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The primary purpose of this thesis is to carefully develop and prove some of the fundamental, classical theorems of the differential calculus for functions of two real variables.

This paper is concerned with equations in which all derivatives are ordinary rather than partial derivatives. The customary meanings of differential order and difference order of an equation are observed.

The purpose of this research study was to determine whether the semantic differential could measure in the college population the variation in meaning of selected masculine and feminine concepts as a function of sex difference.

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

We give an overview of the numerical solution of the initial value differential-algebraic equation (DAB) from the underlying theory, through both the development of numerical techniques and software and a survey of the major areas of application, to the implementation of available software codes in the solution of DABs arising in applications. The experimental part serves to demonstrate the need for specific DAB, rather than simply ordinary differential equation (ODE), methods and the special considerations requited for the successful numerical solution of DAEs, as well.as verifying predictions made in the theory. It is hoped that '.his dissertation can be used as a reference by those working in areas of application, or at least as a pointer to the relevant literature. / GR 2016

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