The Novel Rational Spectral Collocation Method for a Singularly Perturbed Reaction-Diffusion Problem with Nonsmooth Data

Abstract

A singularly perturbed reaction-diffusion problem with a discontinuous source term is considered. A novel rational spectral collocation combined with a singularity-separated method for this problem is presented. The solution is expressed as $u=w+v$ , where $w$ is the solution of corresponding auxiliary boundary value problem and $v$ is a singular correction with direct expressions. The rational spectral collocation method combined with a sinh transformation is applied to solve the weakened singularly boundary value problem. According to the asymptotic analysis, the sinh transformation parameters can be determined by the width and position of the boundary layers. The parameters in the singular correction can be determined by the boundary conditions of the original problem. Numerical experiment supports theoretical results and shows that compared with previous research results, the novel method has the advantages of a high computational accuracy in singularly perturbed reaction-diffusion problems with nonsmooth data.