A novel fitted numerical scheme for singularly perturbed delay parabolic problems with two small parameters

Abstract

In this paper, fitted operator finite difference methods are presented for two-parameter singularly perturbed one-dimensional parabolic partial differential equations with a delay in the time variable. Such boundary value problems are frequently encountered in the spatial diffusion of reactants and in control systems. To approximate the solution of the problem, we consider the Crank–Nicolson method for time discretization and the non-standard scheme for space discretization on uniform meshes. We prove that the proposed numerical method is uniformly convergent, having convergence of order two in time and one in space. Further, we present numerical results for two test problems in support of our theoretical findings and to demonstrate that the non-standard scheme is more effective than the hybrid scheme comprising central difference, upwind, and mid-point.

Keywords

• Singular perturbation problems
• Two parameters
• Time delay parabolic problem
• Non-standard fitting
• Uniform convergence