Singularly perturbed reaction diffusion problem with large spatial delay via non-standard fitted operator method

Abstract

In this article, a numerical method for singularly perturbed partial differential equations of the reaction-diffusion type with a large spatial delay is constructed. The method is developed using the Crank-Nicolson method for time derivative and a non-standard finite difference method for spatial derivative on a uniform mesh. Due to the presence of a small perturbation parameter, which multiplies the highest order derivative term, the solution exhibits twin boundary layers. The solution to the problem also shows an interior layer as a result of the delay in the spatial variable. To enhance the order of convergence, the Richardson extrapolation technique is applied. The convergence analysis of the proposed scheme is given. It is proved that the resulting scheme is uniformly convergent of fourth-order accurate in both space and time variables. Numerical experiments are conducted, and the results are in agreement with the theoretical findings. In addition, comparisons are performed, and the results show that the proposed scheme gives more accurate solutions and a higher rate of convergence than previous findings in the literature.