Philosophiae Doctor - PhD / Fitted operator finite difference methods (FOFDMs) for singularly perturbed
problems have been explored for the last three decades. The construction of
these numerical schemes is based on introducing a fitting factor along with the
diffusion coefficient or by using principles of the non-standard finite difference
methods. The FOFDMs based on the latter idea, are easy to construct and they
are extendible to solve partial differential equations (PDEs) and their systems.
Noting this flexible feature of the FOFDMs, this thesis deals with extension
of these methods to solve interior layer problems, something that was still outstanding.
The idea is then extended to solve singularly perturbed time-dependent
PDEs whose solutions possess interior layers. The second aspect of this work is
to improve accuracy of these approximation methods via methods like Richardson
extrapolation. Having met these three objectives, we then extended our
approach to solve singularly perturbed two-point boundary value problems with
variable diffusion coefficients and analogous time-dependent PDEs. Careful analyses
followed by extensive numerical simulations supporting theoretical findings
are presented where necessary.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uwc/oai:etd.uwc.ac.za:11394/7320 |
| Date | January 2020 |
| Creators | Sayi, Mbani T |
| Contributors | Munyakazi, Justin B. |
| Publisher | University of the Western Cape |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Rights | University of the Western Cape |
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