conditions including the three basic types of linear boundary conditions the quasi linear equation is discretized by the finite difference method and the method of upper lower solutions and its associated monotone iteration are used to compute the solutions of the finite difference system this method leads to monotone iterative schemes for the computation of numerical solutions as well as some comparison results among the monotone iterative schemes it also leads to the existence of a maximal and a minimal finite difference solution including the uniqueness of the solution and the convergence of the finite difference solution to the corresponding continuous solution applications are given to two physical problems in heat conduction and combustion theory and numerical results for the heat conduction problem are given and are compared with the known true continuous solution

stability analysis of numerical methods for linear neutral volterra delay integro differential system this paper is concerned with the stability of numerical methods for linear neutral volterra delay integro differential system a sufficient condition such that the system is asymptotically stable is derived furthermore it is proved that every linear q method with email protected 1 2 1 and a stable bdf method preserve the delay independent stability of its exact solutions finally the numerical experiments are given to demonstrate the conclusions

numerical methods for nonlinear integro parabolic equations of fredholm type abstract this paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction diffusion convection equations under either linear or nonlinear boundary conditions the discrete approximation of the problem is based on the finite difference method and the computation of the finite difference solution is by the method of upper and lower solutions three types of quasi monotone reaction functions are considered and for each type a monotone iterative scheme is obtained each of these iterative schemes yields two sequences which converge monotonically from above and below respectively to a unique solution of the finite difference system this monotone convergence leads to an existence uniqueness theorem as well as a computational algorithm for the computation of the solution an error estimate between the computed approximations and the true finite difference solution is obtained for each iterative scheme these error estimates are given in terms of the strength of the reaction function and the effect of diffusion convection and are independent of the true solution applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm

monotone iterates with quadratic convergence rate for solving weighted average approximations to semilinear parabolic problems this paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems monotone sequences based on the accelerated monotone iterative method are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem this monotone convergence leads to the existence uniqueness theorem an analysis of convergence of the monotone iterative method to the solutions of the nonlinear difference scheme is given numerical experiments are presented

numerical solutions of reaction diffusion equations with nonlocal boundary conditions abstract the purpose of this paper is to present some iterative methods for numerical solutions of a class of nonlinear reaction diffusion equations with nonlocal boundary conditions using the finite difference method and the method of upper and lower solutions we present some monotone iterative schemes for both the time dependent and the steady state finite difference systems each monotone iterative scheme gives a computational algorithm for numerical solutions and an existence comparison theorem for the corresponding finite difference system the existence comparison theorems are used to investigate the asymptotic behavior of the discrete time dependent solution in relation to the discrete maximal and minimal solutions of the steady state problem numerical results are given to a model problem where the solution of the continuous problem is explicitly known and its values at the mesh points are used to compare with the numerical solutions obtained by the monotone iterative schemes

qualitative properties of nonlinear parabolic operators abstract it is a natural expectation that the mathematical models of real life phenomena have to possess some characteristic qualitative properties of the original process for parabolic problems the main known qualitative properties are the maximum minimum principles nonnegativity nonpositivity preservation and maximum norm contractivity these properties have a fundamental relevance concerning the validity of the mathematical or numerical model without them the model might produce unphysical quantities that contradict reality for linear problems with dirichlet boundary conditions these properties have been thoroughly investigated and their relations have been characterized in the present paper we extend the linear results to nonlinear problems with general boundary conditions firstly we characterize various implications between the qualitative properties some of them are given in general and in certain cases we restrict our study to operators with gradient dependent principal part or to operators with heat conduction coefficient secondly we give general sufficient conditions to ensure these qualitative properties both separately and all of them together the relations are illustrated with several examples

nonstationary monotone iterative methods for nonlinear partial differential equations a simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations with the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system picard gauss seidel and jacobi monotone iterative methods are presented and analyzed for the adaptive solutions the adaptive meshes are generated by the 1 irregular mesh refinement scheme which together with the m matrix of the finite element stiffness matrix lead to existence uniqueness comparison theorems with simple upper and lower solutions as initial iterates some numerical examples including a test problem with known analytical solution are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties numerical results of simulations on a mosfet with the gate length down to 34 nm are also given

monotone finite difference schemes for nonlinear systems with mixed quasi monotonicity monotone finite difference schemes are proposed for nonlinear systems with mixed quasi monotonicity two monotone iteration processes for the corresponding discrete problems are presented which converge monotonically to the quasi solutions of the discrete problems the limits are the exact solutions under some conditions a monotone finite difference scheme on uniform mesh with the accuracy of fourth order is constructed the numerical results coincide with theoretical analysis

finite difference reaction diffusion systems with coupled boundary conditions and time delays abstract this paper is concerned with finite difference solutions of a coupled system of reaction diffusion equations with nonlinear boundary conditions and time delays the system is coupled through the reaction functions as well as the boundary conditions and the time delays may appear in both the reaction functions and the boundary functions the reaction diffusion system is discretized by the finite difference method and the investigation is devoted to the finite difference equations for both the time dependent problem and its corresponding steady state problem this investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions monotone convergence of the time dependent solution to a maximal or a minimal steady state solution for quasimonotone functions and local and global attractors of the time dependent system including the convergence of the time dependent solution to a unique steady state solution also discussed are some computational algorithms for numerical solutions of the steady state problem when the reaction function and the boundary function are quasimonotone all the results for the coupled reaction diffusion equations are directly applicable to systems of parabolic ordinary equations and to reaction diffusion systems without time delays

a modified accelerated monotone iterative method for finite difference reaction diffusion convection equations this paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction diffusion convection equations with nonlinear boundary conditions a modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time dependent problem and its corresponding steady state problem this method leads to a simple and yet efficient linear iterative algorithm it yields two sequences of iterations that converge monotonically from above and below respectively to a unique solution of the system the monotone property of the iterations gives concurrently improving upper and lower bounds for the solution it is shown that the rate of convergence for the sum of the two sequences is quadratic under an additional requirement quadratic convergence is attained for one of these two sequences in contrast with the existing accelerated monotone iterative methods our new method avoids computing local maxima in the construction of these sequences an application using a model problem gives numerical results that illustrate the effectiveness of the proposed method

numerical methods for coupled systems of nonlinear parabolic boundary value problems abstract a system of nonlinear finite difference equations corresponding to a class of coupled parabolic equations in a bounded domain is investigated using the method of upper lower solutions we construct two monotone sequences for the finite difference equations it is shown that when the reaction function is quasimonotone nondecreasing nonincreasing or mixed quasimonotone these two sequences converge monotonically to a unique solution of the finite difference system this monotone convergence leads to an existence comparison theorem as well as iterative numerical algorithms for the computation of the solution it also gives error estimates for the finite difference solution in each iteration without explicit knowledge of the solution moreover the monotone convergence property is used to prove the convergence of the finite difference solution to the corresponding solution of the differential system as the mesh size decreases to zero

finite difference solutions of reaction diffusion equations with continuous time delays abstract this paper is an extension of the monotone iterative methods for finite difference equations with discrete time delays to a class of nonlinear finite difference system with continuous time delays the system under consideration is a finite difference approximation of a class of reaction diffusion equations with continuous time delays in the nonlinear reaction under either dirichlet or neumann robin boundary conditions various monotone iterative schemes which depend on the property of the nonlinear reaction mechanism are developed for the computation of numerical solutions it is shown by the method of upper and lower solutions that the two sequences obtained from each iterative scheme converge monotonically from above and below respectively to a unique solution of the finite difference system applications are given to two model problems known as the diffusive logistic equation and the fisher s diffusion equation in population genetics

periodic solutions of parabolic systems with nonlinear boundary conditions this paper is concerned with the existence and stability of periodic solutions for a coupled system of nonlinear parabolic equations under nonlinear boundary conditions the approach to the problem is by the method of upper and lower solutions and its associated monotone iterations this method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for parabolic initial boundary value problems a sufficient condition for the stability of a periodic solution is also given these results are applied to three model problems arising from chemical kinetics ecology and population biology

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

numerical methods for semilinear parabolic equations the method of upper lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions the idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem the aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable these two schemes are modified jacobi method and gauss seidel method for nonlinear algebraic equations an advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration on the other hand the standard picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations

stability analysis of volterra delay integro differential equations and their backward differentiation time discretization we study the stability of the analytical solutions of initial value problems of a general class of systems of volterra delay integro differential equations numerical methods based on backward differentiation formulae and repeated quadrature formulae are suggested nonlinear and linear stability conditions for the presented methods are derived

on the numerical stability of volterra integral equations with delay argument abstract stability properties of numerical methods for volterra integral equations with delay arguments are considered some suitable definitions for the stability of the numerical methods are included and also plots of the stability regions for particular cases are included

time delayed finite difference reaction diffusion systems with nonquasimonotone functions this paper is concerned with finite difference solutions of a system of reaction diffusion equations with coupled nonlinear boundary conditions and time delays the reaction functions and the boundary functions are not necessarily quasimonotone and the time delays may appear in the reaction functions as well as in the boundary functions the investigation is devoted to the finite difference system for both the time dependent problem and its corresponding steady state problem some monotone iteration processes for the finite difference systems are given and the asymptotic behavior of the time dependent solution in relation to the steady state solution is discussed the asymptotic behavior result leads to some local and global attractors of the time dependent problem including the convergence of the time dependent solution to a unique steady state solution an application and some numerical results to an enzyme substrate reaction diffusion problem are given all the results are directly applicable to parabolic ordinary systems and to reaction diffusion systems without time delays

numerical solutions of a three competition lotka volterra system abstract this paper is concerned with finite difference solutions of a lotka volterra reaction diffusion system with three competing species the reaction diffusion system is discretized by the finite difference method and the investigation is devoted to the finite difference system for the time dependent solution and its asymptotic behavior in relation to the corresponding steady state problem three monotone iterative schemes for the computation of the time dependent solution are presented and the sequences of iterations are shown to converge monotonically to a unique positive solution also discussed is the asymptotic behavior of the time dependent solution in relation to various steady state solutions a simple condition on the competing rate constants is obtained which ensures that for every nontrivial nonnegative initial function the corresponding time dependent solution converges either to a unique positive steady state solution or to one of the semitrivial steady state solutions the above results lead to the coexistence and permanence of the competing system as well as computational algorithms for numerical solutions some numerical results from these computational algorithms are given all the conclusions for the reaction diffusion equations are directly applicable to the finite difference solution of the corresponding ordinary differential system

strongly coupled elliptic systems and applications to lotka volterra models with cross diffusion abstract the aim of this paper is to investigate the existence and method of construction of solutions for a general class of strongly coupled elliptic systems by the method of upper and lower solutions and its associated monotone iterations the existence problem is for nonquasimonotone functions arising in the system while the monotone iterations require some mixed monotone property of these functions applications are given to three lotka volterra model problems with cross diffusion and self diffusion which are some extensions of the classical competition prey predator and cooperating ecological systems the monotone iterative schemes lead to some true positive solutions of the competition system and to quasisolutions of the prey predator and cooperating systems also given are some sufficient conditions for the existence of a unique positive solution to each of the three model problems

traveling waves and competitive exclusion in models of resource competition and mating interference abstract in this article we study reaction diffusion models for two closed related biological species u and v under resource competition and mating interference from v species to u species the effect of one sided sexual competition makes the trivial state and u dominance state both unstable while the v dominance state is shown to be asymptotically stable with attraction regions and convergent rates depending on the biological parameters using the upper lower solution method we further prove that for a family of wave speeds there exist traveling wave solutions connecting the u dominance state and the v dominance state at this confirms an earlier conjecture that unbalanced mating interference will lead to competitive exclusion these results can also be obtained on an extended model with instantaneous effects of resource competition and temporal delay on mating interference through a transformation into three equation system we prove that the temporal delay does not affect the stability of the steady states and the existence of the traveling waves but causes changes on the attraction regions and convergence rates finally numerical simulations are also presented to illustrate the theoretical results

finite difference reaction diffusion equations with nonlinear diffusion coefficients a monotone iterative method for numerical solutions of a class of finite difference reaction diffusion equations with nonlinear diffusion coefficient is presented it is shown that by using an upper solution or a lower solution as the initial iteration the corresponding sequence converges monotonically to a unique solution of the finite difference system it is also shown that the solution of the finite difference system converges to the solution of the continuous equation as the mesh size decreases to zero

a singularly perturbed reaction diffusion problem with inhomogeneous environment reaction diffusion problems have been used to describe pattern formation in developmental biology and material science i study the existence and stability of a singularly perturbed reaction diffusion problem with inhomogeneous environment in one dimensional space domain and also high dimensional space domain

stability of bdf methods for nonlinear volterra integral equations with delay abstract this paper deals with the stability analysis of the backward differential formulas or bdf methods for nonlinear volterra integral equations with delay vides the presented approach is based on a nonclassical lipschitz condition in particular the criteria on the global and the asymptotic stability of the methods are given

numerical solving nonlinear integro parabolic equations by the monotone weighted average method the paper deals with numerical solving nonlinear integro parabolic problems of volterra type based on the weighted average method a monotone iterative method is presented construction of initial upper and lower solutions is given existence and uniqueness of a solution to the nonlinear difference scheme are established an analysis of convergence rates of the monotone iterative method is given numerical experiments are presented

a numerical scheme for particle laden thin film flow in two dimensions the physics of particle laden thin film flow is not fully understood and recent experiments have raised questions with current theory there is a need for fully two dimensional simulations to compare with experimental data to this end a numerical scheme is presented for a lubrication model derived for particle laden thin film flow in two dimensions with surface tension the scheme relies on an adi process to handle the higher order terms and an iterative procedure to improve the solution at each timestep this is the first paper to simulate the two dimensional particle laden thin film lubrication model several aspects of the scheme are examined for a test problem such as the timestep runtime and number of iterations the results from the simulation are compared to experimental data the simulation shows good qualitative agreement it also suggests further lines of inquiry for the physical model

monotone iterative methods for numerical solutions of nonlinear integro elliptic boundary problems abstract the aim of this paper is to obtain various monotone iterative schemes for numerical solutions of a class of nonlinear nonlocal reaction diffusion convection equations under linear boundary conditions the boundary value problem under consideration is discretized into a system of nonlinear algebraic equations by the finite difference method and the iterative schemes are given for the finite difference system using upper and lower solutions as the initial iterations the construction of the monotone sequences and the definition of upper and lower solutions depend on the quasimonotone property of the reaction function and the iterative schemes are presented for each of the three types of quasimonotone functions an application of the monotone iterations including some numerical results is given to a modified logistic diffusive equation where the kernel in the integral term may be positive negative or changing sign in its domain

numerical analysis of coupled systems of nonlinear parabolic equations this paper is concerned with numerical solutions of a general class of coupled nonlinear parabolic equations by the finite difference method three monotone iteration processes for the finite difference system are presented and the sequences of iterations are shown to converge monotonically to a unique solution of the system including an existence uniqueness comparison theorem a theoretical comparison result for the various monotone sequences and an error analysis of the three monotone iterative schemes are given also given is the convergence of the finite difference solution to the continuous solution of the parabolic boundary value problem an application to a reaction diffusion model in chemical engineering and combustion theory is given