Robust Fitted Parameter Finite Difference Method for Delay-Influenced Singularly Perturbed Differential-Difference Equations

Abstract

In this study, we present a novel computational method based on the finite difference approach to solve a class of differential–difference equations characterized by a negative shift in the differentiated term. When the shift parameter is of order $O(\varepsilon)$, the proposed scheme demonstrates strong performance and effectively suppresses oscillations in the solution’s boundary-layer region. To achieve this, we introduce a parameter into the numerical scheme, constructed on a specially designed mesh, whose evaluation is guided by the theory of singular perturbations. Numerical experiments are conducted to validate the method, with maximum absolute errors and convergence orders tabulated to illustrate its efficiency and to substantiate the theoretical convergence analysis.