A THETA METHOD BASED APPROACH FOR SINGULARLY PERTURBED DIFFERENTIAL–DIFFERENCE EQUATIONS

Abstract

In this study, we examine a singularly perturbed differential–difference boundary value problem that captures the layer behaviour induced by a small diffusion parameter together with delay or advance terms.. Using Taylor expansions of the shifted arguments we reduce the original differential–difference equation to a second-order singular perturbation problem subject to prescribed data on the delay/advance parameters. By introducing an auxiliary function and a further perturbation transformation we obtain an equivalent first-order linear problem for an auxiliary variable which can be solved sequentially. A θ-method (implicit one-step discretisation) is proposed for the resulting first-order equation; its consistency and stability are analysed and the leading order error terms are estimated. Numerical implementation via the three-step procedure yields an efficient scheme that captures the layer structure when