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Finite difference reaction diffusion equations with nonlinear boundary conditions
Author(s) -
Pao C. V.
Publication year - 1995
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690110405
Subject(s) - mathematics , nonlinear system , monotone polygon , finite difference , boundary value problem , mathematical analysis , finite difference method , thermal conduction , reaction–diffusion system , physics , thermodynamics , geometry , quantum mechanics
This article is concerned with numerical solutions of finite difference systems of reaction diffusion equations with nonlinear internal and boundary reaction functions. The nonlinear reaction functions are of general form and the finite difference systems are for both time‐dependent and steady‐state problems. For each problem a unified system of nonlinear equations is treated by the method of upper and lower solutions and its associated monotone iterations. This method leads to a monotone iterative scheme for the computation of numerical solutions as well as an existence‐comparison theorem for the corresponding finite difference system. Special attention is given to the dynamical property of the time‐dependent solution in relation to the steady‐state solutions. Application is given to a heat‐conduction problem where a nonlinear radiation boundary condition obeying the Boltzmann law of cooling is considered. This application demonstrates a bifurcation property of two steady‐state solutions, and determines the dynamic behavior of the time‐dependent solution. Numerical results for the heat‐conduction problem, including a test problem with known analytical solution, are presented to illustrate the various theoretical conclusions. © 1995 John Wiley & Sons, Inc.