Spline approximation method for singularly perturbed differential-difference equation on nonuniform grids

In this paper, a second-order singularly perturbed differential-difference equation involving mixed shifts is considered. At first, through Taylor series approximation, the original model is reduced to an equivalent singularly perturbed differential equation. Then, the model is treated by using the hybrid finite difference scheme on different types of layer adapted meshes like Shishkin mesh, Bakhvalov–Shishkin mesh and Vulanović mesh. Here, the hybrid scheme consists of a cubic spline approximation in the fine mesh region and a midpoint upwind scheme in the coarse mesh region. The error analysis is carried out and it is shown that the proposed scheme is of second-order convergence irrespective of the perturbation parameter. To display the efficacy and accuracy of the proposed scheme, some numerical experiments are presented which support the theoretical results.