A thesis submitted to the Faculty of Science, University of the Witwatersrand,
Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy.
Johannesburg, August 2015. / We construct, using rst principles, a number of non-trivial conservation
laws of some partial di erence equations, viz, the discrete Liouville equation
and the discrete Sine-Gordon equation. Symmetries and the more recent
ideas and notions of characteristics (multipliers) for di erence equations are
also discussed.
We then determine the symmetry generators of some ordinary di erence
equations and proceed to nd the rst integral and reduce the order of the
di erence equations. We show that, in some cases, the symmetry generator
and rst integral are associated via the `invariance condition'. That is,
the rst integral may be invariant under the symmetry of the original di erence
equation. We proceed to carry out double reduction of the di erence
equation in these cases.
We then consider discrete versions of the Painlev e equations. We assume
that the characteristics depend on n and un only and we obtain a number
of symmetries. These symmetries are used to construct exact solutions in
some cases.
Finally, we discuss symmetries of linear iterative equations and their transformation
properties. We characterize coe cients of linear iterative equations
for order less than or equal to ten, although our approach of characterization
is valid for any order. Furthermore, a list of coe cients of linear iterative
equations of order up to 10, in normal reduced form is given.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/19366 |
| Date | 22 January 2016 |
| Creators | Folly-Gbetoula, Mensah Kekeli |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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