Singularly Perturbed Parabolic Differential Equations with Turning Point and Retarded Arguments

In this article we propose an efficient numerical scheme based on a Shishkin mesh for a class of singularly perturbed parabolic convection-diffusion problems with boundary turning point and retarded arguments. The solution of the considered problem exhibit a boundary layer on the left side of the domain. The continuous problem is semidiscretized by means of backward Euler finite difference method in time to get a system of ordinary differential equations at each time level. This system of differential equations is discretized by using the standard upwind finite difference scheme on a nonuniform mesh of Shishkin type. It has been shown theoretically that the numerical solution generated by the method converges uniformly to the solution of the continuous problem with respect to the singular perturbation parameter. Numerical experiments supporting the theoretical results are given.