Philosophiae Doctor - PhD / Singularly perturbed problems have been studied extensively over the past
few years from different perspectives. The recent research has focussed on the
problems whose solutions possess interior layers. These interior layers appear
in the interior of the domain, location of which is difficult to determine a-priori
and hence making it difficult to investigate these problems analytically. This
explains the need for approximation methods to gain some insight into the behaviour
of the solution of such problems. Keeping this in mind, in this thesis
we would like to explore a special class of numerical methods, namely, fitted
finite difference methods to determine reliable solutions. As far as the fitted
finite difference methods are concerned, they are grouped into two categories:
fitted mesh finite difference methods (FMFDMs) and the fitted operator finite
difference methods (FOFDMs). The aim of this thesis is to focus on the
former. To this end, we note that FMFDMs have extensively been used for
singularly perturbed two-point boundary value problems (TPBVPs) whose
solutions possess boundary layers. However, they are not fully explored for
problems whose solutions have interior layers. Hence, in this thesis, we intend firstly to design robust FMFDMs for singularly perturbed TPBVPs whose solutions
possess interior layers and to improve accuracy of these approximation
methods via methods like Richardson extrapolation. Then we extend these
two ideas to solve such singularly perturbed TPBVPs with variable diffusion
coefficients. The overall approach is further extended to parabolic singularly
perturbed problems having constant as well as variable diffusion coefficients. / 2023-08-31
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uwc/oai:etd.uwc.ac.za:11394/7324 |
| Date | January 2020 |
| Creators | Mbayi, Charles K. |
| Contributors | Munyakazi, Justin B., Patidar, Kailash C. |
| Publisher | University of Western Cape |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Rights | University of Western Cape |
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