Upwind and midpoint upwind difference methods for time-dependent differential difference equations with layer behavior

In this paper we have considered an important class of time-dependent singularly perturbed convection-diffusion problems with retarded terms which often arise in Computational Neuroscience. We used Taylors series to approximate the retarded terms and the resulting time-dependent singularly perturbed differential equation is approximated using parameters uniform numerical methods based on Euler implicit, upwind and midpoint upwind finite difference schemes. We discretize the continuous problem using implicit Euler scheme in the time direction with a constant step size and the resulting system of equations is approximated using upwind and midpoint upwind difference schemes on a piecewise uniform mesh. We will prove the uniform convergence of these two schemes. Numerical experiments support the convergence results.