Philosophiae Doctor - PhD / Fitted Numerical Methods for Delay Di erential Equations Arising in Biology E.B.M. Bashier PhD thesis, Department of Mathematics and Applied Mathematics,Faculty of Natural Sciences, University of the Western Cape.
This thesis deals with the design and analysis of tted numerical methods
for some delay di erential models that arise in biology. Very often such
di erential equations are very complex in nature and hence the well-known
standard numerical methods seldom produce reliable numerical solutions
to these problems. Ine ciencies of these methods are mostly accumulated
due to their dependence on crude step sizes and unrealistic stability conditions.This usually happens because standard numerical methods are
initially designed to solve a class of general problems without considering
the structure of any individual problems. In this thesis, issues like these
are resolved for a set of delay di erential equations. Though the developed
approaches are very simplistic in nature, they could solve very complex
problems as is shown in di erent chapters.The underlying idea behind the construction of most of the numerical methods in this thesis is to incorporate some of the qualitative features of the solution of the problems into the discrete models. Resulting methods are termed as tted numerical methods. These methods have high stability properties, acceptable (better in many cases) orders of convergence, less computational complexities and they provide reliable solutions with less CPU times as compared to most of the other conventional solvers. The results obtained by these methods are comparable to those found in the literature. The other salient feature of the proposed tted methods is that they are unconditionally stable for most of the problems under consideration.We have compared the performances of our tted numerical methods with well-known software packages, for example, the classical fourth-order Runge-Kutta method, standard nite di erence methods, dde23 (a MATLAB routine) and found that our methods perform much better.
Finally, wherever appropriate, we have indicated possible extensions of
our approaches to cater for other classes of problems. May 2009.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uwc/oai:etd.uwc.ac.za:11394/3134 |
| Date | January 2009 |
| Creators | Bashier, Eihab Bashier Mohammed |
| Contributors | Patidar, Kailash C. |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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