Accelerated Finite Difference Method For Singularly Perturbed Two Parameter Boundary Value Problems

Abstract

In this thesis, accelerated finite difference method for solving singularly perturbed boundary value problems with two small parameters is presented. First, the solution domain is discretized. Then, the derivatives in the given boundary value problem are replaced by finite difference approximations.The numerical scheme that provides algebraic systems of equations is obtained. The system can easily be solved by Thomas algorithm. The consistency, stability and convergence of the method have been established. The established convergence of the scheme is further accelerated by applying the Richardsons extrapolation which yields sixth order convergence. To validate the applicability of the proposed method, two model examples have been considered and solved for different values of perturbation parameters and mesh sizes. Both theoretical error bounds and numerical rate of convergence have been established for the method. The numerical results have been presented in tables and graphs to illustrate; the present method approximates the exact solution very well. Moreover, the present method gives better results than some of numerical methods mentioned in the literature.