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A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy to the Faculty of Science, University of the Witwatersrand, Johannesburg, 2019 / We will study the symmetry, invariance properties and conservation laws of partial dif ferential equations (pdes) that arise in a number of situations in mathematical physics. These will be range from Image Processing and noise removal algorithms to Timoshenko beam systems. Furthermore, we will study the invariance properties and approximate conservation laws of some nonlinear Schro¨dinger equation with PT-symmetric potentials with inhomogeneous nonlinearity and some nonlinear Schro¨dinger equation involving a spatially extended system consisting of two coupled elements. / TL (2019)

Degree awarded with distinction on 6 December 1995. A research report submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Masters. Johannesburg, 1995. / In this work we give a brief overview of the existing group classification methods of partial differential equations by means of examples. On top of these methods we introduce another new method which classify according to low-dimensional Lie elgebras, One can ask: What is the aim of introducing a new method whilst there are existing methods? This question is answered in the following paragraph. Firstly we classify our system of non-linear partial differential equations using the preliminary group classification method (one of the existing methods). The results are not different from what; Euler, Steeb and Mulsor have obtained in 1991 and 1992. That is, this method does not yield new information. This new method which classifies according to low-dimensional Lie algebras is used to classify a general system of equations from plasma physics. Finally, using this method we completely classify our system for four-dimensionnl algebras. For a partial differential equation to be completely classified using this method, it must admit a low-dimensional Lie algebra.

Approved for Public Release; Distribution is Unlimited / Child and Langmuir introduced a solution to space charge limited emission in an infinite area planar diode. The solution follows from starting with Poisson's equation, and requires solving a non-linear differential equation. This approach can also be applied to cylindrical and spherical geometries, but only for one-dimensional cases. By approaching the problem from Coulomb's law and applying the effect of an assumed charge distribution, it is possible to solve for space charge limited emission without solving a non-linear differential equation, and to limit the emission area to two-dimensional geometries. Using a Mathcad worksheet to evaluate Coulomb's law, it is possible to show correlation between the solution derived by Child and Langmuir and Coulomb's law. / Ensign, United States Navy

by Chi-chung Lee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves 45-47). / INTRODUCTION --- p.1 / Chapter CHAPTER 1 --- RADIAL SYMMETRY OF GROUND STATES --- p.7 / Chapter CHAPTER 2 --- EXISTENCE OF A GROUND STATE --- p.14 / Chapter CHAPTER 3 --- UNIQUENESS OF GROUND STATE I --- p.23 / Chapter CHAPTER 4 --- UNIQUENESS OF GROUND STATE II --- p.35 / REFERENCES --- p.45

by Huang Yu. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 145-150). / Introduction --- p.1 / Chapter 1 --- Global Attractors of Semigroups --- p.9 / Introduction --- p.9 / Chapter 1.1 --- Basic Notions --- p.9 / Chapter 1.2 --- Semigroup of Class K --- p.11 / Chapter 1.3 --- Semigroup of Class AK --- p.15 / Chapter 1.4 --- Hausdorff and Fractal Dimensions of Attractors --- p.19 / Chapter 1.4.1 --- Hausdorff and Fractal dimensions --- p.20 / Chapter 1.4.2 --- The Dimensions of Invariant Sets --- p.22 / Chapter 1.4.3 --- An Application to Evolution Equations --- p.35 / Notes --- p.39 / Chapter 2 --- Invariant Manifolds and Inertial Manifolds --- p.40 / Introduction --- p.40 / Chapter 2.1 --- Preliminary --- p.41 / Chapter 2.1.1 --- Notions --- p.41 / Chapter 2.1.2 --- Nemytskii Operator --- p.43 / Chapter 2.1.3 --- Contractions on Embedded Banach Spaces --- p.47 / Chapter 2.2 --- Linear and Nonlinear Integral Equations --- p.49 / Chapter 2.3 --- Invariant Manifolds --- p.55 / Chapter 2.4 --- Inertial Manifolds --- p.59 / Notes --- p.63 / Chapter 3 --- Semilinear Parabolic Variational Inequalities --- p.64 / Introduction --- p.64 / Chapter 3.1 --- Existence Results --- p.66 / Chapter 3.2 --- The Existence of Global Attractors --- p.69 / Chapter 3.3 --- The Weakly Approximating Inertial Manifolds --- p.76 / Chapter 3.4 --- An Application: The Obstacle Problem --- p.87 / Chapter 4 --- Semilinear Wave Equations with Damping and Critical Expo- nent --- p.91 / Introduction --- p.91 / Chapter 4.1 --- Existence Results --- p.93 / Chapter 4.2 --- The Global Attractor for the Problem --- p.96 / Chapter 4.2.1 --- A Proposition on Uniform Decay --- p.98 / Chapter 4.2.2 --- Compactness of the Trajectories of (4.2.7) --- p.102 / Chapter 4.3 --- A Particular Case-Linear Damping --- p.105 / Chapter 4.4 --- Estimate of the Dimensions of the Global Attractor --- p.111 / Chapter 4.4.1 --- The Linearized Equation --- p.114 / Chapter 4.4.2 --- The Hausdorff and Fractal Dimensions of the Attractor --- p.117 / Chapter 5 --- Partially Dissipative Evolution Equations --- p.123 / Introduction --- p.123 / Chapter 5.1 --- Basic Notions --- p.124 / Chapter 5.2 --- Semilinear Parabolic Equations and Systems --- p.128 / Chapter 5.3 --- Semilinera Hyperbolic Equation with Damping --- p.136 / Reference --- p.145

by Ho Wing Kin. / Thesis submitted in: July 1997. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 89-92). / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- The Euler-Lagrange Equation for Capillary Surfaces --- p.6 / Chapter 1.2 --- The Capillary Tube --- p.12 / Chapter 2 --- Comparison Principles --- p.16 / Chapter 2.1 --- The General Comparison Principle --- p.16 / Chapter 2.2 --- Applications --- p.24 / Chapter 3 --- Existence Criteria --- p.30 / Chapter 3.1 --- Necessary Conditions --- p.31 / Chapter 3.2 --- Sufficient Conditions --- p.35 / Chapter 4 --- Uniqueness --- p.53 / Chapter 4.1 --- General Bounded Domains Case --- p.53 / Chapter 4.2 --- Infinite Strip Case --- p.56 / Chapter 5 --- Gradient Estimate of Surfaces of Constant Mean Curvature --- p.66 / Chapter 5.1 --- The Gradient Estimate --- p.67 / Chapter 5.2 --- Behavior as R→ 1 --- p.70 / Chapter 5.3 --- Existence of the Comparison Surfaces --- p.76 / Chapter 5.4 --- Ro is Best Possible --- p.82 / Appendix Mean Curvature --- p.85

Keung Yee Lo. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 109-111). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Inverse problems and Parameter Identification --- p.1 / Chapter 1.2 --- Examples in inverse problems --- p.2 / Chapter 1.3 --- Applications in parameter identifications --- p.5 / Chapter 1.4 --- Difficulties arising in inverse problems --- p.7 / Chapter 2 --- Identifying Parameters in Parabolic Systems --- p.9 / Chapter 2.1 --- Introduction --- p.9 / Chapter 2.2 --- An averaging-terminal status formulation and existence of its solutions --- p.12 / Chapter 2.3 --- Optimization approach and its convergence --- p.17 / Chapter 2.4 --- Unconstrained minimization problems --- p.26 / Chapter 2.5 --- Armijo algorithm --- p.28 / Chapter 2.6 --- Numerical experiments --- p.32 / Chapter 2.6.1 --- Convergence of the minimization problem --- p.40 / Chapter 2.7 --- Noisy data --- p.59 / Chapter 3 --- Identifying Parameters in Elliptic Systems --- p.68 / Chapter 3.1 --- Augmented Lagrangian Method --- p.68 / Chapter 3.2 --- The discrete saddle-point problem --- p.70 / Chapter 3.3 --- An Uzawa algorithm --- p.71 / Chapter 3.4 --- Formulation of the algorithm --- p.73 / Chapter 3.5 --- Numerical experiments --- p.76 / Chapter 3.6 --- Alternative formulation of the cost functional --- p.90 / Chapter 3.7 --- Iterative GMRES method --- p.102 / Bibliography --- p.109

Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Ao, Weiwei. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 178-186). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Interior Spike Solutions for Lin-Ni-Takagi Problem --- p.7 / Chapter 1.1.1 --- Background and Main Results --- p.7 / Chapter 1.1.2 --- Sketch of the Proof of Theorem 1.1.1 --- p.12 / Chapter 1.2 --- The A2 and B2 Chern-Simons System --- p.14 / Chapter 1.2.1 --- Background --- p.14 / Chapter 1.2.2 --- Previous Results --- p.19 / Chapter 1.2.3 --- Main Results --- p.20 / Chapter 1.2.4 --- Sketch of the Proof for A₂ Case --- p.21 / Chapter 1.2.5 --- Sketch of the Proof for B₂ Case --- p.26 / Chapter 1.3 --- Organization of the Thesis --- p.27 / Chapter 2 --- The Lin-Ni-Takagi Problem --- p.29 / Chapter 2.1 --- Notation and Some Preliminary Analysis --- p.29 / Chapter 2.2 --- Linear Theory --- p.35 / Chapter 2.3 --- The Non Linear Projected Problem --- p.40 / Chapter 2.4 --- An Improved Estimate --- p.43 / Chapter 2.5 --- The Reduced Problem: A Maximization Procedure --- p.50 / Chapter 2.6 --- Proof of Theorem 1.1.1 --- p.58 / Chapter 2.7 --- More Applications and Some Open Problems --- p.60 / Chapter 3 --- The Chern-Simons System --- p.66 / Chapter 3.1 --- Proof of Theorem 1.2.1 in the A₂ Case --- p.66 / Chapter 3.1.1 --- Functional Formulation of the Problem --- p.66 / Chapter 3.1.2 --- First Approximate Solution --- p.68 / Chapter 3.1.3 --- Invertibility of Linearized Operator --- p.72 / Chapter 3.1.4 --- Improvements of the Approximate Solution: O(ε) Term --- p.76 / Chapter 3.1.5 --- Next Improvement of the Approximate Solution: O(ε²) Term --- p.78 / Chapter 3.1.6 --- A Nonlinear Projected Problem --- p.82 / Chapter 3.1.7 --- Proof of Theorem 1.2.1 for A₂ under Assumption (i) --- p.85 / Chapter 3.1.8 --- Proof of Theorem 1.2.1 for A₂ under Assumption (ii) --- p.94 / Chapter 3.1.9 --- Proof of Theorem 1.2.1 for A₂ under Assumption (iii) --- p.99 / Chapter 3.2 --- Proof of Theorem 1.2.1 in the B₂ Case --- p.100 / Chapter 3.2.1 --- Functional Formulation of the Problem for B₂ Case --- p.100 / Chapter 3.2.2 --- Classi cation and Non-degeneracy for B₂ Toda system --- p.101 / Chapter 3.2.3 --- Invertibility of Linearized Operator --- p.105 / Chapter 3.2.4 --- Improvements of the Approximate Solution --- p.106 / Chapter 3.2.5 --- Proof of Theorem 1.2.1 for B₂ under Assumption (i) --- p.112 / Chapter 3.2.6 --- Proof of Theorem 1.2.1 for B₂ under Assumption (ii) --- p.122 / Chapter 3.2.7 --- Proof of Theorem 1.2.1 for B₂ under Assumption (iii) --- p.127 / Chapter 3.3 --- Open Problems --- p.128 / Chapter 4 --- Appendix --- p.129 / Chapter 4.1 --- B₂ and G₂ Toda System with Singular Source --- p.129 / Chapter 4.1.1 --- Case 1: B₂ Toda system with singular source --- p.130 / Chapter 4.1.2 --- Case 2: G₂ Toda system with singular source --- p.136 / Chapter 4.2 --- The Calculations of the Matrix Q₁ --- p.148 / Chapter 4.3 --- The Calculations of the Matrix Q₁ --- p.169 / Bibliography --- p.178

Geng Di. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 116-119). / Acknowledgement --- p.i / Abstract --- p.ii / Introduction --- p.iii / Part I --- p.1 / Chapter 1 --- Critical Dimension of a Semilinear Degenerate Elliptic Equation Involving Critical Sobolev-Hardy Exponent --- p.2 / Chapter 1.1 --- Introduction --- p.2 / Chapter 1.2 --- Non-existence (I) --- p.5 / Chapter 1.3 --- Non-existence (II) --- p.11 / Chapter 1.4 --- Existence --- p.13 / Chapter 1.5 --- Radial Symmetry of Solutions --- p.16 / Appendix A --- p.20 / Appendix B --- p.23 / Chapter 2 --- Critical Dimension of a Hessian Equation Involving Critical Ex- ponent --- p.27 / Chapter 2.1 --- Introduction --- p.27 / Chapter 2.2 --- Preliminary Results --- p.29 / Chapter 2.3 --- Existence Results --- p.32 / Chapter 2.4 --- Non-existence Results --- p.43 / Chapter 3 --- Absence of Critical Dimension for the Subelliptic Laplacian on the Heisenberg Group --- p.48 / Chapter 3.1 --- Introduction and Main Result --- p.48 / Chapter 3.2 --- Proof of the Theorem --- p.49 / Part2 --- p.55 / Chapter 4 --- Asymptotic Behavior for Weighted p-Laplace Equations Involv- ing Critical Growth on the Ball --- p.56 / Chapter 4.1 --- Introduction --- p.56 / Chapter 4.2 --- A Crucial Lemma --- p.59 / Chapter 4.3 --- Proof of the Main Theorems --- p.61 / Chapter 5 --- Asymptotics for a Semilinear Weighted Elliptic Equation In- volving Critical Sobolev-Hardy Exponent --- p.71 / Chapter 5.1 --- Introduction --- p.71 / Chapter 5.2 --- Some Preliminary Results --- p.73 / Chapter 5.3 --- A Crucial Estimate --- p.80 / Chapter 5.4 --- Proof of the Main Theorem --- p.85 / Appendix --- p.88 / Chapter 6 --- Asymptotics for Positive Solutions for a Biharmonic Equation Involving Critical Exponent --- p.93 / Chapter 6.1 --- Introduction --- p.93 / Chapter 6.2 --- Preliminary Results --- p.94 / Chapter 6.3 --- Pohozaev's identity and Green's Function --- p.98 / Chapter 6.4 --- A Crucial Lemma --- p.103 / Chapter 6.5 --- Proof of Main Theorem --- p.112 / Bibliography --- p.115