A dissertation submitted to the Faculty of Science, School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 2015. / Invariant solutions for two-dimensional free and wall jets are derived by consid- ering the Lie point symmetry associated with the appropriate conserved vectors of Prandtl's boundary layer equations for the jets. For the two-dimensional jets we also consider the comparison, advantages and disadvantages between the standard method that uses a linear combination of all the Lie point symme- tries of Prandtl's boundary layer equations to generate the invariant solution with the new method explored in this paper which uses the Lie point sym- metry associated with a conserved vector to generate the invariant solution. Invariant solutions for two-dimensional classical and self-propelled wakes are also derived by considering the Lie point symmetry associated with the appro- priate conserved vectors of Prandtl's boundary layer equations for the wakes. We also consider and discuss the standard method that uses a linear combi- nation of all the Lie point symmetry of Prandtl's boundary layer equations to generate the invariant solutions for the classical and self-propelled wakes.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Master of Science, May 14, 2013. / In this dissertation, we investigate the parabolic mixed derivative diffusion equation modeling the viscous and viscoelastic effects in a non-Newtonian viscoelastic fluid. The model is analytically considered using Fourier and Laplace transformations. The main focus of the dissertation, however, is the implementation of the Peaceman-Rachford Alternating Direction Implicit method. The one-dimensional parabolic mixed derivative diffusion equation is extended to a two-dimensional analog. In order to do this, the two-dimensional analog is solved using a Crank-Nicholson method and implemented according to the Peaceman- Rachford ADI method. The behaviour of the solution of the viscoelastic fluid model is analysed by investigating the effects of inertia and diffusion as well as the viscous behaviour, subject to the viscosity and viscoelasticity parameters. The two-dimensional parabolic diffusion equation is then implemented with a high-order method to unveil more accurate solutions. An error analysis is executed to show the accuracy differences between the numerical solutions of the general ADI and high-order compact methods. Each of the methods implemented in this dissertation are investigated via the von-Neumann stability analysis to prove stability under certain conditions.

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. July 2014. / This is the study of Lie algebraic properties of first integrals of scalar second-, third and higher-order ordinary differential equations (ODEs). The Lie algebraic classification of such differential equations is now well-known from the works of Lie [10] as well as recently Mahomed and Leach [19]. However, the algebraic properties of first integrals are not known except in the maximal cases for the basic first integrals and some of their quotients. Here our intention is to investigate the complete problem for scalar second-order and maximal symmetry classes of higher-order ODEs using Lie algebras and Lie symmetry methods. We invoke the realizations of low-dimensional Lie algebras. Symmetries of the fundamental first integrals for scalar second-order ODEs which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the point symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classi cation of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0, 1, 2 or 3 point symmetry cases. It is proved that the maximal algebra case is unique. By use of Lie symmetry group methods we further analyze the relationship between the first integrals of the simplest linear third-order ODEs and their point symmetries. It is well-known that there are three classes of linear third-order ODEs for maximal and submaximal cases of point symmetries which are 4, 5 and 7. The simplest scalar linear third-order equation has seven point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equation y000 = 0 is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs of maximal type. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals. The relationship between rst integrals of sub-maximal linearizable third-order ODEs and their symmetries are investigated as well. All scalar linearizable third-order equations can be reduced to three classes by point transformations. We obtain the classifying relations between the symmetries and the first integral for sub-maximal cases of linear third-order ODEs. It is known, from the above, that the maximum Lie algebra of the first integral is achieved for the simplest equation. We show that for the other two classes they are not unique. We also obtain counting theorems of the symmetry properties of the rst integrals for these classes of linear third-order ODEs. For the 5 symmetry class of linear third-order ODEs, the first integrals can have 0, 1, 2 and 3 symmetries and for the 4 symmetry class of linear third-order ODEs they are 0, 1 and 2 symmetries respectively. In the case of sub-maximal linear higher-order ODEs, we show that their full Lie algebras can be generated by the subalgebras of certain basic integrals. For the n+2 symmetry class, the symmetries of the rst integral I2 and a two-dimensional subalgebra of I1 generate the symmetry algebra and for the n + 1 symmetry class, the full algebra is generated by the symmetries of I1 and a two-dimensional subalgebra of the quotient I3=I2. Finally, we completely classify the first integrals of scalar nonlinear second-order ODEs in terms of their Lie point symmetries. This is performed by first obtaining the classifying relations between point symmetries and first integrals of scalar nonlinear second order equations which admit 1, 2 and 3 point symmetries. We show that the maximum number of symmetries admitted by any first integral of a scalar second-order nonlinear (which is not linearizable by point transformation) ODE is one which in turn provides reduction to quadratures of the underlying dynamical equation. We provide physical examples of the generalized Emden-Fowler, Lane-Emden and modi ed Emden equations.

Estudamos a representação das equações diferenciais funcionais (EDF) lineares autônomas do tipo neutro como uma classe de equações de renovação, isto é, equações do tipo convolução. Utilizando ferramentas como a transformada de Laplace-Stieltjes, estudamos o comportamento assintótico das soluções desta equações quando t \'SETA\' \' INFINITO\' / We studied the representation of linear autonomous functional differential equations (FDE) as a class of renewall equations, that is, convolution-type equations. Using tools like the Laplace-Stieltjes trnsform, we obtained the asymptotic behaviour of those solutions as t \'ARROW\' \'INFINITY

In this thesis we study the thin-film type equations in one spatial dimension. These equations arise from the lubrication approximation to the thin films of viscous fluids which is described by the Navier-Stokes equations with free boundary. From the structural point of view, they are fourth-order degenerate nonlinear parabolic equations, with principal term from diffusion and lower order term from external forces. In Chapter one we study the dynamics of the equations when the external force is given by a power law. Classification of steady states of this equation, which is important for the dynamics, was already known. Previous numerical studies show that there is a mountain pass scenario among the steady states. We shall provide a rigorous justification to these numerical results. As a result, a rather complete picture of the dynamics of the thin film is obtained when the power law is in the range (1,3). In Chapter two we turn to the special case of the equation where the external force is the gravity. This is important, but, unfortunately not a power law. We study and classify the steady states of this equation as well as compare their energy levels. Some numerical results are also present. / Zhang, Zhenyu. / Asviser: Kai Seng Chou. / Source: Dissertation Abstracts International, Volume: 73-06, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 103-107). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

by Leung Tin Chi. / Bibliography: leaf 32 / Thesis (M.Phil.)--Chinese University of Hong Kong, 1982

Chan Yuet Tai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 107-111). / Chapter 1 --- Introduction to Sturm-Liouville Problem --- p.1 / Chapter 1.1 --- What are inverse problems? --- p.1 / Chapter 1.2 --- Introductory background --- p.2 / Chapter 1.3 --- The Liouville transformation --- p.3 / Chapter 1.4 --- The Sturm-Liouville problem 一 A historical look --- p.4 / Chapter 1.5 --- Where Sturm-Liouville problems come from? --- p.6 / Chapter 1.6 --- Inverse problems of interest --- p.8 / Chapter 2 --- Reconstruction Method I --- p.10 / Chapter 2.1 --- Perturbative inversion --- p.10 / Chapter 2.1.1 --- Inversion problem via Fredholm integral equation --- p.10 / Chapter 2.1.2 --- Output least squares method for ill-posed integral equations --- p.15 / Chapter 2.1.3 --- Numerical experiments --- p.17 / Chapter 2.2 --- Total inversion --- p.38 / Chapter 2.3 --- Summary --- p.45 / Chapter 3 --- Reconstruction Method II --- p.46 / Chapter 3.1 --- Computation of q --- p.47 / Chapter 3.2 --- Computation of the Cauchy data --- p.48 / Chapter 3.2.1 --- Recovery of Cauchy data for K --- p.51 / Chapter 3.2.2 --- Numerical implementation for computation of the Cauchy data . --- p.51 / Chapter 3.3 --- Recovery of q from Cauchy data --- p.52 / Chapter 3.4 --- Iterative procedure --- p.53 / Chapter 3.5 --- Numerical experiments --- p.60 / Chapter 3.5.1 --- Eigenvalues without noised data --- p.64 / Chapter 3.5.2 --- Eigenvalues with noised data --- p.69 / Chapter 4 --- Appendices --- p.79 / Chapter A --- Tikhonov regularization --- p.79 / Chapter B --- Basic properties of the Sturm-Liouville operator --- p.80 / Chapter C --- Asymptotic formulas for the eigenvalues --- p.86 / Chapter C.1 --- Case 1: h ≠ ∞ and H ≠ ∞ --- p.87 / Chapter C.2 --- Case 2: h= ∞ and H ≠∞ --- p.90 / Chapter C.3 --- Case 3: h = ∞ and H = ∞ --- p.91 / Chapter D --- Completeness of the eigenvalues --- p.92 / Chapter E --- d'Alembert solution formula for the wave equation --- p.97 / Chapter E.1 --- "The homogeneous solution uH(x,t)" --- p.98 / Chapter E.2 --- "The particular solution up(x, t)" --- p.99 / Chapter E.3 --- "The standard d'Alembert solution u(x,t)" --- p.101 / Chapter E.4 --- Applications to our problem --- p.101 / Chapter F --- Runge-Kutta method for solving eigenvalue problems --- p.104 / Bibliography --- p.107

Lee Wai Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 129-130). / Text in English; abstract also in Chinese. / Lee Wai Shing. / Contents --- p.i / List of Figures --- p.v / Abstract --- p.vii / Acknowledgement --- p.ix / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- What are inverse problems? --- p.1 / Chapter 1.2 --- Background of this research project --- p.2 / Chapter 1.3 --- Conservative systems and open systems -normal modes (NM's) vs quasi-normal modes (QNM's) --- p.3 / Chapter 1.4 --- Appetizer ´ؤ What our problems are like --- p.6 / Chapter 1.5 --- A brief overview of the following chapters --- p.7 / Chapter Chapter 2. --- Inversion of conservative systems- perturbative inversion --- p.9 / Chapter 2.1 --- Overview --- p.9 / Chapter 2.2 --- Way to introduce the additional information --- p.9 / Chapter 2.3 --- General Formalism --- p.11 / Chapter 2.4 --- Example --- p.15 / Chapter 2.5 --- Further examples --- p.19 / Chapter 2.6 --- Effects of noise --- p.23 / Chapter 2.7 --- Conclusion --- p.25 / Chapter Chapter 3. --- Inversion of conservative systems - total inversion --- p.26 / Chapter 3.1 --- Overview --- p.26 / Chapter 3.2 --- Asymptotic behaviour of the eigenfrequencies --- p.26 / Chapter 3.3 --- General formalism --- p.28 / Chapter 3.3.1 --- Evaluation of V(0) --- p.28 / Chapter 3.3.2 --- Squeezing the interval - evaluation of the potential at other positions --- p.32 / Chapter 3.4 --- Remarks --- p.36 / Chapter 3.5 --- Conclusion --- p.37 / Chapter Chapter 4. --- Theory of Quasi-normal Modes (QNM's) --- p.38 / Chapter 4.1 --- Overview --- p.38 / Chapter 4.2 --- What is a Quasi-normal Mode (QNM) system? --- p.38 / Chapter 4.3 --- Properties of QNM's in expectation --- p.40 / Chapter 4.4 --- General Formalism --- p.41 / Chapter 4.4.1 --- Construction of Green's function and the spectral represen- tation of the delta function --- p.42 / Chapter 4.4.2 --- The generalized norm --- p.45 / Chapter 4.4.3 --- Completeness of QNM's and its justification --- p.46 / Chapter 4.4.4 --- Different senses of completeness --- p.48 / Chapter 4.4.5 --- Eigenfunction expansions with QNM's 一 the two-component formalism --- p.49 / Chapter 4.4.6 --- Properties of the linear space Γ --- p.51 / Chapter 4.4.7 --- Klein-Gordon equation - The delta-potential system --- p.54 / Chapter 4.5 --- Studies of other QNM systems --- p.54 / Chapter 4.5.1 --- Wave equation - dielectric rod --- p.55 / Chapter 4.5.2 --- Wave equation ´ؤ string-mass system --- p.57 / Chapter 4.6 --- Summary --- p.58 / Chapter Chapter 5. --- Inversion of open systems- perturbative inversion --- p.59 / Chapter 5.1 --- Overview --- p.59 / Chapter 5.2 --- General Formalism --- p.59 / Chapter 5.3 --- Example 1. Klein-Gordon equation ´ؤ delta-potential system --- p.66 / Chapter 5.3.1 --- Model perturbations --- p.66 / Chapter 5.4 --- Example 2. Wave equation ´ؤ dielectric rod --- p.72 / Chapter 5.5 --- Example 3. Wave equation ´ؤ string-mass system --- p.76 / Chapter 5.5.1 --- Instability of the matrix [d] = [c]-1 upon truncation --- p.79 / Chapter 5.6 --- Large leakage regime and effects of noise --- p.81 / Chapter 5.7 --- Conclusion . . . --- p.84 / Chapter Chapter 6. --- Transition from open systems to conservative counterparts --- p.85 / Chapter 6.1 --- Overview --- p.85 / Chapter 6.2 --- Anticipations of what is going to happen --- p.86 / Chapter 6.3 --- Some computational experiments --- p.86 / Chapter 6.4 --- Reason of breakdown - An intrinsic error of physical systems --- p.87 / Chapter 6.4.1 --- Mathematical derivation of the breakdown behaviour --- p.90 / Chapter 6.4.2 --- Two verifications --- p.93 / Chapter 6.5 --- Another source of errors - An intrinsic error of practical computations --- p.95 / Chapter 6.5.1 --- Vindications --- p.96 / Chapter 6.5.2 --- Mathematical derivation of the breakdown --- p.98 / Chapter 6.6 --- Further sources of errors --- p.99 / Chapter 6.7 --- Dielectric rod --- p.100 / Chapter 6.8 --- String-mass system --- p.103 / Chapter 6.9 --- Conclusion --- p.105 / Chapter Chapter 7. --- A first step to Total Inversion of QNM systems? --- p.106 / Chapter 7.1 --- Overview --- p.106 / Chapter 7.2 --- Derivation for F(0) --- p.106 / Chapter 7.3 --- Example 一 delta potential system --- p.108 / Chapter Chapter 8. --- Conclusion --- p.111 / Chapter 8.1 --- A summary on what have been achieved --- p.111 / Chapter 8.2 --- Further directions to go --- p.111 / Appendix A. A note on notation --- p.113 / Appendix B. Asymptotic series of NM eigenvalues --- p.114 / Appendix C. Evaluation of functions related to RHS(x) --- p.117 / Appendix D. Asymptotic behaviour of the Green's function --- p.119 / Appendix E. Expansion coefficient an --- p.121 / Appendix F. Asymptotic behaviour of QNM eigenvalues --- p.123 / Appendix G. Properties of the inverse matrix [d] = [c]-1 --- p.125 / Appendix H. Matrix inverse through the LU decomposition method --- p.127 / Bibliography --- p.129

This dissertation presents new numerical methods for the solution of fractional differential equations of single and distributed order that find application in the different fields of physics and engineering. We start by presenting the relationship between fractional derivatives and processes like anomalous diffusion, and, we then develop new numerical methods for the solution of the time-fractional diffusion equations. The first numerical method is developed for the solution of the fractional diffusion equations with Neumann boundary conditions and the diffusivity parameter depending on the space variable. The method is based on finite differences, and, we prove its convergence (convergence order of O(Δx² + Δt²^-α), 0 < α < 1) and stability. We also present a brief description of the application of such boundary conditions and fractional model to real world problems (heat flux in human skin). A discussion on the common substitution of the classical derivative by a fractional derivative is also performed, using as an example the temperature equation. Numerical methods for the solution of fractional differential equations are more difficult to develop when compared to the classical integer-order case, and, this is due to potential singularities of the solution and to the nonlocal properties of the fractional differential operators that lead to numerical methods that are computationally demanding. We then study a more complex type of equations: distributed order fractional differential equations where we intend to overcome the second problem on the numerical approximation of fractional differential equations mentioned above. These equations allow the modelling of more complex anomalous diffusion processes, and can be viewed as a continuous sum of weighted fractional derivatives. Since the numerical solution of distributed order fractional differential equations based on finite differences is very time consuming, we develop a new numerical method for the solution of the distributed order fractional differential equations based on Chebyshev polynomials and present for the first time a detailed study on the convergence of the method. The third numerical method proposed in this thesis aims to overcome both problems on the numerical approximation of fractional differential equations. We start by solving the problem of potential singularities in the solution by presenting a method based on a non-polynomial approximation of the solution. We use the method of lines for the numerical approximation of the fractional diffusion equation, by proceeding in two separate steps: first, spatial derivatives are approximated using finite differences; second, the resulting system of semi-discrete ordinary differential equations in the initial value variable is integrated in time with a non-polynomial collocation method. This numerical method is further improved by considering graded meshes and an hybrid approximation of the solution by considering a non-polynomial approximation in the first sub-interval which contains the origin in time (the point where the solution may be singular) and a polynomial approximation in the remaining intervals. This way we obtain a method that allows a faster numerical solution of fractional differential equations (than the method obtained with non-polynomial approximation) and also takes into account the potential singularity of the solution. The thesis ends with the main conclusions and a discussion on the main topics presented along the text, together with a proposal of future work.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Bibliography: leaves 179-181. / by Stephen Mark Paneitz. / Ph.D.