Parameter-uniform numerical methods for general nonlinear singularly perturbed reaction diffusion problems having a stable reduced solution

Abstract

A general nonlinear singularly perturbed reaction diffusion differential equation with solutions exhibiting boundary layers is analysed in this paper. The problem is considered as having a stable (attractive) reduced solution that satisfies any one of a comprehensive set of conditions for stable reduced solutions of reaction diffusion problems. A numerical method is presented consisting of a finite difference scheme to be solved over a Shishkin mesh. It is shown that suitable transition points for the Shishkin mesh and the error of the numerical method depend on which stability condition the reduced solution satisfies. Moreover, we show that the error may be affected adversely depending on the stability condition satisfied. Numerical experiments are presented to demonstrate the convergence rate established.