A Parameter-Uniform Numerical Scheme for Solving Singularly Perturbed Parabolic Reaction-Diffusion Problems with Delay in the Spatial Variable

Abstract

The objective of this research work is to develop and analyse a numerical scheme for solving singularly perturbed parabolic reaction-diffusion problems with large spatial delay. The presence of the small positive parameter on the term with the highest order of derivative exhibits two strong boundary layers in the solution of the problem, and the large delay term gives rise to a strong interior layer. The layers’ behavior makes it difficult to solve the problem analytically. To treat such a problem, we developed a numerical scheme using the Crank–Nicolson method in the time direction and the central difference method in the spatial direction via nonstandard finite difference methods on uniform meshes. Stability and convergence analyses for the obtained scheme have been established, which show that the developed numerical scheme is uniformly convergent. To confirm the theoretical analysis, model numerical examples are considered and demonstrated.