a second order numerical method for two-parameter singularly perturbed time-delay parabolic problems

In this article, a time delay parabolic convection-reaction-diffusion singularly perturbed  problem with two small parameters is considered. we investigate the layer behavior of the  solution for both smooth and non-smooth data. a numerical method to solve the problems described is developed using the crank-nicolson scheme to discretize the time-variable on a  uniform mesh while a hybrid finite difference is applied for the space-variable. the hybrid  scheme is a combination of the central, upwind and mid-point differencing on a piecewise uniform mesh of shishkin type. the convergence analysis shows that the proposed method is  uniformly convergent of second order in both  space and   time. numerical experiments  conducted on some test examples confirm the theoretical results.