A First-Order Convergent Parameter-Uniform Numerical Method for a Singularly Perturbed Second-Order Delay-Differential Equation of Reaction–Diffusion Type with a Discontinuous Source Term

Abstract

In this paper, a boundary value problem for a second-order singularly perturbed delay differential equation of reaction–diffusion type with a discontinuous source term is considered on the interval [0, 2]. A single discontinuity in the source term is assumed to occur at a point \(\;d \in (0,2).\;\) The leading term of the equation is multiplied by a small positive parameter. The solution of this problem exhibits boundary layers at \(x = 0\) and \(x = 2\) and interior layers at \(x = 1\) and/or at \(x = d\) and \(x = 1+d\) with respect to the position of d in (0, 2). A numerical method composed of a classical finite difference scheme applied on a piecewise uniform Shishkin mesh is suggested to solve the problem. The method is proved to be first-order convergent uniformly in the perturbation parameter. Numerical illustrations provided support the theory.