NUMERICAL SOLUTION OF SECOND ORDER SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS

Abstract

The purpose of this study is to investigate the numerical solution of second order singularly perturbed differential-difference equations by using second order finite difference methods. Taylor series expansion is used to expand the terms containing the shift parameters and Finite Difference Method is applied to discretize the given equations to provide the derivation of the numerical scheme. The second order singularly perturbed differential difference equations is replaced by an asymptotically equivalent singularly perturbed boundary value problem. A fitting factor is introduced in the finite difference scheme which takes care of the rapid changes that occur in the boundary layer and is obtained from the theory of singular perturbations. The convergence of the numerical scheme is discussed. Thomas Algorithm is used to solve the tri-diagonal system and its stability investigated. To validate the applicability of the method, four test examples have been solved by taking different values for the delay parameter , advanced parameter and the perturbation parameter . The MATLAB code is developed and all the computations are performed using MATLAB.