Fourth-Order Stable Central Difference Method For Self-Adjoint Singular Perturbation Problems

Abstract

In this thesis, fourth order stable central difference method has been presented for solving selfadjoint singular perturbation problems for small values of perturbation parameter . First, the given interval is discritized and the derivative of the given differential equation is replaced by the finite difference approximations. Then, the given differential equation is transformed to linear system of algebraic equations. Further, these algebraic equations are transformed into a three-term recurrence relation, which can easily be solved by using Thomas Algorithm. To validate the applicability of the proposed method, four model examples with and without exact solution have been considered and solved for different values of perturbation parameter and mesh sizes. Both theoretical error bounds and numerical rate of convergence have been established for the method. As it can be observed from the numerical results presented in tables compared to the numerical solution by Kadalbajoo and Kumar [17], Kumar and Kadalbajoo [19] and Patidar and Kadalbajoo [29] from literature and graphs, the present method approximates the exact solution very well.