No description available.

No description available.

No description available.

Two methods are presented for use on an electronic computer for the solution of partial differential systems. The first is concerned with accurate solutions of differential equations. It is equally applicable to ordinary differential equations and partial differential equations, and can be used for parabolic, hyperbolic or elliptic systems, and also for non-linear and mixed systems. It can be used in conjunction with existing schemes. Conversely, the method can be used as a very fast method of obtaining a rough solution of the system. It has an additional advantage over traditional higher order methods in that it does not require extra boundary conditions. The second method is concerned with the acceleration of the convergence rate in the solution of hyperbolic systems. The number of iterations has been reduced from tens of thousands with the traditional Lax-Wendroff methods to the order of twenty iterations. Analyses for both the differential and the difference systems are presented. Again the method is easily added to existing programs. The two methods may be used together to give one fast and accurate method.

D.Sc. (Mathematics) / In this thesis aspects of continuous symmetries of differential equations are studied. In particular the following aspects are studied in detail: Lie algebras, the Lie derivative, the jet bundle formalism for differential equations, Lie point and Lie-Backlund symmetry vector fields, recursion operators, conservation laws, Lax pairs, the Painlcve test, Lie algebra valued differenmtial forms and Dose operators as a representation of differential operators. The purpose of the study is to gain a better understanding of complicated nonlinear dirrerential equations that describe nature and to construct solutions. The differential equations under consideration were derived [rom physics and engineering. They are the following: the Kortcweg-dc Vries equation, Burgers' cquation , the sine-Gordon equation, nonlinear diffusion equations, the Klein Gordon equation, the Schrodinger equation, nonlinear Dirac equations, Yang-Mills equations, the Lorentz model, the Lotka-Volterra model, damped unharrnonic oscillators, and others. The newly found results and insights are discussed in chapters 8 to 17. Details on the COli tents of each chapter and rcfernces to some of my articles arc given in chapter 1.

M.Sc. (Theoretical Physics) / Please refer to full text to view abstract

No description available.

In the numerical solution of the two point "boundary value problem, [ equation omitted ] (1) the usual method is to approximate the problem by a finite difference analogue of the form [ equation omitted ] (2) with k = 2, and the truncation error T.E. = O(h⁴) or O(h⁶), where h is the step-size. Varga (1962) has obtained error bounds for the former when the problem (1) is linear and of class M . In this thesis, more accurate finite difference methods are considered. These can be obtained in essentially two different ways, either by increasing the value k in difference equations (2), or by introducing higher order derivatives. Several methods of both types have been derived. Also, it is shown how the initial value problem y' = ϕ(x,y) can be formulated as a two point boundary value problem and solved using the latter approach. Error bounds have been derived for all of these methods for linear problems of class M . In particular, more accurate bounds have been derived than those obtained by Varga (1962) and Aziz and Hubbard (1964). Some error estimates are suggested for the case where [ equation omitted ], but these are not accurate bounds, especially when [ equation omitted ] not a constant. In the case of non-linear differential equations, sufficient conditions are derived for the convergence of the solution of the system of equations (2) by a generalized Newton's method. Some numerical results are included and the observed errors compared with theoretical error bounds. / Science, Faculty of / Mathematics, Department of / Graduate

The first chapter of the thesis is concerned with the construction of finite difference approximations to boundary value problems in linear nth order ordinary differential equations. This construction is based upon a local collocation procedure with polynomials, which is equivalent to a method of undetermined coefficients. It is shown that the coefficients of these finite difference approximations can be expressed as the determinants of matrices of relatively small dimension. A basic theorem states that these approximations are consistent, provided only that a certain normalization factor does not vanish. This is the case for compact difference equations and for difference equations with only one collocation point. The order of consistency may be improved by suitable choice of the collocation points. Several examples of known, as well as new difference approximations are given. Approximations to boundary conditions are also treated in detail. The stability theory of H. O. Kreiss is applied to investigate the stability of finite difference schemes based upon these approximations. A number of numerical examples are also given. In the second chapter it is shown how the construction method of the first chapter can be extended to initial value problems for systems of linear first order ordinary differential equations. Specific examples are 'included and the well-known stability theory for these difference equations is summarized. It is then shown how these difference methods may be applied to linear parabolic partial differential equations in one space variable after first discretizing in space by a suitable method from the first chapter. The stability of such difference schemes for parabolic equations is investigated using an eigenvalue-eigenvector analysis. In particular, the effect of various approximations to the boundary conditions is considered. The relation of this analysis to the stability theory of J. M. Varah is indicated. Numerical examples are also included. / Science, Faculty of / Mathematics, Department of / Graduate

A research thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfillment of the requirement for the degree of Doctor of Philosophy. August 7, 2013. / In this thesis we consider the construction of exact solutions for models describing heat transfer through extended surfaces (fins). The interest in the solutions of the heat transfer in extended surfaces is never ending. Perhaps this is because of the vast application of these surfaces in engineering and industrial processes. Throughout this thesis, we assume that both thermal conductivity and heat transfer are temperature dependent. As such the resulting energy balance equations are nonlinear. We attempt to construct exact solutions for these nonlinear models using the theory of Lie symmetry analysis of differential equations. Firstly, we perform preliminary group classification of the steady state problem to determine forms of the arbitrary functions appearing in the considered equation for which the principal Lie algebra is extended by one element. Some reductions are performed and invariant solutions that satisfy the Dirichlet boundary condition at one end and the Neumann boundary condition at the other, are constructed. Secondly, we consider the transient state heat transfer in longitudinal rectangular fins. Here the imposed boundary conditions are the step change in the base temperature and the step change in base heat flow. We employ the local and nonlocal symmetry techniques to analyze the problem at hand. In one case the reduced equation transforms to the tractable Ermakov-Pinney equation. Nonlocal symmetries are admitted when some arbitrary constants appearing in the governing equations are specified. The exact steady state solutions which satisfy the prescribed boundary conditions are constructed. Since the obtained exact solutions for the transient state satisfy only the zero initial temperature and adiabatic boundary condition at the fin tip, we sought numerical solutions. Lastly, we considered the one dimensional steady state heat transfer in fins of different profiles. Some transformation linearizes the problem when the thermal conductivity is a differential consequence of the heat transfer coefficient, and exact solutions are determined. Classical Lie point symmetry methods are employed for the problem which is not linearizable. Some reductions are performed and invariant solutions are constructed. The effects of the thermo-geometric fin parameter and the power law exponent on temperature distribution are studied in all these problems. Furthermore, the fin efficiency and heat flux are analyzed.