This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work. It uses a historical perspective to highlight many of the concepts underlying modern equation solving.

Several methods to solve a system of linear equations with real and complex coefficients exist. The most popular methods are Gauss-Jordan, L-U Decomposition, Gauss-Seidel, and Matrix Reduction. These methods are utilized to optimize run-time of the DLANET circuit analysis program. As concluded by this study, the Matrix Reduction method which is presently utilized in the DLANET program, results in run-times which are faster than the other solution methods studied in this paper for lower order systems. Similarly, the L-U Decomposition and Gauss-Jordan methods result in faster run-times than the other techniques for higher order systems. Finally, the Gauss-Seidel Iterative method, when incorporated into the DLANET program, has proven to be much slower than the other solution methods considered in this paper.

In this dissertation we obtain an e cient hybrid numerical method for the solution of stochastic di erential equations (SDEs). Speci cally, our method chooses between two numerical methods (Euler and Milstein) over a particular discretization interval depending on the value of the simulated Brownian increment driving the stochastic process. This is thus a new1 adaptive method in the numerical analysis of stochastic di erential equation. Mauthner (1998) and Hofmann et al (2000) have developed a general framework for adaptive schemes for the numerical solution to SDEs, [30, 21]. The former presents a Runge-Kutta-type method based on stepsize control while the latter considered a one-step adaptive scheme where the method is also adapted based on step size control. Lamba, Mattingly and Stuart, [28] considered an adaptive Euler scheme based on controlling the drift component of the time-step method. Here we seek to develop a hybrid algorithm that switches between euler and milstein schemes at each time step over the entire discretization interval, depending on the outcome of the simulated Brownian motion increment. The bias of the hybrid scheme as well as its order of convergence is studied. We also do a comparative analysis of the performance of the hybrid scheme relative to the basic numerical schemes of Euler and Milstein. / Mathematical Sciences / M.Sc. (Applied Mathematics)

In this dissertation we obtain an e cient hybrid numerical method for the solution of stochastic di erential equations (SDEs). Speci cally, our method chooses between two numerical methods (Euler and Milstein) over a particular discretization interval depending on the value of the simulated Brownian increment driving the stochastic process. This is thus a new1 adaptive method in the numerical analysis of stochastic di erential equation. Mauthner (1998) and Hofmann et al (2000) have developed a general framework for adaptive schemes for the numerical solution to SDEs, [30, 21]. The former presents a Runge-Kutta-type method based on stepsize control while the latter considered a one-step adaptive scheme where the method is also adapted based on step size control. Lamba, Mattingly and Stuart, [28] considered an adaptive Euler scheme based on controlling the drift component of the time-step method. Here we seek to develop a hybrid algorithm that switches between euler and milstein schemes at each time step over the entire discretization interval, depending on the outcome of the simulated Brownian motion increment. The bias of the hybrid scheme as well as its order of convergence is studied. We also do a comparative analysis of the performance of the hybrid scheme relative to the basic numerical schemes of Euler and Milstein. / Mathematical Sciences / M.Sc. (Applied Mathematics)

The principal theorem of this thesis is a theorem by Peano on the existence of a solution to a certain integral equation. The two primary notions underlying this theorem are uniform convergence and equi-continuity. Theorems related to these two topics are proved in Chapter II. In Chapter III we state and prove a classical existence and uniqueness theorem for an integral equation. In Chapter IV we consider the approximation on certain functions by means of elementary expressions involving "bent line" functions. The last chapter, Chapter V, is the proof of the theorem by Peano mentioned above. Also included in this chapter is an example in which the integral equation has more than one solution. The first chapter sets forth basic definitions and theorems with which the reader should be acquainted.

M.Sc. (Applied Mathematics) / Numerous studies on linear and nonlinear Volterra integral equations (VIEs), have been performed. These studies mainly considered the existence and uniqueness of the solution, and numerical solutions of these equations. In this work, a class of nonlinear (nonstandard) Volterra integral equation that has received very little attention in the literature is considered. The existence and uniqueness of the solution for the nonlinear VIE is proved using the contraction mapping theorem in the space C[0; d]. Collocation methods, repeated trapezoidal rule and repeated Simpson's rule are used to solve the nonlinear (nonstandard) VIE. For the collocation solutions we considered two cases: implicit Euler method and implicit midpoint method. Examples are used to compare the performance of these methods and the results show that the repeated Simpson's rule performs better than the other methods. An analysis of the collocation solution and the solution by the repeated trapezoidal rule is performed. Su cient conditions for existence and uniqueness of the numerical solution are given. The collocation methods and repeated trapezoidal rule yield convergence of order one.

We study a nonlinear kinetic model describing the interactions of particles in a granular medium, i.e. inelastic systems where kinetic energy is not conserved due to internal friction. Examples of particles that fall into this category are sand, ground coffee and many others. Originally studied by Benedetto, Caglioti and Pulvirenti in the one-dimensional setting (RAIRO Model. Math. Anal. Numer, 31(5): 615-641, (1997)) the original model contained inconsistencies later accounted for and corrected by invoking a mollifier (Modelisation Mathematique et Analyse Numerique, M2AN, Vol. 33, No 2, pp. 439–441 (1999)). This thesis approximates the generalized model presented by Agueh (Arch. Rational Mech., Anal. 221, pp. 917-959 (2016)) with the added assumption of a spatial mollifier present in the kinetic equation. In dimension d ≥ 1 this model reads as ∂tf + v · ∇xf = divv(f([ηα∇W] ∗(x,v) f)) where f is a non-negative particle density function, W is a radially symmetric class C2 velocity interaction potential, and and ηα is a mollifier. A physical interpretation of this approximation is that the particles are spheres of radius α > 0 as opposed to the original assumption of being point-masses. Properties lost by this approximation and macroscopic quantities that remain conserved are discussed in greater detail and contrasted. The main result of this thesis is a proof of the weak global existence and uniqueness. An argument utilizing the tools of Optimal Transport allows simple construction of a weak solution to the kinetic model by transporting an initial measure under the characteristic flow curves. Concluding regularity arguments and restrictions on the velocity interaction potential ascertain that global classical solutions are obtained. / Graduate

A thesis submitted to the faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Doctor of Philosophy, / The inverse problem in the Calculus of Variations involves determining the Lagrangians, if any, associated with a given (system of) differential equation(s). One can classify Lagrangians according to the Lie algebra of symmetries of the Action integral (the Noether algebra). We give a complete classification of first-order Lagrangians defined on the line and produce results pertaining to the dimensionality of the algebra of Noether symmetries and compare and contrast these with similar results on the algebra of Lie symmetries of the corresponding Euler-Lagrange .equations. It is proved that the maximum dimension of the Noether point symmetry algebra of a particle Lagrangian. is five whereas it is known that the maximum dimension Qf the Lie algebra of the corresponding scalar second-order Euler-Lagrange equation is eight. Moreover, we show th'a.t a particle Lagrangian does not admit a maximal four-dimensional Noether point symmeiry algebra and consequently a particle Lagrangian admits the maximal r E {O, 1,2,3, 5}-dimensional Noether point symmetry algebra, It is well .known that an important means of analyzing differential equations lies in the knowledge of the first integrals of the equation. We deliver an algorithm for finding first integrals of partial differential equations and show how some of the symmetry properties of the first integrals help to 'further' reduce the order of the equations and sometimes completely solve the equations. Finally, we discuss some open questions. These include the inverse problem and classification of partial differential equations. ALo, there is the question of the extension of the results to 'higher' dimensions. / Andrew Chakane 2018

A research project submitted in partial fulfilment of the requirements of the degree of Masters of Science (Mathematics Education) University of the Witwatersrand Johannesburg, South Africa. 10 October 2016 / This research investigates three different kinds of errors made by learners using a discursive approach when dealing with equations in grade 10. The data for the study is made up of Grade 10 learners' responses to a pre-test and post-test, and is part of a much larger data set that was collected by the Wits Maths Connect Secondary Project (WMCS) in 2013. Questions like substitution and multiplying and simplifying as well as factorization are given attention as they are related to equations. The data for this study is collected by coding pre-test and post-test scripts. The data is analyzed using codes and using commognition. The role played by the equal sign in equations and operations that are performed on symbols as well as well as errors that creep in executing these operations are given attention in the study. Special attention is given to linear equations in this study. Errors identified by Brodie and Berger (2010) are confirmed. The finding is that extent to which errors made by grade 10 learners when dealing with equations stem from errors in arithmetic is substantial. The other finding is that extent to which grade 10 learners made errors related to basic algebra, when dealing with equations, is substantial. / TG2016

PhD., Faculty of Science, University of the Witwatersrand, 2011 / The construction of conserved vectors using Noether’s theorem via a knowledge of a Lagrangian (or via the recently developed concept of partial Lagrangians) is well known. The formulae to determine these for higher-order flows is somewhat cumbersome and becomes more so as the order increases. We carry out these for a class of fourth, fifth and sixth order PDEs. In the latter case, we involve the fifth-order KdV equation using the concept of ‘weak’ Lagrangians better known for the third-order KdV case. We then consider the case of a mixed ‘high-order’ equations working on the Shallow Water Wave and Regularized Long Wave equations. These mixed type equations have not been dealt with thus far using this technique. The construction of conserved vectors using Noether’s theorem via a knowledge of a Lagrangian is well known. In some of the examples, our focus is that the resultant conserved flows display some previously unknown interesting ‘divergence properties’ owing to the presence of the mixed derivatives. We then analyse the conserved flows of some multi-variable equations that arise in Relativity. In addition to a larger class of conservation laws than those given by the isometries or Killing vectors, we may conclude what the isometries are and that these form a Lie subalgebra of the Noether symmetry algebra. We perform our analysis on versions of the Vaidya metric yielding some previously unknown information regarding the corresponding manifold. Lastly, with particular reference to this metric, we also show the variations that occur for the unknown functions. We discuss symmetries of classes of wave equations that arise as a consequence of the Vaidya metric. The objective of this study is to show how the respective geometry is responsible for giving rise to a nonlinear inhomogeneous wave equation as an alternative to assuming the existence of nonlinearities in the wave equation due to physical considerations. We find Lie and Noether point symmetries of the corresponding wave equations and give some reductions. Some interesting physical 4 conclusions relating to conservation laws such as energy, linear and angular momenta are also determined. We also present some interesting comparisons with the standard wave equations (on a ‘flat geometry’). Finally, we pursue the nature of the flow of a third grade fluid with regard to its underlying conservation laws. In particular, the fluid occupying the space over a wall is considered. At the surface of the wall, suction or blowing velocity is applied. By introducing a velocity field, the governing equations are reduced to a class of PDEs. A complete class of conservation laws for the resulting equations are constructed and analysed using the invariance properties of the corresponding multipliers/characteristics.