| The main purposes of this dissertation are to study nonlinear singular perturbation problems and to provide robust and efficient numerical solutions. We construct a new method by detecting a boundary layer for the solution of a singular perturbation problem. Subsequently, the domain of a singular perturbation problem is divided into two parts: a boundary layer and a non boundary layer. On the non boundary layer, the singular perturbation problem is dominated by the reduced differential equation, which results when the singular perturbation parameter vanishes. While on the boundary layer, it is controlled by the singular perturbation, which is the original differential equation with a perturbed boundary value. In addition, we develop high order finite difference methods, up to 6th order. The boundary layer detection and the stability of a singular perturbation BVP with a perturbed boundary value are among the original contribution of this dissertation. The new method is robust in the sense of computing cost and error estimation. The numerical error is maintained at the same level with the same number of mesh points for a family of singular perturbation problems with diminishing values of the singular perturbation parameter. Numerical results are also presented which verify the theory. |
