Lie group analysis is arguably the most systematic vehicle for finding exact solutions of differential equations. Using this approach one has at one's disposal a variety of algorithms that make the solution process of many differential equations algorithmic. Vital properties of a given differential equation can often be inferred from the symmetries admitted by the equation. However, Lie group analysis has not enjoyed wide-spread application to systems of first-order ordinary differential equations. This is because such systems typically admit an infinite number of Lie point symmetries, and there is no systematic way to find even a single nontrivial one-dimensional Lie symmetry algebra. In the few applications available, the approach has been to circumvent the problem by transforming a given system of first-order ordinary differential equations into one in which at least one of the equations is of order two or greater. It is therefore fair to say that the full power of Lie group analysis has not been sufficiently harnessed in the solution of systems of first-order ordinary differential equations. In this dissertation we review some applications of Lie group analysis to systems of first order ordinary differential equations. We shed light on the integration procedure for first-order systems of ordinary differential equations admitting a solvable Lie algebra. We do this via instructive examples drawn from mathematical epidemiology models. In particular we revisit the work of Nucci and Torrisi [54] and improve the exposition of the Lie-symmetry-inspired solution of a mathematical model which describes a HIV transmission. To aid implementation of the integration strategy for systems of ordinary differential equations, we have developed ad-hoc routines for finding particular types of admitted symmetries and checking if a given symmetry is indeed admitted by a system of ordinary differential equations.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wsu/vital:18486 |
| Date | January 2013 |
| Creators | Otieno, Andrew Alex Omondi |
| Publisher | Walter Sisulu University, Faculty of Science, Engineering and Technology |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis, Masters, MSc |
| Format | ix, 79 leaves, pdf |
| Rights | Walter Sisulu University |
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