Fitted Finite Difference Method for Solving Singularly Perturbed Differential-Difference Equations

Abstract

In this thesis, fitted fourth order finite difference method is presented for solving singularly perturbed differential-difference equations. First the given singularly perturbed differentialdifference equations are transformed into an asymptotically equivalent singularly perturbed boundary value problem. Using fitted finite difference approximation, the given differential equation is transformed into a three-term recurrence relation, which can easily be solved by Thomas Algorithm. The stability and convergence of the method have been investigated. To validate the applicability of the proposed method three model examples have been considered and solved for different values of parameters and mesh size h. Both theoretical error bounds and numerical rate of convergence have been established for the method. The numerical results have been presented in tables and further to examine the effect of delay and advance parameters on the left and right boundary layer of the solution; graphs have been given for different values of parameters. Concisely, the present method gives better result than some existing numerical methods reported in the literature.