A parameter-uniform scheme for singularly perturbed partial differential equations with a time lag

A numerical scheme for a class of singularly perturbed delay parabolic partial differential equations which has wide applications in the various branches of science and engineering is suggested. The solution of these problems exhibits a parabolic boundary layer on the lateral side of the rectangular domain which continuously depends on the perturbation parameter. For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer the extended cubic B-spline basis functions consisting of a free parameter λ are used on a fitted-mesh. The extended B-splines are the extension of classical B-splines. To find the best value of λ the optimization technique is adopted. The extended cubic B-splines are an advantage over the classical B-splines as for some optimized value of λ the solution obtained by the extended B-splines is better than the solution obtained by classical B-splines. The method is shown to be first-order accurate in t and almost the second-order accurate in x. It is also shown that this method is better than some existing methods. Several test problems are encountered to validate the theoretical results.