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Modeling of patch antennas and resonators on arbitrary dielectric substrates using surface RWG and volume edge based basis functions and the Method of Moments is implemented. The performance of the solver is studied for different mesh configurations. The results obtained are tested by comparison with experiments and Ansoft HFSS v9 simulator. The latter uses a large number of finite elements (up to 200K) and adaptive mesh refinement, thus providing the reliable data for comparison. The error in the resonant frequency is estimated for canonical resonator structures at different values of the relative dielectric constant ƒÕr, which ranges from 1 to 200. The reported results show a near perfect agreement in the estimation of resonant frequency for all the metal-dielectric resonators. Behavior of the antenna input impedance is tested, close to the first resonant frequency for the patch antenna. The error in the resonant frequency is estimated for different structures at different values of the relative dielectric constant ƒÕr, which ranges from 1 to 10. A larger error is observed in the calculation of the resonant frequency of the patch antenna. Moreover, this error increases with increase in the dielectric constant of the substrate. Further scope for improvement lies in the investigation of this effect.

Bibliography: leaves 95-97 / iv, 97 leaves : ill ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1986

The purpose of this paper is to generalize a theorem that characterizes absolute continuity of bounded finitely additive set functions in the form of an integral approximation. We show that his integral exists if the condition of absolute continuity is removed.

In this dissertation we obtain an e cient hybrid numerical method for the solution of stochastic di erential equations (SDEs). Speci cally, our method chooses between two numerical methods (Euler and Milstein) over a particular discretization interval depending on the value of the simulated Brownian increment driving the stochastic process. This is thus a new1 adaptive method in the numerical analysis of stochastic di erential equation. Mauthner (1998) and Hofmann et al (2000) have developed a general framework for adaptive schemes for the numerical solution to SDEs, [30, 21]. The former presents a Runge-Kutta-type method based on stepsize control while the latter considered a one-step adaptive scheme where the method is also adapted based on step size control. Lamba, Mattingly and Stuart, [28] considered an adaptive Euler scheme based on controlling the drift component of the time-step method. Here we seek to develop a hybrid algorithm that switches between euler and milstein schemes at each time step over the entire discretization interval, depending on the outcome of the simulated Brownian motion increment. The bias of the hybrid scheme as well as its order of convergence is studied. We also do a comparative analysis of the performance of the hybrid scheme relative to the basic numerical schemes of Euler and Milstein. / Mathematical Sciences / M.Sc. (Applied Mathematics)

In this dissertation we obtain an e cient hybrid numerical method for the solution of stochastic di erential equations (SDEs). Speci cally, our method chooses between two numerical methods (Euler and Milstein) over a particular discretization interval depending on the value of the simulated Brownian increment driving the stochastic process. This is thus a new1 adaptive method in the numerical analysis of stochastic di erential equation. Mauthner (1998) and Hofmann et al (2000) have developed a general framework for adaptive schemes for the numerical solution to SDEs, [30, 21]. The former presents a Runge-Kutta-type method based on stepsize control while the latter considered a one-step adaptive scheme where the method is also adapted based on step size control. Lamba, Mattingly and Stuart, [28] considered an adaptive Euler scheme based on controlling the drift component of the time-step method. Here we seek to develop a hybrid algorithm that switches between euler and milstein schemes at each time step over the entire discretization interval, depending on the outcome of the simulated Brownian motion increment. The bias of the hybrid scheme as well as its order of convergence is studied. We also do a comparative analysis of the performance of the hybrid scheme relative to the basic numerical schemes of Euler and Milstein. / Mathematical Sciences / M.Sc. (Applied Mathematics)

This thesis consists of three parts. The first part is an introduction and referencessome recent work on 2D electromagnetic scattering problems at high frequencies. It alsopresents the basic integral equation types for impenetrable objects. A brief discussionof the standard elements of the method of moments is followed by summaries of thepapers.Paper I presents an accurate implementation of the method of moments for a perfectlyconducting cylinder. A scaling for the rapid variation of the solution improves accuracy.At high frequencies, the method of moments leads to a large dense system of equations.Sparsity in this system is obtained by modifying the integration path in the integralequation. The modified path reduces the accuracy in the deep shadow.In paper II, a hybrid method is used to handle the standing waves that are prominentin the shadow for the TE case. The shadow region is treated separately, in a hybridscheme based on a priori knowledge about the solution. An accurate method to combinesolutions in this hybrid scheme is presented.

The principal theorem of this thesis is a theorem by Peano on the existence of a solution to a certain integral equation. The two primary notions underlying this theorem are uniform convergence and equi-continuity. Theorems related to these two topics are proved in Chapter II. In Chapter III we state and prove a classical existence and uniqueness theorem for an integral equation. In Chapter IV we consider the approximation on certain functions by means of elementary expressions involving "bent line" functions. The last chapter, Chapter V, is the proof of the theorem by Peano mentioned above. Also included in this chapter is an example in which the integral equation has more than one solution. The first chapter sets forth basic definitions and theorems with which the reader should be acquainted.

M.Sc. (Applied Mathematics) / Numerous studies on linear and nonlinear Volterra integral equations (VIEs), have been performed. These studies mainly considered the existence and uniqueness of the solution, and numerical solutions of these equations. In this work, a class of nonlinear (nonstandard) Volterra integral equation that has received very little attention in the literature is considered. The existence and uniqueness of the solution for the nonlinear VIE is proved using the contraction mapping theorem in the space C[0; d]. Collocation methods, repeated trapezoidal rule and repeated Simpson's rule are used to solve the nonlinear (nonstandard) VIE. For the collocation solutions we considered two cases: implicit Euler method and implicit midpoint method. Examples are used to compare the performance of these methods and the results show that the repeated Simpson's rule performs better than the other methods. An analysis of the collocation solution and the solution by the repeated trapezoidal rule is performed. Su cient conditions for existence and uniqueness of the numerical solution are given. The collocation methods and repeated trapezoidal rule yield convergence of order one.

Lee Joy Yan Agatha. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves [112]-114). / Text in English; abstracts in English and Chinese. / Lee Joy Yan Agatha. / Abstract --- p.ii / Acknowledgement --- p.iii / Contents --- p.iv / List of Figures --- p.viii / List of Tables --- p.xii / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- The Transfer Integral Equation Method --- p.3 / Chapter 2.1 --- The System --- p.3 / Chapter 2.1.1 --- The Hamiltonian --- p.4 / Chapter 2.1.2 --- The length parameter --- p.5 / Chapter 2.1.3 --- The temperature parameter --- p.5 / Chapter 2.2 --- The Transfer Integral Equation --- p.6 / Chapter 2.2.1 --- The partition function --- p.6 / Chapter 2.2.2 --- The transfer integral equation --- p.6 / Chapter 2.2.3 --- The pseudo-Schrodinger equation approximation --- p.7 / Chapter 2.2.4 --- Distribution function of the displacements --- p.9 / Chapter 2.3 --- The Thermodynamics --- p.10 / Chapter 2.3.1 --- Internal energy and heat capacity --- p.10 / Chapter 2.3.2 --- Displacement fluctuation --- p.12 / Chapter 2.3.3 --- Displacement correlation function --- p.12 / Chapter Chapter 3. --- The Φ4 Chain --- p.14 / Chapter 3.1 --- Soliton In The Chain --- p.15 / Chapter 3.1.1 --- Kink soliton and antikink soliton --- p.15 / Chapter 3.1.2 --- Energy of a static kink --- p.18 / Chapter 3.2 --- Low Temperature WKB Approximation for the Φ4 Chain --- p.20 / Chapter 3.2.1 --- The ground state energy ε0 and tunneling-splitting contribution --- p.20 / Chapter 3.2.2 --- First order WKB approximation of ΨRo( φ) --- p.22 / Chapter 3.2.3 --- Second order WKB wavefunction ΨRo( φ)） --- p.26 / Chapter 3.2.4 --- Third order WKB wavefunction for ΨRo( φ) --- p.27 / Chapter 3.3 --- Thermodynamics --- p.28 / Chapter 3.3.1 --- Ground state energy ε0 and wavefunction Ψo( φ) --- p.28 / Chapter 3.3.2 --- Internal energy and heat capacity --- p.33 / Chapter 3.3.3 --- Displacement correlation function --- p.36 / Chapter Chapter 4. --- Other Nonlinear Klein-Gordon Models --- p.42 / Chapter 4.1 --- The φ8 Chain --- p.42 / Chapter 4.1.1 --- The potential --- p.42 / Chapter 4.1.2 --- The ground state energy εo and wavefunction Ψo( φ) --- p.44 / Chapter 4.1.3 --- Internal energy and heat capacity --- p.49 / Chapter 4.1.4 --- Displacement correlation function cyy(n) --- p.51 / Chapter 4.2 --- The Gaussian-Double-Well Chains --- p.53 / Chapter 4.2.1 --- The potential --- p.53 / Chapter 4.2.2 --- The ground state energy εo and wavefunction ψo --- p.55 / Chapter 4.2.3 --- Internal energy and heat capacity --- p.58 / Chapter 4.2.4 --- Displacement correlation function cyy(n) --- p.59 / Chapter 4.3 --- Comparison Between Different NKG Models --- p.61 / Chapter 4.3.1 --- The potentials --- p.61 / Chapter 4.3.2 --- Ground state energy εo and wavefunction ψo(ψ) --- p.65 / Chapter 4.3.3 --- Internal energy and heat capacity --- p.68 / Chapter 4.3.4 --- Displacement fluctuation --- p.70 / Chapter 4.3.5 --- Displacement correlation function cyy(n) --- p.71 / Chapter 4.4 --- Linear Response of a NKG Chain to a Static Perturbing Field --- p.75 / Chapter 4.4.1 --- The external perturbing field --- p.75 / Chapter 4.4.2 --- The linear response --- p.75 / Chapter 4.4.3 --- Linear response of an array of weakly coupled NKG chains --- p.80 / Chapter Chapter 5. --- Quantum Corrections --- p.86 / Chapter 5.1 --- The Effective Potential --- p.86 / Chapter 5.1.1 --- The smearing parameter --- p.86 / Chapter 5.1.2 --- The effective potential --- p.88 / Chapter 5.2 --- Quantum Corrections on Thermodynamics --- p.90 / Chapter 5.2.1 --- The ground state energy εo and wavefunction ψo(ψ) --- p.90 / Chapter 5.2.2 --- The heat capacity --- p.94 / Chapter 5.2.3 --- Displacement correlation function and displacement fluctuation --- p.97 / Chapter Chapter 6. --- Conclusion --- p.103 / Appendix A. Infinite-Square-Well Basis Diagonalization --- p.105 / Appendix B. Oscillator Basis Diagonalization --- p.110 / Bibliography --- p.112