3. Electronic Theses and Dissertations (ETDs) - All submissions

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    Symmetry classifications on a curved geometry
    (2020) Mathebula, Agreement
    In this thesis, we consider one-parameter point transformations that leave a differential equation invariant. In particular, we show that Noether symmetry classifications of any diagonal metric may be simplified by geometric criteria. We describe the Klein-Gordon equation for some general spaces and deal with the corresponding Killing algebra. Moreover, our investigation consists of several metrics, their Lie algebras, the point generators of the Klein-Gordon equation and their associated potential functions. Finally, we study a class of ecological diffusive equations and determine higher-order symmetries of non-linear diffusion equations
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    Symmetry analysis and invariant properties of some partial differential equations
    (2017) Mathebula, Agreement
    This dissertation contains evolutionary partial differential equations (PDEs). The PDEs are used to investigate ecological phenomena. The main goal is to determine Lie point symmetries, perform Lie reduction, obtain analytical solutions and visualize the solutions in 3D plots using the help of Mathematica. Drift diffusion, biased diffusion and the Kierstead, Slobodkin and Skellam (KiSS) models arising in population ecology are discussed. The importance of these PDEs in ecology is to analyse the movements of organisms and their long-term existence especially in heterogeneous environments.
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