Numerical Treatment for a Class of Singularly Perturbed Parabolic Partial Differential Equations

Abstract

This dissertation focused on the development of efficient numerical methods for singularly perturbed parabolic partial differential equations. A singularly perturbed parabolic partial differential equation is a partial differential equation in which the highest space derivative of the differential equation is multiplied by a small parameter ε. When such a differential equation contains at least one delay parameter on the term different from the highest space derivative, then the differential equation is said to be a singularly perturbed delay differential equation. On the other hand, when the convection coefficient of the differential equation vanishes at a point in the domain, the problem is said to be a turning point problem. Due to the presence of the parameter ε, the solution of such differential equations exhibits a narrow region in which the solution varies rapidly near the layer region while changing slowly and smoothly away from it as ε approaches zero. This multi-scale nature of the solution makes it difficult to solve such differential equations analytically or standard numerical methods. As a result, it is essential to develop ε-uniform numerical methods to overcome the limitation. The main aim of this study is to develop, analyze and improve the ε-uniform numerical methods for solving a class of singu larly perturbed parabolic partial differential equations having a time delay and a degenerating coefficient. Specifically, the two numerical approaches such as fitted operator finite difference method or fitted mesh finite difference method were used to develop the ε-uniform numeri cal methods. In the first approach, the construction of a fitted operator ε-uniform numerical scheme involves introducing an exponential fitting factor using the standard finite difference operator on a uniform mesh. The other successful approach of formulating a ε-uniform nu merical scheme involves the use of standard finite difference operators on special piecewise uniform meshes condensed around the layer region. The stability and uniform convergence of these numerical methods were thoroughly analyzed and established. Extensive numerical com putations were conducted to validate the theoretical findings. The results obtained using the proposed methods demonstrate higher accuracy compared to existing methods in the literature. Additionally, graphical representations were used to depict the boundary layer behaviour of the solutions.