Perturbed asymptotic expansions for interior-layer solutions of a semilinear reaction-diffusion problem with small diffusion

TL;DR

This paper develops perturbed asymptotic expansions for interior-layer solutions of a semilinear reaction-diffusion boundary value problem with small diffusion, aiding in the analysis of numerical methods.

Contribution

It introduces a novel perturbed asymptotic expansion approach for interior-layer solutions in reaction-diffusion problems with small diffusion.

Findings

Constructed asymptotic expansion for interior-layer solutions

Established properties of perturbations of the expansion

Applied results to finite difference method convergence

Abstract

A semilinear reaction-diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter $\eps^2$, is considered. It can have multiple solutions. An asymptotic expansion is constructed for a solution that has an interior layer. Further properties are then established for a perturbation of this expansion. These are used in\cite{KoStMain} to obtain discrete sub-solutions and super-solutions for certain finite difference methods described there, and in this way yield convergence results for those methods.