Numerical Treatment of Singularly Perturbed Delay Differential Equations using Hybrid Finite Difference Scheme

Abstract

This study introduces an efficient numerical method tailored for singularly perturbed delay differential equations (SPDDEs), leveraging a hybrid finite difference framework. Such equations, prevalent in scientific and engineering contexts, often display intricate solution features like sharp boundary and interior layers due to the interplay of small parameters and delay effects. The developed approach utilizes a third-order Adams–Moulton scheme, integrated with a specially designed fitting parameter, to accurately capture these rapid transitions. Through theoretical justification, the method is shown to convert the SPDDE into a parameter-uniform boundary value problem, which is then solved using a tridiagonal matrix algorithm. Extensive computational experiments on standard test cases with varying delay and perturbation values demonstrate that the proposed technique consistently delivers high accuracy and uniform convergence, outperforming established methods. These findings underscore the method’s reliability, stability, and broad applicability for challenging delay differential equations exhibiting layer phenomena.