This thesis is concerned with the solution of large systems of linear algebraic equations in which the matrix of coefficients is sparse. Such systems occur in the numerical solution of elliptic partial differential equations by finite-difference methods. By applying some well-known iterative methods, usually used to solve linear PDE systems, the thesis investigates their applicability to solve a set of four mildly nonlinear test problems. In Chapter 4 we study the basic iterative methods and semiiterative methods for linear systems. In particular, we derive and apply the CS, SOR, SSOR methods and the SSOR method extrapolated by the Chebyshev acceleration strategy. In Chapter 5, three ways of accelerating the SOR method are described together with the applications to the test problems. Also the Newton-SOR method and the SOR-Newton method are derived and applied to the same problems. In Chapter 6, the Alternating Directions Implicit methods are described. Two versions are studied in detail, namely, the Peaceman-Rachford and the Douglas-Rachford methods. They have been applied to the test problems for cycles of 1, 2 and 3 parameters. In Chapter 7, the conjugate gradients method and the conjugate gradient acceleration procedure are described together with some preconditioning techniques. Also an approximate LU-decomposition algorithm (ALUBOT algorithm) is given and then applied in conjunction with the Picard and Newton methods. Chapter 8 contains the final conclusions.

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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.

Les propriétés des supraconducteurs sont à la base d’une multitude d’applications dans les domaines tels que l’ingénierie, la médecine, ou encore la recherche fondamentale. Les travaux de caractérisation réalisés depuis la découverte de la supraconductivité ont permis d’introduire des lois d’évolution macroscopiques. Elles sont aujourd’hui très utilisées pour dimensionner les nouvelles applications de ces matériaux. L’une d’elle est une relation de type puissance qui relie la densité de courant au champ électrique E Jn. Lorsqu’elle est associée aux équations de Maxwell, on obtient des problèmes différentiels complexes dont la résolution est devenu un axe de recherche très important pour la caractérisation des ces matériaux. Les travaux présentés dans ce manuscrit s’articulent autour de la résolution du problème de diffusion non linéaire satisfait par le champ électrique. Dans ceux-ci, on utilise d’abord une approche analytique basée sur le principe d’auto-similarité pour caractériser la pénétration de la densité de courant dans une plaque supraconductrice bornée. Cette solution nous permet de valider la méthode numérique mixte éléments finis-volumes finis (FEM-FVM) proposée pour faire face aux difficultés qu’introduit l’exposant n de la caractéristique E(J). L’exploitation des calculs numériques effectués pour n 2 [1, 200], nous ont notamment permis de mettre en évidence l’influence de la vitesse de variation d’une source imposée sur l’aimantation et la dissipation dans le matériau. / The properties of superconductors are behind of many applications in fields such as engineering, medicine or yet fundamental research.The works of characterization performed since the discovery of superconductivity has enabled the introduction of macroscopic evolution laws. They are now widely used to size the new applications of these materials. One of them is a power law linking the current density the electric field E Jn. When it is coupled with Maxwell equations, we obtain complex differentials problems, whose resolution has become an very important axis of research for the characterization of these materials. The work presented in this manuscript are focused on solving the nonlinear diffusion equation satisfied by the electric field. In these, one first uses an analytical approach based on the principle of self-similarity in order to characterize the penetration of the current density in a superconducting bounded plate. This solution allows us to validate the mixed finite element-finite volume (FVM-FEM) method proposed in order to face up the difficulties introduced by the exponent n of power law E(J). The use of numerical calculations performed for n in[1, 200] allowed us to highlight the influence of the rate of change of imposed source on the magnetization and dissipation in the material.

The primary intent of this thesis is to uncover the presence of matrices in polynomials and also to demonstrate that wherever there are vast numbers of interlocking relationships that must be handled, it is reasonable to guess that matrices will appear on the scene and lend their strength to facilitate the process. Most important of all, I seriously expose this omnipresent ability of matrices in polynomials and matrix equations. Matrix polynomials and equations are, in the main, a part of algebra, but it has become increasingly clear that they possess a utility that extends beyond the domain of algebra into other regions of mathematics. More than this, we have discovered that they are exactly the means necessary for expressing many ideas of applied mathematics. My thesis illustrates this for polynomials and equations of matrices.

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

University Of The Witwatersrand Department Of Computational And Applied Mathematics Masters’ Dissertation 2015 / The ideal hitting location on a cricket bat, the ‘sweet-spot’, is taken to be defined in two parts: 1) the Location of Impact on a cricket bat that transfers the maximum amount of energy into the batted ball and 2) the Location of Impact that transfers the least amount of energy to the batsman’s hands post-impact with the ball; minimizing the unpleasant stinging sensation felt by the batsman in his hands. An analysis of di↵erent hitting locations on a cricket bat is presented with the cricket bat modelled as a one dimensional beam which is approximated by the Euler-Lagrange Beam Equation. The beam is assumed to have uniform density and constant flexural rigidity. These assumptions allow for the Euler-Lagrange Beam Equation to be simplified considerably and hence solved numerically. The solution is presented via both a Central Time, Central Space finite di↵erence scheme and a Crank-Nicolson scheme. Further, the simplified Euler-Lagrange Beam Equation is solved analytically using a Separation of Variables approach. Boundary conditions, initial conditions and the framework of various collision scenarios between the bat and ball are structured in such a way that the model approximates a batsman playing a defensive cricket shot in the first two collision scenarios and an aggressive shot in the third collision scenario. The first collision scenario models a point-like, impulsive, perpendicular collision between the bat and ball. A circular Hertzian pressure distribution is used to model an elastic, perpendicular collision between the bat and ball in the second collision scenario, and an elliptical Hertzian pressure distribution does similarly for an elastic, oblique collision in the third collision scenario. The pressure distributions are converted into initial velocity distributions through the use of the Lagrange Field Equation. The numerical solution via the Crank-Nicolson scheme and the analytical solution via the Separation of Variables approach are analysed. For di↵erent Locations of Impact along the length on a cricket bat, a post-impact analysis of the displacement of points along the bat and the strain energy in the bat is conducted. Further, through the use of a Fourier Transform, a post-impact frequency analysis of the signals travelling in the cricket bat is performed. Combining the results of these analyses and the two-part definition of a ‘sweet-spot’ allows for the conclusion to be drawn that a Location of Impact as close as possible to the fixed-end of the cricket bat (a point just below the handle of the bat) results in minimum amount of energy transferred to the hands of the batsman. This minimizes the ‘stinging’ sensation felt by the batsman in his hands and satisfies the second part of the definition of a sweet-spot. Due to the heavy emphasis of the frequency analysis in this study, the conclusion is drawn that bat manufacturers should consider the vibrational properties of bats more thoroughly in bat manufacturing. Further, it is concluded that the solutions from the numerical Crank-Nicolson scheme and the analytical Separation of Variables approach are in close agreement.