efficient numerical methods on modified graded mesh for singularly perturbed parabolic problem with time delay

In this article, we develop an efficient numerical method for one-dimensional time-delayed singularly perturbed parabolic problems. the proposed numerical approach comprises an upwind difference scheme with modified graded mesh in the spatial direction and a backward euler scheme on uniform mesh in the temporal direction. in order to capture the local behavior of the solutions, stability and error estimations are obtained with respect to the maximum norm. the proposed numerical method converges uniformly with first-order up to logarithm in the spatial variable and also first-order in the temporal variable. finally, the outcomes of the numerical experiments are included for two test problems to validate the theoretical findings.