Finite difference scheme for singularly perturbed reaction diffusion problem of partial delay differential equation with nonlocal boundary condition

Sekar Elango,Ayyadurai Tamilselvan,Haitao Zhu,Jinde Cao,Nallappan Gunasekaran,R Vadivel,Xiaodi Li

doi:10.1186/s13662-021-03296-x

Abstract

This paper investigates singularly perturbed parabolic partial differential equations with delay in space, and the right end plane is an integral boundary condition on a rectangular domain. A small parameter is multiplied in the higher order derivative, which gives boundary layers, and due to the delay term, one more layer occurs on the rectangle domain. A numerical method comprising the standard finite difference scheme on a rectangular piecewise uniform mesh (Shishkin mesh) of N_{r} times N_{t} elements condensing in the boundary layers is suggested, and it is proved to be parameter-uniform. Also, the order of convergence is proved to be almost two in space variable and almost one in time variable. Numerical examples are proposed to validate the theory.

Highlights

Perturbed parabolic partial differential equations have received an increasing research attention over the past few decades
5 Error estimate we prove parameter uniform convergence of the numerical solution
We considered a class of singularly perturbed partial differential equations with delay in space and integral boundary condition

Summary

Introduction

Perturbed parabolic partial differential equations have received an increasing research attention over the past few decades (see [1,2,3,4,5,6,7,8,9] and the references therein). The authors in [10] and [11] studied a finite difference scheme for solving singularly perturbed parabolic time delay differential equation with Dirichlet and Robin boundary conditions on Shishkin mesh. They made use of the results of Miller et al [12]. ), note that (r, t) ⇒ (r, t) and the parameter ε is independent of problem (18)–(19), suitable estimates in [26, Sect.

The discretised problem

Discussion