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This article deals with an efficient approximation method named successive complementary expansion method (SCEM) for solving singularly perturbed differential-difference equations with mixed shifts. It is compared with the method of matched asymptotic expansion (MMAE) and the parameter uniform upwind finite difference scheme for solving such a model. The comparison shows, unlike the MMAE, the SCEM method requires no matching procedure. It requires less computation when compared to the upwind finite difference scheme on the Shishkin mesh. The error analysis is carried out to prove the robustness of the method. Some numerical experiments are provided, which show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, a second-order singularly perturbed initial value problem is considered. A hybrid scheme which is a combination of a cubic spline and a modified midpoint upwind scheme is proposed on various types of layer-adapted meshes. The error bounds are established for the numerical solution and for the scaled numerical derivative in the discrete maximum norm. It is observed that the numerical solution and the scaled numerical derivative are of second-order convergence on the layer-adapted meshes irrespective of the perturbation parameter. To show the performance of the proposed method, it is applied on few test examples which are in agreement with the theoretical results. Furthermore, existing results are also compared to show the robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]