A thesis submitted to the Faculty of Science, University of the Witwatersrand, in requirement for the degree Doctor of Philosophy, Johannesburg, 2013. / The (1+3) dimensional wave and Klein-Gordon equations are constructed using the covariant d'Alembertian operator on several spacetimes of interest. Equations on curved geometry inherit the nonlinearities of the geometry. These equations display interesting properties in a number of ways. In particular, the number of symmetries and therefore, the conservation laws reduce depending on how curved the manifold is. We study the symmetry properties and conservation laws of wave equations on Freidmann-Robertson-Walker, Milne, Bianchi, and de Sitter universes. Symmetry structures are used to reduce the number of unknown functions, and hence contribute to nding exact solutions of the equations. As expected, properties of reduction procedures using symmetries, variational structures and conservation laws are more involved than on the well known at (Minkowski) manifold.

Ke, Ting. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 30-32). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction and Main results --- p.1 / Chapter 2 --- Preliminaries --- p.7 / Chapter 3 --- Finite Speed of Propagation Property --- p.11 / Chapter 4 --- Proof of the Main Results --- p.19 / Chapter 4.1 --- Proof of Theorem 1.0.1 --- p.19 / Chapter 4.2 --- Proof of Theorem 1.0.2 --- p.24 / Chapter 5 --- Discussions --- p.26 / Bibliography --- p.30

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 185-191) and index.

Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (p. 214-234).

Thesis (Ph.D.)--Brown University, 2005. / Vita. Thesis advisor: Chi-Wang Shu. Includes bibliographical references (leaves 83-87). Also available online.

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 101-104).

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, 2014. / Symmetries and conservation laws of partial di erential equations (pdes) have been instrumental in giving new approaches for reducing pdes. In this dissertation, we study the symmetries and conservation laws of the two-dimensional Schr odingertype equation and the Benney-Luke equation, we use these quantities in the Double Reduction method which is used as a way to reduce the equations into a workable pdes or even an ordinary di erential equations. The symmetries, conservation laws and multipliers will be determined though di erent approaches. Some of the reductions of the Schr odinger equation produced some famous di erential equations that have been dealt with in detail in many texts.

M.Sc. / The innovative work of Black and Scholes [1, 2] extended the mathematical understanding of the options pricing model, beginning the deliberate study of the theory of option pricing. Its impact on the nancial markets was immediate and unprecedented and is arguably one of the most important discoveries within nance theory to date. By just inserting a few variables, which include the stock price, risk-free rate of return, option's strike price, expiration date, and an estimate of the volatility of the stock's price, the option-pricing formula is easily used by nancial investors. It allows them to price various derivatives ( nancial instrument whose price and value are derived from the value of assets underlying them), including options on commodities, nancial assets and even pricing of employee stock options. Hence, European1 and American2 call or put options on a non-dividend-paying stock can be valued using the Black-Scholes model. All further advances in option pricing since the Black-Scholes analysis have been re nements, generalisations and expansions of the original idea presented by them.

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, 2017 / Killing vectors are widely used to study conservation laws admitted by spacetime metrics or to determine exact solutions of Einstein field equations (EFE) via Killing’s equation. Its solutions on a manifold are in one-to-one correspondence with continuous symmetries of the metric on that manifold. Two well known spherically symmetric static spacetime metrics in Relativity that admit maximal symmetry are given by Minkowski and de-Sitter metrics. Some other spherically symmetric metrics forming interesting solutions of the EFE are known as Schwarzschild, Kerr, Bertotti-Robinson and Einstein metrics. We study the symmetry properties and conservation laws of the geodesic equations following these metrics as well as the wave and Klein-Gordon (KG) type equations constructed using the covariant d’Alembertian operator on these manifolds. As expected, properties of reduction procedures using symmetries are more involved than on the well known flat (Minkowski) manifold. / XL2017

Zhang Hedan. / Thesis submitted in: September 2004. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 78-81). / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Convex Scalar Conservation Laws --- p.9 / Chapter 2.1 --- Cauchy Problems and Weak Solutions --- p.9 / Chapter 2.2 --- Rankine-Hugoniot Condition --- p.11 / Chapter 2.3 --- Entropy Condition --- p.13 / Chapter 2.4 --- Uniqueness of Weak Solution --- p.15 / Chapter 2.5 --- Riemann Problems --- p.17 / Chapter 3 --- General Scalar Conservation Laws --- p.21 / Chapter 3.1 --- Entropy-Entropy Flux Pairs --- p.21 / Chapter 3.2 --- Admissibility Conditions --- p.22 / Chapter 3.3 --- Kruzkov Theory --- p.23 / Chapter 4 --- Elementary waves and Riemann Problems for Nonconvex Scalar Conservation Laws --- p.35 / Chapter 4.1 --- Basic Facts --- p.35 / Chapter 4.2 --- Riemann Solutions --- p.36 / Chapter 5 --- Asymptotic Behavior --- p.46 / Chapter 5.1 --- Periodic Asymptotic Behavior --- p.46 / Chapter 5.2 --- Asymptotic Behavior of Convex Conservation Law --- p.49 / Chapter 5.3 --- Asymptotic Behavior of Non-convex case --- p.52 / Chapter 5.3.1 --- L∞ Behavior --- p.53 / Chapter 5.3.2 --- Wave-Interactions and Asymptotic Behavior Toward Shock Waves --- p.55 / Bibliography --- p.78