published_or_final_version / abstract / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

In this research report we explore some applications of symmetry methods for boundary value problems in the pricing of barrier options. Various nancial instruments satisfy the Black-Scholes partial di erential equation (pde) but with di erent domain, maturity date and boundary conditions. We nd Lie symmetries that leave the Black-Scholes (pde) invariant and will guarantee that the relevant solutions satisfy the boundary conditions. Using these sym- metries, we can thus generate group-invariant solutions to the boundary value problem.

There is a well established connection between one parameter Lie groups of transformations and conservation laws for differential equations. In this thesis, we construct conservation laws via the invariance and multiplier approach based on the wellknown result that the Euler-Lagrange operator annihilates total divergences. This technique will be applied to some plasma physics models. We show that the recently developed notion of the association between Lie point symmetry generators and conservation laws lead to double reductions of the underlying equation and ultimately to exact/invariant solutions for higher-order nonlinear partial di erential equations viz., some classes of Schr odinger and KdV equations.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2012. / Unable to load abstract.

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.

Guo Bao Zhu. / Thesis (Ph.D.)--Chinese University of Hong Kong. / Includes bibliographical references. / Abstract --- p.1 / Introduction --- p.3 / Chapter Chapter 1. --- Semigroup for Age-Dependent Population Equations with Time Delay / Chapter 1.1 --- Introduction --- p.13 / Chapter 1.2 --- Problem Statement and Linear Theory --- p.14 / Chapter 1.3 --- Spectral Properties of the Infinitesimal Generator --- p.20 / Chapter 1.4 --- A Nonlinear Semigroup of the Logistic Age-Dependent Model with Delay --- p.26 / References --- p.34 / Chapter Chapter 2. --- Global Behaviour of Logistic Model of Age-Dependent Population Growth / Chapter 2.1 --- Introduction --- p.35 / Chapter 2.2 --- Global Behaviour of the Solutions --- p.37 / Chapter 2.3 --- Oscillatory Properties --- p.47 / References --- p.51 / Chapter Chapter 3. --- Semigroups for Age-Size Dependent Population Equations with Spatial Diffusion / Chapter 3. 1 --- Introduction --- p.52 / Chapter 3.2 --- Properties of the Infinitesimal Generator --- p.54 / Chapter 3.3 --- Properties of the Semigroup --- p.59 / Chapter 3.4 --- Dynamics with Age-Size Structures --- p.61 / Chapter 3.5 --- Logistic Model with Diffusion --- p.66 / References --- p.70 / Chapter Chapter 4. --- Semi-Discrete Population Equations with Time Delay / Chapter 4. 1 --- Introduction --- p.72 / Chapter 4.2 --- Linear Semi-Discrete Model with Time Delay --- p.74 / Chapter 4.3 --- Nonlinear Semi-Discrete Model with Time Delay --- p.88 / References --- p.98 / Chapter Chapter 5. --- A Finite Difference Scheme for the Equations of Population Dynamics / Chapter 5.1 --- Introduction --- p.99 / Chapter 5.2 --- The Discrete System --- p.102 / Chapter 5.3 --- The Main Results --- p.107 / Chapter 5.4 --- A Finite Difference Scheme for Logistic Population Model --- p.113 / Chapter 5.5 --- Numerical Simulation --- p.116 / References --- p.119 / Chapter Chapter 6. --- Optimal Birth Control Policies I / Chapter 6. 1 --- Introduction --- p.120 / Chapter 6.2 --- Fixed Horizon and Free Point Problem --- p.120 / Chapter 6.3 --- Time Optimal Control Problem --- p.129 / Chapter 6.4 --- Infinite Horizon Problem --- p.130 / Chapter 6.5 --- Results of the Nonlinear System with Logistic Term --- p.143 / Reference --- p.148 / Chapter Chapter 7. --- Optimal Birth Control Policies II / Chapter 7. 1 --- Free Final Time Problems --- p.149 / Chapter 7.2 --- Systems with Phase Constraints --- p.160 / Chapter 7.3 --- Mini-Max Problems --- p.166 / References --- p.168 / Chapter Chapter 8. --- Perato Optimal Birth Control Policies / Chapter 8.1 --- Introduction --- p.169 / Chapter 8.2 --- The Duboviskii-Mi1yutin Theorem --- p.171 / Chapter 8.3 --- Week Pareto Minimum Principle --- p.172 / Chapter 8.4 --- Problem with Nonsmooth Criteria --- p.175 / References --- p.181 / Chapter Chapter 9. --- Overtaking Optimal Control Problems with Infinite Horizon / Chapter 9. 1 --- Introduction --- p.182 / Chapter 9.2 --- Problem Statement --- p.183 / Chapter 9.3 --- The Turnpike Property --- p.190 / Chapter 9.4 --- Existence of Overtaking Optimal Solutions --- p.196 / References --- p.198 / Chapter Chapter 10. --- Viable Control in Logistic Populatiuon Model / Chapter 10. 1 --- Introduction --- p.199 / Chapter 10. 2 --- Viable Control --- p.200 / Chapter 10.3 --- Minimum Time Problem --- p.205 / References --- p.208 / Author's Publications During the Candidature --- p.209

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

Mak, Tsz Fan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 33-36). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Wang Yi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 61-63). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Some Preparations --- p.10 / Chapter 3 --- Proof of Theorem 1.1 --- p.24 / Chapter 4 --- Location of Blow-up Points (for n=2) --- p.26 / Chapter 5 --- Location of Blow-up Points (for General n) --- p.35 / Chapter 6 --- Asymptotic behavior of solutions near blow-up point --- p.46 / Chapter 7 --- Appendix --- p.57 / Bibliography --- p.61