An effective numerical approach for two parameter time-delayed singularly perturbed problems

A numerical scheme for the two-parameter singularly perturbed parabolic initial-boundary-value problems with a delay in time is considered. The solution to these problems exhibits twin boundary layers near the endpoints of the spatial domain. An appropriate piecewise-uniform mesh is constructed to resolve these layers. First, the given problem is semi-discretized in the temporal direction by employing the Crank-Nicolson scheme resulting in a system of ordinary differential equations at each time level. Then, to solve these systems, B-spline basis functions with the piecewise-uniform mesh leading to a tri-diagonal system of algebraic equations are used. The tri-diagonal system of algebraic equations is solved using the Thomas algorithm. Through rigorous analysis, we have shown that the scheme is second-order accurate in time and almost second-order accurate in space. Four test problems are solved to validate the theoretical results.