GRADEDMESHMETHODSFORTWOPARAMETER SINGULARLYPERTURBEDSECONDORDER BOUNDARYVALUEPROBLEMS

Abstract

This dissertation presents fitted mesh numerical methods for the solution of a specific class of two-parameter singularly perturbed second-order boundary value problems in ordinary differential equations. These problems occur in several areas, such as chemical reactor theory, lubrication theory, the theory of plates and shells, and D-C motors. The solutions to these problems exhibit multi-scale characteristics, featuring a narrow region known as the boundary layer where solutions change rapidly, and an outer region where solutions change more gradually. Due to the presence of boundary layers, standard numerical methods are inadequate for solving such multi-scale problems. This work develops graded mesh B-spline collocation and finite difference methods designed to capture solutions within the boundary layer, providing stable and accurate approximations of the entire solution. A detailed stability analysis is conducted for each method, along with a comprehensive error analysis to demonstrate parameter uniform convergence. To validate the theoretical f indings, numerical experiments were performed using known test problems in the literature. Additionally, the developed methods were compared with other existing methods in the literature to showcase their superior efficiency in terms of accuracy and rate of convergence. The results indicate that the theoretical and experimental findings align; the methods achieve parameter uniform convergence of order two, and outperform certain existing methods. Overall, this dissertation makes a significant contribution to the numerical treatment of two-parameter singularly perturbed second-order boundary value problems and serves as a valuable resource for researchers and practitioners in various applied mathematics fields.