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Xiao, Jingjing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 46-50). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Backgrounds and Our Main Results --- p.4 / Chapter 2.1 --- Backgrounds --- p.4 / Chapter 2.1.1 --- The Scalar Case --- p.4 / Chapter 2.1.2 --- 2x2 Systems --- p.5 / Chapter 2.1.3 --- General n x n(n ≥ 3) Systems --- p.9 / Chapter 2.2 --- Our Main Results --- p.18 / Chapter 3 --- Lifespan of Periodic Solutions to Gas Dynamics Systems --- p.21 / Chapter 3.1 --- Riemann Invariant Formulation --- p.21 / Chapter 3.2 --- Calculation along Characteristics --- p.26 / Chapter 3.3 --- Estimate of the Global Wave Interaction --- p.35 / Chapter 3.4 --- Proof of Theorem 2.2.1 --- p.38 / Chapter 4 --- Proof of Theorem 2.2.2 and a Special Case --- p.40 / Chapter 4.1 --- Proof of Theorem 2.2.2 --- p.40 / Chapter 4.2 --- A Special Case --- p.43 / Chapter 5 --- Appendix --- p.45

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.

Thesis (Ph.D.)--City University of Hong Kong, 2009. / "Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves [90]-94)

Thesis (Ph. D.)--West Virginia University, 2010. / Title from document title page. Document formatted into pages; contains vii, 48 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 48).

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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.

There is a well established connection between one parameter Lie groups of transformations and conservation laws for differential equations. In this thesis, we construct conservation laws via the invariance and multiplier approach based on the wellknown result that the Euler-Lagrange operator annihilates total divergences. This technique will be applied to some plasma physics models. We show that the recently developed notion of the association between Lie point symmetry generators and conservation laws lead to double reductions of the underlying equation and ultimately to exact/invariant solutions for higher-order nonlinear partial di erential equations viz., some classes of Schr odinger and KdV equations.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2012. / Unable to load abstract.

PhD., Faculty of Science, University of the Witwatersrand, 2011 / The construction of conserved vectors using Noether’s theorem via a knowledge of a Lagrangian (or via the recently developed concept of partial Lagrangians) is well known. The formulae to determine these for higher-order flows is somewhat cumbersome and becomes more so as the order increases. We carry out these for a class of fourth, fifth and sixth order PDEs. In the latter case, we involve the fifth-order KdV equation using the concept of ‘weak’ Lagrangians better known for the third-order KdV case. We then consider the case of a mixed ‘high-order’ equations working on the Shallow Water Wave and Regularized Long Wave equations. These mixed type equations have not been dealt with thus far using this technique. The construction of conserved vectors using Noether’s theorem via a knowledge of a Lagrangian is well known. In some of the examples, our focus is that the resultant conserved flows display some previously unknown interesting ‘divergence properties’ owing to the presence of the mixed derivatives. We then analyse the conserved flows of some multi-variable equations that arise in Relativity. In addition to a larger class of conservation laws than those given by the isometries or Killing vectors, we may conclude what the isometries are and that these form a Lie subalgebra of the Noether symmetry algebra. We perform our analysis on versions of the Vaidya metric yielding some previously unknown information regarding the corresponding manifold. Lastly, with particular reference to this metric, we also show the variations that occur for the unknown functions. We discuss symmetries of classes of wave equations that arise as a consequence of the Vaidya metric. The objective of this study is to show how the respective geometry is responsible for giving rise to a nonlinear inhomogeneous wave equation as an alternative to assuming the existence of nonlinearities in the wave equation due to physical considerations. We find Lie and Noether point symmetries of the corresponding wave equations and give some reductions. Some interesting physical 4 conclusions relating to conservation laws such as energy, linear and angular momenta are also determined. We also present some interesting comparisons with the standard wave equations (on a ‘flat geometry’). Finally, we pursue the nature of the flow of a third grade fluid with regard to its underlying conservation laws. In particular, the fluid occupying the space over a wall is considered. At the surface of the wall, suction or blowing velocity is applied. By introducing a velocity field, the governing equations are reduced to a class of PDEs. A complete class of conservation laws for the resulting equations are constructed and analysed using the invariance properties of the corresponding multipliers/characteristics.