Uniform Hybrid Difference Scheme for Singularly Perturbed Differential-Difference Turning Point Problems Exhibiting Boundary Layers

Abstract

In this paper, a class of linear second-order singularly perturbed differential-difference turning point problems with mixed shifts exhibiting two exponential boundary layers is considered. For the numerical treatment of these problems, first we employ a second-order Taylor’s series approximation on the terms containing shift parameters and obtain a modified singularly perturbed problem which approximates the original problem. Then a hybrid finite difference scheme on an appropriate piecewise-uniform Shishkin mesh is constructed to discretize the modified problem. Further, we proved that the method is almost second-order ɛ-uniformly convergent in the maximum norm. Numerical experiments are considered to illustrate the theoretical results. In addition, the effect of the shift parameters on the layer behavior of the solution is also examined.