A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation

Abstract

We design a robust fitted operator finite difference method for the numerical solution of a singularly perturbed delay parabolic partial differential equation. This method is unconditionally stable and is convergent with order O ( k + h 2 ) , where k and h are respectively the time and space step-sizes, which is better than the one obtained by Ansari et al. [A.R. Ansari, S.A. Bakr, G.I. Shishkin, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. Comput. Appl. Math. 205 (2007) 552–566] where they have used a fitted mesh finite difference method. Their method was of the order O N t - 1 + N x - 2 ln 2 N x , where N t and N x denote the total number of sub-intervals in the time and space directions. The performance of our method is illustrated through some numerical experiments. We also compare our results with those obtained by a standard finite difference method as well as other works seen in the literature. In addition, we provide a novel proof for the bounds on partial derivatives of the solution of the continuous problem.