A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand, Johannesburg, 2015. / This dissertation analyses the reaction-di usion equations, in particular the modi ed
Huxley model, arising in population dynamics. The focus is on determining the classical
Lie point symmetries, and the construction of the conservation laws and group-invariant
solutions for reaction-di usion equations. The invariance criterion for determination
of classical Lie point symmetries results in a system of linear determining equations
which can be solved analytically. Furthermore, the Lie point symmetries associated
with the conservation laws are determined. Reductions by associated Lie point symmetries
are carried out. Nonclassical symmetry techniques are also employed. Here
the invariance criterion for symmetry determination results in a system of nonlinear
determining equations which may be solved albeit di cult. Nonclassical symmetries
results in exact solutions which may not be constructed by classical Lie point symmetries.
The highlight in construction of exact solution using nonclassical symmetries
is the introduction of the modi ed Hopf-Cole transformation. In this dissertation, the
di usion term and the coe cient of the source term are given as quadratic functions
of space variable in one case, and the coe cient as the generalised power law in the
other. These equations admit a number of classical Lie point symmetries. The genuine
nonclassical symmetries are admitted when the source term of the reaction-di usion
equation is a cubic.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/18556 |
| Date | 29 May 2015 |
| Creators | Louw, Kirsten |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf, application/pdf |
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