Numerical Solutions For Singularly Perturbed Differential-Difference Equations: A Dual-Layer Fitted Approach

Abstract

Singularly perturbed differential-difference equations (SPDDE) pose significant challenges due to their oscillatory behavior and unsatisfactory results when traditional numerical methods are applied with large step sizes relative to the perturbation parameter ε. In this study, we propose a novel computational approach for solving SPDDE, leveraging a dual-layer fitted method and Taylor series expansion. First, the given SPDDE is reduced to an ordinary singularly perturbed problem using Taylor series expansion to handle terms involving negative and positive shifts. Subsequently, a three-term numerical scheme is derived using finite differences, augmented by a fitting factor derived from singular perturbation theory to enhance accuracy and stability. The resulting tridiagonal system of equations is efficiently solved using the Thomas algorithm. To validate the proposed method, we solve model problems with varying values of ε, delay parameter δ, and advance parameter η. The computational results are compared with existing literature, presenting maximum absolute errors and graphical representations. Our approach demonstrates significant improvements in accuracy and stability, making it a valuable tool for researchers tackling SPDDE in diverse applications.