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A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in requirement for the degree Master of Science, January 2017. / 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. / XL2018

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulflment of the requirements for the degree of Master of science. Johannesburg, December 2012. / Solutions of linear iterative equations and expressions for these solutions in terms of the parameters of the source equation are obtained. Based on certain properties of iterative equations, nding the solutions is reduced to nding group-invariant solutions of the second-order source equation. We have therefore found classes of solutions to the source equations. Regarding the expressions of the solutions in terms of the parameters of the source equation, an ansatz is made on the original parameters r and s, by letting them be functions of a speci c type such as monomials, functions of exponential and logarithmic type. We have also obtained an expression for the source parameters of the transformed equation under equivalence transformations and we have looked for the conservation laws of the source equation. We conducted this work with a special emphasis on second-, third- and fourth-order equations, although some of our results are valid for equations of a general order.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2015. / This work deals with conditional symmetries of ordinary di erential equations (ODEs). We re ne the de nition of conditional symmetries of systems of ODEs in general and provide an algorithmic viewpoint to compute such symmetries subject to root di erential equations. We prove a proposition which gives important and precise criteria as to when the derived higher-order system inherits the symmetries of the root system of ODEs. We rstly study the conditional symmetry properties of linear nth-order (n 3) equations subject to root linear second-order equations. We consider these symmetries for simple scalar higherorder linear equations and then for arbitrary linear systems. We prove criteria when the derived scalar linear ODEs and even order linear system of ODEs inherit the symmetries of the root linear ODEs. There are special symmetries such as the homogeneity and solution symmetries which are inherited symmetries. We mention here the constant coe cient case as well which has translations of the independent variable symmetry inherited. Further we show that if a system of ODEs has exact solutions, then it admits a conditional symmetry subject to the rst-order ODEs related to the invariant curve conditions which arises from the known solution curves. This is even true if the system has no Lie point sym

University of Macau / Faculty of Science and Technology. / Department of Mathematics

by Li Leong Kwan. / Bibliography: leaves [38]-39 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1984

by Chin Yu-Tung. / Thesis (M.Phil)--Chinese University of Hong Kong, 1981. / Bibliography: leaf 32.

Wudu Lu. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 114-116). / Abstract --- p.ii / Introduction --- p.iv / Chapter 1 --- The Fundamental Theory of NFDEs with Infinite Delay --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Phase spaces and NFDEs with infinite delay --- p.2 / Chapter 1.3 --- Local theory --- p.4 / Chapter 2 --- Periodicity and Bp -Boundedness in Neutral Systems of Non- linear D-operator with Infinite Delay --- p.12 / Chapter 2.1 --- Introduction --- p.12 / Chapter 2.2 --- Preliminaries --- p.15 / Chapter 2.3 --- Existence of periodic solutions --- p.22 / Chapter 2.4 --- BP-U.B and Bp -U.U.B of solutions --- p.29 / Chapter 2.5 --- Applications --- p.42 / Chapter 3 --- Stability in Neutral Differential Equations of Nonlinear D- operator with Infinite Delay --- p.47 / Chapter 3.1 --- Introduction --- p.47 / Chapter 3.2 --- Preliminaries --- p.49 / Chapter 3.3 --- Uniformly Asymptotic Stability --- p.57 / Chapter 3.4 --- Applications --- p.74 / Chapter 4 --- Nonoscillation and Oscillation of First Order Linear Neutral Equations --- p.79 / Chapter 4.1 --- Introduction --- p.79 / Chapter 4.2 --- Existence of Nonoscillatory Solutions --- p.80 / Chapter 4.3 --- Oscillation --- p.90 / Chapter 5 --- Nonoscillation and Oscillation of First Order Nonlinear Neu- tral Equations --- p.94 / Chapter 5.1 --- Introduction --- p.94 / Chapter 5.2 --- Existence of Nonoscillatory Solutions --- p.95 / Chapter 5.3 --- Oscillation --- p.102 / Bibliography --- p.108 / List of Author's Publications --- p.114