A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. September 26, 2014. / The aim of the dissertation is to extend on the work done by Hydon in [17]. We only consider second order ordinary difference equations and calculate their symmetry generators, first integrals and reduce their order, that is, find a general solution. We investigate the association between a symmetry generator and a first integral. Furthermore, we investigate when a reduced equation may be further reduced and lead to a double reduction. The examples considered are obtained from [17]. ii

by Yu Wai Kuen. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves 71-76). / Preface --- p.1 / PART I / Chapter Chapter 1 --- Inverse Scattering Method / Chapter §1 --- Introduction --- p.5 / Chapter §2 --- Rapidly decreasing solutions of the GNLSE --- p.6 / Chapter Chapter 2 --- Modified Inverse Scattering Method / Chapter §1 --- Introduction --- p.25 / Chapter §2 --- Singular solutions of the KdV equation --- p.25 / PART II / Chapter Chapter 3 --- Backlund Transformation Method / Chapter §1 --- Introduction --- p.37 / Chapter §2 --- Solution by Backlund transformation --- p.37 / Chapter §3 --- Clairin's method for finding Backlund transformations --- p.46 / Chapter §4 --- Construction of multi-soliton solutions --- p.48 / Chapter Chapter 4 --- Dressing Method And Hirota Direct Method / Chapter §1 --- Introduction --- p.51 / Chapter §2 --- Zakharov-Shabat's dressing method --- p.52 / Chapter §3 --- Hirota direct method --- p.57 / Chapter Chapter 5 --- Group Reduction Method / Chapter §1 --- Introduction --- p.61 / Chapter §2 --- Method of group reduction --- p.61 / Bibliography --- p.71

This thesis is concerned with stochastic heat equation with memory and nonlinear energy supply. The main motivation to study such systems comes from Thermodynamics, see [85]. The main objective of this work is to study the existence and uniqueness of solutions to such equations and to investigate some fundamental properties of solutions like continuous dependence on initial conditions. In our approach we follow the seminal papers by Da Prato and Clement [10], where the stochastic heat equation with memory is tranformed into an integral equation in a function space and the so-called mild solutions are studied. In the aforementioned papers only linear equations with additive noise were investigated. The main contribution of this work is the extension of this approach to nonlinear equations. Our main tools are the theory of stochastic convolutions as developed in [33] and the theory of resolvent kernels for deterministic linear heat equations with memory, see[10]. Since the solution at time t depends on the whole history of the process up to time t, the resolvent kernel does not define a semigroup of operators in the state space of the process and therefore a ???standard??? theory of stochastic evolution equations as presented in the monograph [33] does not apply. A more delicate analysis of the resolvent kernles and the associated stochastic convolutions is needed. We will describe now content of this thesis in more detail. Introductory Chapters 1 and 2 collect some basic and essentially well known facts about the Wiener process, stochastic integrals, stochastic convolutions and integral kernels. However, some results in Chapter 2 dealing with stochastic convolution with respect to non-homogenous Wiener process are extensions of the existing theory. The main results of this thesis are presented in Chapters 3 and 4. In Chapter 3 we prove the existence and uniqueness of solutions to heat equations with additive noise and either Lipschitz or dissipative nonlinearities. In both cases we prove the continuous dependence of solutions on initial conditions. In Chapter 4 we prove the existence and uniqueness of solutions and continuous dependence on initial conditions for equations with multiplicative noise. The diffusion coefficients defined by unbounded operators are allowed.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1928. / "Sonderdruck aus "Mathematische Annalen", Bd. 101, Heft 1"--T.p. verso. Vita. Includes bibliographical references.

Thesis (M.S. in Engineering Acoustics)--Naval Postgraduate School, December 2003. / Thesis advisor(s): S.E. Forsythe, S.R. Baker. Includes bibliographical references (p. 167). Also available online.

Thesis (Ph. D.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains ix, 118 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 115-118).

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