Exponentially Fitted Initial Value Technique for Singularly Perturbed Differential-Difference Equations

Abstract

In this paper, an exponentially fitted initial value technique is presented for solving singularly perturbed differential-difference equations with delay as well as advance terms whose solutions exhibit boundary layer on one (left/right) of the interval. It is distinguished by the following fact that the original second order differential-difference equation is replaced by an asymptotically equivalent singular perturbation problem and in turn the singular perturbation problem is replaced by an asymptotically equivalent first order problem and solved as an initial value problem using exponential fitting factor. To validate the method, model examples with boundary layers have been solved by taking different values for the delay parameter δ, advance parameter η and the perturba- tion parameter ɛ. The effect of the small shifts on the boundary layer has been investigated and presented in graphs. Theoretical convergence of the scheme has also been investigated.