| Singular perturbation problems appear in many areas of applied mathematics. In this dissertation, we consider several approximation methods for the numerical integration of a class of linear, singular perturbation two-point boundary value problem in ordinary differential equations with a boundary layer. Many techniques such as finite difference and asymptotic expansions have been used to solve this class of problem. However, in some cases, these methods do not apply; in other cases, they only give second order approximation.;In Chapter II, we will extend a method given by Kadalbajoo and Reddy that handles only linear cases with boundary layer at the left end of the interval to solve cases with the boundary layer at either end. Moreover, we will give a new method to solve this class of problems. Furthermore we will extend some schemes to handle nonlinear problems. Numerical results and copter codes for these methods will be given. Both the pivot method and a Gaussian elimination algorithm are used in solving a tridiagonal system resulting from using these methods on linear problems.;In Chapter III, fourth-order methods for solving two-point boundary value problems are given and refinement step size methods are applied to improve the approximation. In the final chapter, a new method for solving an integral equation is given. We demonstrate its use by solving two-point boundary value problems. |
