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A Finite Difference Method for Time-periodic Solutions of Reaction-diffusion Equations

Author: ShuAXiu
Tutor: WangYuanMing
School: East China Normal University
Course: Computational Mathematics
Keywords: nonlinear parabolic equations with time delays time-periodic solutions finite difference method high accuracy monotone iteration method of upper and lower solutions
CLC: O241.82
Type: Master's thesis
Year: 2009
Downloads: 28
Quote: 0
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Abstract


Time-periodic behavior of solutions arises from many problems in various field of applied sciences, such as biology,ecology,biochemistry and physics,and many of these phenomena are usually described by a coupled system of nonlinear parabolic equation or equations,and they can described by equations with time delays if the present solutions depend on the past time solutions.It is of certain practical interests to give an efficient numerical method for such systems.In this paper,a finite difference method with high accuracy is established for solving time-periodic solutions of a class of nonlinear parabolic systems.This method has second order accuracy in time and fourth order accuracy in space.Some qualitative analyses are given for the nonlinear finite difference scheme.This includes the existence-uniqueness of finite difference solution and the convergence of the finite difference solution to the analytical solution.To solve the nonlinear finite difference scheme,an efficient monotone iterative algorithm is developed.The sequences of iterations converge monotonically to an unique solution of nonlinear finite difference system,and the initial iteration can be explicitly constructed without any knowledge of the solution.The numerical results demonstrate the advantages of the method,including the monotone convergence property of iterative sequences and the high accuracy of the method.

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CLC: > Mathematical sciences and chemical > Mathematics > Computational Mathematics > Numerical Analysis > The numerical solution of differential equations, integral equations > Numerical Solution of Partial Differential Equations
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