A NUMERICAL APPROACH WITH MODIFIED GRADED MESH FOR SINGULARLY PERTURBED DELAY REACTION-DIFFUSION PROBLEMS

This paper aims to analyze a singularly perturbed reaction-diffusion problem with time delay in one dimension using numerical techniques. We use a finite difference method, specifically the upwind scheme, to obtain the numerical solution. The solutions are calculated on a specially designed graded mesh, and error bounds are carefully assessed in the discrete minimum norm. The main goals of this study are to emphasize the importance of using a modified graded mesh for achieving improved convergence in numerical solutions and to enhance the convergence of the upwind difference scheme via Richardson extrapolation. Our theoretical framework is well-supported by numerical simulations. This study contributes to a deeper understanding of singularly perturbed time-delayed reaction-diffusion problems and highlights the advantages of modified graded meshes and Richardson extrapolation in numerical analysis of such problems.