Parameter uniform numerical methods for some singularly perturbed problems

Abstract

In this thesis, efficient numerical techniques have been proposed to solve different types of singularly perturbed problems. That is problems whose solutions are characterized with rapid oscillation usually known as layers. The presence of these layers makes the use of classical numerical methods unfit to solve such problems since they are unable to mimic the behaviour of the exact solution in the part of the domain where the layers are present. This thesis seeks to address this issue by modifying the classical numerical methods such that they will be able to capture the behaviour of the exact solution in the part of the domain where the layers are present. Thus we design, analyse, implement and enhance the accuracy of efficient numerical methods to solve various forms of singularly perturbed problems. The scheme is based on the Non-standard finite difference schemes and used with other time integrators to solve parabolic problems either via the full discrteisation schemes or the step by step discretisation schemes (Rothe's method or the method of lines). For stationary integro-differential problems, the scheme is employed along with the Simpson's rule to provide the numerical solution. The mathematical software Matlab has been used for all simulations in this thesis.