A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem

Abstract

In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be $\epsilon$-uniformly convergent of $O\left(\Delta t+N^{-2}{\ln }^{2}N\right)$, where $\Delta t$ and $N$ denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter $\lambda$. Some numerical results are carried out to support the theory.