Numerical methods for coupled systems of nonlinear parabolic boundary value problems
type: Item
instance of: scholarly article
title: Numerical methods for coupled systems of nonlinear parabolic boundary value problems
zbMATH Open document ID: 0713.65053
DOI: 10.1016/0022-247X(90)90167-E
author: C. V. Pao
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: The coupled system of nonlinear equations \(u_ t-D^{(1)}\nabla^ 2u=f^{(1)}(x,t,u,v),\) \(v_ t-D^{(2)}\nabla^ 2v=f^{(2)}(x,t,u,v),\) (t\(\in [0,T]\), \(x\in \Omega)\) with the initial- boundary conditions \(\alpha^{(1)}\partial u/\partial \mu +\beta^{(1)}u=g^{(1)}(x,t),\alpha^{(2)}\partial v/\partial \nu +\beta^{(2)}v=g^{(2)}(x,t),\) \(u(x,0)=\psi^{(1)}_{(x)},\) \(v(x,0)=\psi^{(2)}(x)\) is considered. Here \(\Omega\) is a bounded domain in \(R^ p\), \(D^{(\ell)}\equiv D^{(\ell)}(x,t)>0\), \(\alpha^{(\ell)}=\alpha^{(\ell)}(x)\geq 0\), \(\beta^{(\ell)}=\beta^{(\ell)}(x)\geq 0\) with \(\alpha^{(\ell)}+\beta^{(\ell)}>0\) on \(\partial \Omega\) and \(f^{(\ell)}\), \(g^{(\ell)}\), \(\psi^{(\ell)}\) are Hölder continuous functions of their respective arguments. Using the method of upper-lower solutions monotone sequences for the finite difference equations are constructed. The convergence of the finite difference solution to the differential solution as the mesh size decreases to zero is proved.
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 4174255
zbMATH Keywords: reaction-diffusion equation
zbMATH Keywords: coupled system of nonlinear equations
zbMATH Keywords: method of upper-lower solutions
zbMATH Keywords: finite difference equations
zbMATH Keywords: convergence
MaRDI profile type: Publication
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full work available at URL: 0022-247x(90)90167-e
OpenAlex ID: W2093512373
description: scientific article; zbMATH DE number 4174255