by Chong Man Yuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 125-128). / Dedication / Acknowledgements / List of Figure / List of Table / List of Appendix / Chapter 1. --- Introduction / Chapter 1.1 --- Background / Chapter 1.2 --- Ultra-Wide Band Antenna / Chapter 1.3 --- Finite Element Method (FEM) / Chapter 1.3.1 --- Domain Discretization / Chapter 1.3.2 --- Formulation of Variational Method / Chapter 2 --- Theory / Chapter 2.1 --- Variational principles for electromagnetics / Chapter 2.1.1 --- Construction of Functional / Chapter 2.2 --- Artificial Boundary / Chapter 2.2.1 --- Absorbing Boundary Conditions / Chapter 2.2.2 --- Perfectly Matched Layer (PML) / Chapter 2.3 --- Edge Basis Function / Chapter 2.4 --- Slotline Analysis / Chapter 3 --- Implementation of FEM / Chapter 3.1 --- Formulation of Element matrix / Chapter 3.2 --- Mesh Generation / Chapter 3.3 --- Assembly / Chapter 3.4 --- Incorporation of Boundary Conditions / Chapter 3.5 --- Code Implementation / Chapter 4 --- Finite Element Simulations / Chapter 4.1 --- Slotline / Chapter 4.2 --- Artificial Boundary of the domain / Chapter 4.3 --- Slotline Taper Junction / Chapter 4.4 --- Slotline Bowtie Junction / Chapter 5 --- Conclusion / Appendix A1 / Appendix A2 / Appendix A3 / Bibliography

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This thesis is about using appropriate tools in functional analysis arid classical analysis to tackle the problem of existence and uniqueness of nonlinear partial differential equations. There being no unified strategy to deal with these equations, one approaches each equation with an appropriate method, depending on the characteristics of the equation. The correct setting of the problem in appropriate function spaces is the first important part on the road to the solution. Here, we choose the setting of Sobolev spaces. The second essential part is to choose the correct tool for each equation. In the first part of this thesis (Chapters 3 and 4) we consider a variety of nonlinear hyperbolic partial differential equations with mixed boundary and initial conditions. The methods of compactness and monotonicity are used to prove existence and uniqueness of the solution (Chapter 3). Finding a priori estimates is the main task in this analysis. For some types of nonlinearity, these estimates cannot be easily obtained, arid so these two methods cannot be applied directly. In this case, we first linearise the equation, using linear recurrence (Chapter 4). In the second part of the thesis (Chapter 5), by using an appropriate tool in functional analysis (the Sobolev Imbedding Theorem), we are able to improve previous results on a posteriori error estimates for the finite element method of lines applied to nonlinear parabolic equations. These estimates are crucial in the design of adaptive algorithms for the method, and previous analysis relies on, what we show to be, unnecessary assumptions which limit the application of the algorithms. Our analysis does not require these assumptions. In the last part of the thesis (Chapter 6), staying with the theme of choosing the most suitable tools, we show that using classical analysis in a proper way is in some cases sufficient to obtain considerable results. We study in this chapter nonexistence of positive solutions to Laplace's equation with nonlinear Neumann boundary condition. This problem arises when one wants to study the blow-up at finite time of the solution of the corresponding parabolic problem, which models the heating of a substance by radiation. We generalise known results which were obtained by using more abstract methods.

by John R. Pellam. / "November 7, 1947." / Bibliography: p. 17. / Army Signal Corps Contract No. W-36-039 sc-32037.

Thesis (Ph.D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (February 27, 2007) Vita. Includes bibliographical references.

Thesis (Ph. D.)--Oregon State University, 1983. / Typescript (photocopy). Includes bibliographical references (leaves 130-132). Also available on the World Wide Web.

The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points.

The paper is devoted to pseudodifferential boundary value problems in domains with cuspidal wedges. Concerning the geometry we even admit a more general behaviour, namely oscillating cuspidal wedges. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to edges.

Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbol structure is responsible or ellipicity and for the nature of parametrices within an algebra of "edge-degenerate" pseudo-differential operators. The edge symbol component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operatot-valued Mellin symbols. We establish a calculus in s framework of "twisted homogenity" that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We consider smooth symbols and ellipticity without additional boundary conditions which is the relevant case on a manifold with boundary. Starting from a class of symbols that has been investigated before for integer orders in boundary value problems with the transmission property we study operators of arbitrary real orders that play a similar role for operators without the transmission property. Moreover, we show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We finally investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary.