Layer-Adapted Mesh Methods For Singularly Perturbed Parabolic Partial Differential Equations with Robin Boundary Conditions

Abstract

The main purpose of this dissertation is to present layer-adapted mesh methods for singularly perturbed parabolic partial differential equations of convection-diffusion and reaction-diffusion types with Robin boundary conditions. A singularly perturbed parabolic differential equation with Robin boundary conditions is a partial differential equation in which the highest space derivative in the differential equation and the first derivatives in the boundary conditions are multiplied by a small parameter (0 < 1). The parameter is known as the perturbation parameter. Because of the presence of , the solution of such differential equations exhibits a thin layer in which the solution varies rapidly near the layer while changing slowly and smoothly away from it. The presence of the layer phenomenon makes it difficult to solve such differential equations analytically. Thus, it is desirable to develop parameter-uniform numerical methods that help to solve singularly perturbed parabolic differential equations with Robin boundary conditions. As a result, this dissertation presents some parameter-uniform numerical methods for singu larly perturbed parabolic partial differential equations with Robin boundary conditions on well-known layer-adapted meshes of Shishkin, Bakhvalov-Shishkin and Vulanović-Shishkin types. Furthermore, the stability and convergence analysis of the present numerical meth ods are well established. To support the theoretical findings, extensive numerical com putations are carried out in all the chapters. The numerical results using the present methods improved the existing methods in the literature. At the end of the dissertation, a brief summary, conclusions and possible future scope are provided.