AN INITIAL VALUE TECHNIQUE FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS WITH A SMALL NEGATIVE SHIFT

Abstract

Abstract. In this paper, we present an initial value technique for solvingsingularly perturbed differential difference equations with a boundary layerat one end point. Taylor’s series is used to tackle the terms containing shiftprovided the shift is of small order of singular perturbation parameter andobtained a singularly perturbed boundary value problem. This singularlyperturbed boundary value problem is replaced by a pair of initial valueproblems. Classical fourth order Runge–Kutta method is used to solvethese initial value problems. The effect of small shift on the boundarylayer solution in both the cases, i.e., the boundary layer on the left sideas well as the right side is discussed by considering numerical experiments.Several numerical examples are solved to demonstate the applicability ofthe method.AMS Mathematics Subject Classification : 65L11.Key words and phrases : Initial value technique, singular perturbation,differential–difference equation, boundary layer. 1. IntroductionThe numerical treatment of singular perturbations is far from the trivial be-cause of the boundary layer behavior. In recent decades this is a field of increas-ing interest to applied mathematicians and numerical analysts in view of thechallenges the problems there in pose to the researchers. For a detailed theoryand analytical discussion on singular perturbation problems one may refer tothe books and high level monographs: O’ Malley [13], Nayfeh [12], Kevorkianand Cole [11], Bender and Orszag [10]. A Singularly perturbed delay differentialequation is an ordinary differential equation in which the highest derivative ismultiplied by a small parameter and involving atleast one delay term. In 1979,the Japanese physicist Kensuke Ikeda considered a nonlinear absorbing medium