Numerical Treatment of a Class of Singularly Perturbed Differential Equations with Nonlocal Boundary Conditions

Abstract

This dissertation presents a numerical treatment of a class of singularly perturbed problems with nonlocal boundary conditions. The study of these singular perturbation problems has a prominent role in many areas of science and engineering. Such problems are often described by differential equations. These differential equations become singularly perturbed when the highest order derivative is multiplied by a small perturbation parameter (ε). This leads to the presence of boundary layers which are essentially small regions in the neighborhood of the boundary of the domain and/or interior layer, where the solution changes rapidly. Due to the existence of the singular perturbation parameter ε, classical numerical methods for solv ing singularly perturbed problems are not well-posed and fail to give an accurate solution. To overcomes this limitations, it is essential to formulate numerical methods that are uni formly convergent independent of the values of perturbation parameter. The main aim of this dissertation is to develop, analyze and improve the ε-uniform numerical methods for solving a class of singularly perturbed differential equations with nonlocal boundary conditions. In this study, exponentially fitted finite difference method, fitted mesh finite difference method, nonstandard finite difference method and hybrid numerical method were formulated to solve a class of SPDEs with nonlocal boundary conditions. As a result, the developed numerical methods were shown to be stable, efficient, and uniformly convergent independent of the val ues of perturbation parameter. The nonlocal boundary conditions given at the right end of the spatial domain was treated using Simpson’s rule of integration. The uniform stability and convergence analysis of the developed numerical methods were analyzed. In order to validate the applicabilty of the developed methods, numerical examples were considered and solved for different perturbation parameters and mesh sizes. The numerical algorithm was written using Matlab Software. The maximum pointwise error of the developed numerical methods was computed using double mesh principle. The rate of convergence of the devel oped numerical methods was also computed. The developed numerical methods are efficient and it achieves higher accuracy comparing with other methods. The numerical results are in agreement with the theoretical estimates. The dissertation concludes with a brief discussion of potential directions for future research.