Fitted Numerical Scheme for Second-Order Singularly Perturbed Differential-Difference Equations with Mixed Shifts

Abstract

This paper presents the study of singularly perturbed differential-difference equations of delay and advance parameters. The proposed numerical scheme is a fitted fourth-order finite difference approximation for the singularly perturbed differential equations at the nodal points and obtained a tridiagonal scheme. This is significant because the proposed method is applicable for the perturbation parameter which is less than the mesh size , where most numerical methods fail to give good results. Moreover, the work can also help to introduce the technique of establishing and making analysis for the stability and convergence of the proposed numerical method, which is the crucial part of the numerical analysis. Maximum absolute errors range from 10 − 03 up to 10 − 10 , and computational rate of convergence for different values of perturbation parameter, delay and advance parameters, and mesh sizes are tabulated for the considered numerical examples. Concisely, the present method is stable and convergent and gives more accurate results than some existing numerical methods reported in the literature.

Computational Approach for a Singularly Perturbed Differential Equations With Mixed Shifts Using a Non-Polynomial Spline

“…Using MATLAB, the MAEs in the solutions of the Examples 5.1, 5.2 and 5.3 are listed in comparison to the method given in [21] in Tables 1, 2, 3 and 4. Tables 5 and 6 compare the MAEs in Example 5.4 solution to the method described in [16].…”

“…In the articles [21], [22] the authors suggested an exponentially fitted FDM to solve SPDDEs with delay and advanced terms and turning point problems. The authors advised a fourth order FDM with a fitting factor in [10], [16] to solve the considered problem. In [8], [9] the authors developed some numerical techniques to solve SPDDEs with mixed shifts.…”