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.
2002 Elsevier Science (USA). All rights reserved
