M.Sc. (Mathematics) / Please refer to full text to view abstract

A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy to the Faculty of Science, University of the Witwatersrand, Johannesburg, 2019 / We will study the symmetry, invariance properties and conservation laws of partial dif ferential equations (pdes) that arise in a number of situations in mathematical physics. These will be range from Image Processing and noise removal algorithms to Timoshenko beam systems. Furthermore, we will study the invariance properties and approximate conservation laws of some nonlinear Schro¨dinger equation with PT-symmetric potentials with inhomogeneous nonlinearity and some nonlinear Schro¨dinger equation involving a spatially extended system consisting of two coupled elements. / TL (2019)

This research studies two nonlinear problems arising in mathematical physics. Firstly the Korteweg-de Vrics-Burgers equation is considered. Lie symmetry method is used to obtain t he exact solutions of Korteweg-de Vries-Burgers equation. Also conservation laws are obtained for this equation using the new conservation theorem. Secondly, we consider the generalized (2+ 1)-dimensional Zakharov-Kuznctsov (ZK) equation of time dependent variable coefficients from the Lie group-theoretic point of view. We classify the Lie point symmetry generators to obtain the optimal system of one-dimensional subalgebras of t he Lie symmetry algebras. These subalgebras arc then used to construct a number of symmetry reductions and exact group-invariant solutions of the ZK equation. We utilize the new conservation theorem to construct the conservation laws of t he ZK equation. / Thesis (M. Sci in Applied Mathematics) North-West University, Mafikeng Campus, 2011

This research studies two nonlinear differential equations arising in fluid mechanics. Firstly, the Zakharov-Kuznetsov's equation in (3+1) dimensions with an arbitrary power law nonlinearity is considered. The method of Lie symmetry analysis is used to carry out the integration of Zakharov-Kuznetsov's equation. Also, the extended tanh-function method and t he G'/G method are used to integrate the Zakharov-Kuznetsov's equation. The non-topological soliton solution is obtained by the aid of solitary wave ansatz method. Numerical simulation is given to support the analytical development. Secondly. the nonlinear flow problem of an incompressible viscous fluid is considered. The fluid is taken in a channel having two weakly permeable moving porous walls. An incompressible fluid fills the porous space inside the channel. The fluid is magnetohydrodynamic in the presence of a time-dependent magnetic field. Lie group method is applied along with perturbation method in the derivation of analytic solution. The effects of the magnetic field, porous medium, permeation Reynolds number and wall dilation rate on the axial velocity arc shown and discussed. / Thesis (M.Sc.(Applied Mathematics) North-West University, Mafikeng Campus, 2010

published_or_final_version / Mathematics / Master / Master of Science

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The pervading theme of this thesis is the development of insights that contribute to the understanding of whether certain classes of functional differential equation have solutions that are all oscillatory. The starting point for the work is the analysis of simple (linear autonomous) ordinary differential equations where existing results allow a full explanation of the phenomena. The Laplace transform features as a key tool in developing a theoretical background. The thesis goes on to explore the corresponding theory for delay equations, advanced equations and functional di erential equations of mixed type. The focus is on understanding the links between the characteristic roots of the underlying equation, and the presence or otherwise of oscillatory solutions. The linear methods are used as a class of numerical schemes which lead to discrete problems analogous to each of the classes of functional differential equation under consideration. The thesis goes on to discuss the insights that can be obtained for discrete problems in their own right, and then considers those new insights that can be obtained about the underlying continuous problem from analysis of the oscillatory behaviour of the analogous discrete problem. The main conclusions of the work are some semi-automated computational approaches (based upon the Principle of the Argument) which allow the prediction of oscillatory solutions to be made. Examples of the effectiveness of the approach are provided, and there is some discussion of its theoretical basis. The thesis concludes with some observations about further work and some of the limitations of existing analytical insights which restrict the reliability with which the approach developed can be applied to wider classes of problem.

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