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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. Johannesburg, February 15, 2015. / In this thesis, the forward and inverse Spectral Theory for first order Hermitian systems with complex potentials and periodic boundary conditions are studied. The aim of this work is to prove two inverse periodicity Theorems and two uniqueness results for determinants of quasiperiodic boundary value problems.

The bending of thin flat plates has occupied the interests of mechanicians and applied mathematicians since J. L. Lagrange discovered the differential equation characterizing the behavior of such structural members. One particular phase of investigation in this field concerns itself with the solution of the differential equation subject to given boundary conditions. Indeed, it may be safely stated that the bulk of the literature on the subject of flat plates is concerned with the solution of problems involving the specification of the transverse loading on the plate and the conditions at the boundary of the plate. Various mathematical techniques are available for the solution of such problems. Among these, the most prominent are, a.) the method of series, b) the method of singularities, and c) the complex variable techniques. A survey of the literature in this area has revealed a paucity of solutions of certain types of problems; notably, those problems in which boundary conditions are mixed a.long a portion of the edge of the plate which ha.s a continuously turning tangent. By mixed boundary conditions, we mean a. change in condition from prescription of bending moment and vertical shear to assignment of slope and deflection along a portion of the edge which has a continuously turning tangent. In the first section of this thesis, a number of problems are considered for the half-plane. The attendant boundary conditions considered a.re combinations of clamping and simple support. The second portion consists of a number of problems associated with the quarter-plane. Solutions for these problems are obtained by utilizing the method of images in conjunction with the solutions presented in the first section. After this, we examine some problems connected with the circular plate. In particular, a numerical solution is given for a uniformly loaded circular plate simply-supported over half of its boundary and clamped over the remaining portion. The last chapter is a brief discussion of plates in the form of rectangles. Here, a closed solution is presented for the bending moments in terms of Weierstrassian elliptic functions. Another numerical example is included for a uniformly loaded plate clamped over a portion of one edge and simply- supported over the remainder of its boundary. / Doctor of Philosophy

M.S.

In this paper results concerning a semilinear elliptic boundary value problem are proven. This problem has five solutions when the range of the derivative of the nonlinearity ƒ includes the first two eigenvalues. The existence and multiplicity or radially symmetric solutions under suitable conditions on the nonlinearity when Ω is a ball in R^N.

We study the linear abstract kinetic equation T𝜑(x)′=-A𝜑(x) in the half space {x≥0} with partial range boundary conditions. The function <i>ψ</i> takes values in a Hilbert space H, T is a self adjoint injective operator on H and A is an accretive operator. The first step in the analysis of this boundary value problem is to show that T⁻¹A generates a holomorphic bisemigroup. We prove two theorems about perturbation of bisemigroups that are interesting in their own right. The second step is to obtain a special decomposition of H which is equivalent to a Wiener-Hopf factorization. The accretivity of A is crucial in this step. When A is of the form "identity plus a compact operator", we work in the original Hilbert space. For unbounded A’s we consider weak solutions in a larger space H_T, which has a natural Krein space structure. Using the Krein space geometry considerably simplifies the analysis of the question of unique solvability. / Ph. D. / incomplete_metadata

An accuracy study is made of central finite difference methods for solving boundary value problems which are governed by second order differential equations with variable coefficients leading to odd order derivatives. Three methods are studied through applications to selected problems. Definitive expressions for the error in each method are obtained by using Taylor series to derive the differential equations which exactly represent the finite difference approximations. The resulting differential equations are accurately solved by a perturbation technique which yields the error directly. A half station method, which corresponds to making finite difference approximations before expanding derivatives of function products in the differential equations, was found superior to two whole station methods which correspond to expanding such products first. / M.S.

Li Kin-kuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 52-54). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Variational Formulation --- p.10 / Chapter 3 --- The Existence Of A Mountain Pass Solution --- p.12 / Chapter 4 --- Ground States --- p.21 / Chapter 5 --- The Projections Of v And w --- p.35 / Chapter 6 --- Computation Of The Energy: An Upper Bound --- p.37 / Chapter 7 --- Convergence: The First Approximation --- p.40 / Chapter 8 --- Convergence: The Second Approximation --- p.44 / Chapter 9 --- Computation Of The Energy: A Lower Bound --- p.48 / Chapter 10 --- Comparing The Energy: Completion Of The Proof Of Theorem 1.2 --- p.51 / Bibliography --- p.52

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