Numerical Solution of Singularly Perturbed Delay Differential Equations with Layer Behavior

We present a numerical method to solve boundary value problems (BVPs) for singularly perturbed differential-difference equations with negative shift. In recent papers, the term negative shift has been used for delay. The Bezier curves method can solve boundary value problems for singularly perturbed differential-difference equations. The approximation process is done in two steps. First we divide the time interval, into subintervals; second we approximate the trajectory and control functions in each subinterval by Bezier curves. We have chosen the Bezier curves as piecewise polynomials of degree and determined Bezier curves on any subinterval by control points. The proposed method is simple and computationally advantageous. Several numerical examples are solved using the presented method; we compared the computed result with exact solution and plotted the graphs of the solution of the problems. 1. Introduction In recent years, there has been a growing interest in the singularly perturbed delay differential equation (see [1–4]). A singularly perturbed delay differential equation is an ordinary differential equation in which the highest derivative is multiplied by a small parameter and involving at least one delay term. Such types of differential equations arise frequently in applications, for example, the first exit time problem in modeling of the activation of neuronal variability [5], in a variety of models for physiological processes or diseases [6], to describe the human pupil-light reflex [7], and variational problems in control theory and depolarization in Stein’s model [8]. Investigation of boundary value problems for singularly perturbed linear second-order differential-difference equations was initiated by Lange and Miura [5, 9, 10]; they proposed an asymptotic approach in study of linear second-order differential-difference equations in which the highest order derivative is multiplied by small parameters. Kadalbajoo and Sharma [11–14] discussed the numerical methods for solving such type of boundary value problems. Amiraliyev and Erdogan [15] and Amiraliyeva and Amiraliyev [16] developed robust numerical schemes for dealing with singularly perturbed delay differential equation. In the present work we suggest a technique similar to the one which was used in [17, 18] for solving singularly perturbed differential-difference equation with delay in the following form (see [13]): where is small parameter, , and is also a small shifting parameter, , , , , and are assumed to be smooth, and is a constant. For , the problem is a boundary value problem