A numerical scheme to solve boundary value problems involving singular perturbation

Abstract

In this study, a numerical method is presented in order to approximately solve singularly perturbed second-order differential equations givenwith boundary conditions. Themethod uses the set ofmonomialswhose degrees do not exceed a prescribed N as the set of base functions, resulting from the supposition that the approximate solution is a polynomial of degree N whose coefficients are to be determined. Then, following Galerkin’s approach, inner product with the base functions are applied to the residual of the approximate solution polynomial. This process, with a suitable incorporation of the boundary conditions, gives rise to an algebraic linear system of size N þ 1. The approximate polynomial solution is then obtained from the solution of this resulting system. Additionally, a technique, called residual correction,which exploits the linearity of the problem to estimate the error of any computed approximate solution is discussed briefly. The numerical scheme and the residual correction technique are illustrated with two examples.