Ph.D. (Applied Mathematics) / In this thesis we introduce new numerical methods for solving nonlinear ordinary and partial differential equations. These methods solve differential equations in a manner similar to the Gauss Seidel approach of solving linear systems of algebraic equations. First the nonlinear differential equations are linearized by simply evaluating nonlinear terms at previous iterations. To solve the linearized iteration schemes obtained we use either the spectral method or higher order compact finite difference schemes and we call the resulting methods the spectral relaxation method (SRM) and the compact finite difference relaxation method (CFD-RM) respectively. We test the applicability of these methods in a wide variety of ODEs and PDEs. The accuracy and computational efficiency in terms of CPU time is compared against other methods as well as other results from literature. We solve a range of chaotic and hyperchaotic systems of equations. Chaotic and hyperchaotic are complex dynamical systems which are characterised by rapidly changing solutions and high sensitivity to small perturbations of the initial data. As a result finding their solutions is a challenging task. We modify the proposed SRM to be able to deal with such systems of equations. We also consider chaos control and synchronization between too identical chaotic systems. We also make a comparison between the SRM and CFD-RM and between the spectral quasilinearization method (SQLM) and the compact finite difference quasilinearization method (CFD-QLM). The aim is to compare the performance between the spectral and the compact finite difference approaches in solving similarity boundary layer problems. We consider two examples. First, we consider the flow of a viscous incompressible electrically conducting fluid over a continuously shrinking sheet. We also consider a three-equation system that models the problem of unsteady free convective heat and mass transfer on a stretching surface in a porous medium in the presence of a chemical reaction. We extend the application of the SRMand SQLMto PDEs. In particular we consider two unsteady boundary layer flow problems modelled by a PDE or a system of PDEs. We solve a one dimensional unsteady boundary layer flow due to an impulsively stretching surface and the problem of unsteady three-dimensional MHD flow and mass transfer in a porous space. Results are compared with results obtained using the Keller-box method which is popular in solving unsteady boundary layer problems. We also extend the application of the CFD-RM to PDEs modelling unsteady boundary layer flows and again compare results to Keller-box results. We consider two examples, the unsteady one dimensional MHD laminar boundary layer flow due to an impulsively stretching surface, and the unsteady three-dimensional MHD flow and heat transfer over an impulsively stretching plate.

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering, 2016 / The work presented extends and contributes to the research of modelling lightning return stroke (RS) electromagnetic (EM) fields in the time domain. Although previous work in this area has focused on individual lightning electromagnetic pulse (LEMP) modelling techniques, there has not been an investigation into the strengths and weaknesses of different methods, as well as the implementation considerations of the models. This work critically compares three unique techniques (Finite Antenna, FDTD, and Single Cell FDTD) under the same ideal simulation parameters. The research presented will evaluate the EM fields in the range of 50m to 500m from the lightning channel. This range, often referred to as the near field distance, has a significant effect on lightning induced overvoltages on distribution lines, which are primarily created by the horizontal EM fields of the RS channel. These close distances have a significant effect on the model implementations, especially with the FDTD method. Each of these modelling methods is explained and tested through examples. The models are implemented in C++ and have been included in the Appendix to aid in future implementation. From the model simulations it is clear that the FDTD method is the most comprehensive model available. It allows for non-ideal ground planes, as well as complex simulation environments. However, FDTD has a number of numerical related errors that the Finite Antenna method does not suffer from. The Single Cell FDTD method is simple to implement and does not suffer from the same numerical errors as a full FDTD implementation, but is limited to simple simulation environments. This work contributes to the research field by comparing and evaluating three techniques and giving consideration to the implementation and the applicability to lightning EM simulations. / MT2017

Ho Kwok Ching. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 107-111). / Abstracts in English and Chinese. / Content --- p.5 / Chapter 1 --- Introduction --- p.7 / Chapter 2 --- Background Theory --- p.9 / Chapter 2.1 --- Background history --- p.9 / Chapter 2.2 --- Finite Difference Time Domain Method --- p.10 / Chapter 2.2.1 --- Basic Formulation --- p.10 / Chapter 2.2.2 --- Finite Difference Expression: --- p.11 / Chapter 2.2.3 --- Courant Stability Criterion --- p.13 / Chapter 2.3 --- Absorbing Boundary Condition (PML) --- p.13 / Chapter 2.3.1 --- "Field -Splitting Modification of Maxwell's equation, TE case" --- p.14 / Chapter 2.3.2 --- Propagation of a TE Plane Wave in a PML Medium --- p.15 / Chapter 2.3.3 --- Transmission of a wave through PML-PML Interfaces --- p.19 / Chapter 2.3.4 --- PML for FDTD in 2D domain --- p.23 / Chapter 2.3.5 --- Extension to Three Dimension Case --- p.25 / Chapter 2.3.6 --- Obtaining S-parameters for General Microwave circuit --- p.26 / Chapter 2.4 --- Extended Finite Difference Time Domain Method --- p.29 / Chapter 2.4.1 --- Direct Implementation of Lumped Elements --- p.30 / Chapter 2.4.2 --- Equivalent-Source Techniques --- p.31 / Chapter 2.5 --- EMC --- p.37 / Chapter 3 --- Novel Techniques for Extended FDTD Method --- p.38 / Chapter 3.1 --- Introduction --- p.38 / Chapter 3.2 --- The Improved FDTD-SPICE Interface --- p.38 / Chapter 3.3 --- The Improved DC Bias Source --- p.48 / Chapter 3.4 --- The Improved DC Biasing Component --- p.50 / Chapter 3.5 --- Example --- p.51 / Chapter 3.6 --- Program Architecture --- p.55 / Chapter 3.7 --- Conclusion --- p.57 / Chapter 4 --- Example Design --- p.58 / Chapter 4.1 --- Introduction --- p.58 / Chapter 4.2 --- Internal Antenna Design --- p.58 / Chapter 4.2.1 --- Half-wavelength Patch --- p.58 / Chapter 4.2.2 --- Quarter-wavelength patch --- p.63 / Chapter 4.3 --- RF Power Amplifier Circuit Design --- p.73 / Chapter 4.4 --- Active Internal Antenna Design --- p.88 / Chapter 4.4.1 --- Design --- p.88 / Chapter 4.4.2 --- Surface Wave Analysis 一 Transient state analysis --- p.91 / Chapter 4.4.3 --- Surface wave analysis -AC analysis --- p.95 / Chapter 4.4.4 --- Far Field Pattern --- p.101 / Chapter 4.5 --- Conclusion --- p.105 / Chapter 5 --- Conclusion: --- p.106 / Chapter 6 --- Reference List --- p.107 / Publication --- p.111

by Wai-Chung Fung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 121-124). / Abstract also in Chinese. / Acknowledgement --- p.i / Abstract --- p.ii / Table of contents / Chapter Chapter 1: --- Introduction --- p.1 / Chapter Chapter 2: --- Background Theories --- p.4 / Chapter 2.1 --- Introduction --- p.4 / Chapter 2.2 --- Maxwell's Equations --- p.5 / Chapter 2.3 --- Basic Formulation --- p.8 / Chapter 2.4 --- Plane Wave Formulation --- p.13 / Chapter 2.4.1 --- Total-Field / Scattered-Field Algorithm --- p.14 / Chapter 2.4.2 --- Pure Scattered-Field Algorithm --- p.16 / Chapter 2.4.2.1 --- Application to PEC Structures --- p.16 / Chapter 2.4.2.2 --- Application to Lossy Dielectric Structures --- p.17 / Chapter 2.5 --- Incident Plane Wave Components Generation --- p.20 / Chapter 2.6 --- Source and Termination Modeling in FD-TD model --- p.24 / Chapter 2.6.1 --- Resistive source --- p.25 / Chapter 2.6.2 --- Resistor Formulation --- p.27 / Chapter 2.7 --- PML Formulation --- p.28 / Chapter 2.7.1 --- Two-Dimensional TE Case --- p.28 / Chapter 2.7.2 --- Extension to the Full-vector Three-Dimension Case --- p.32 / Chapter 2.8 --- Time Domain Extrapolation --- p.33 / Chapter 2.8.1 --- Prony's Model --- p.34 / Chapter 2.8.2 --- Auto-regressive Model and Performance Comparison with Prony's Method --- p.36 / Chapter 2.9 --- Summary --- p.42 / Chapter Chapter 3: --- Verification of FD-TD Method --- p.43 / Chapter 3.1 --- Introduction --- p.43 / Chapter 3.2 --- Microstrip Patch Antenna: An Introduction --- p.44 / Chapter 3.2.1 --- Direct Fed Patch --- p.45 / Chapter 3.2.2 --- EMC Patch --- p.50 / Chapter 3.2.3 --- Aperture-Coupled Patch --- p.53 / Chapter 3.3 --- Verification of FD-TD: S11 Analysis --- p.55 / Chapter 3.3.1 --- Analysis of Direct Fed Rectangular Patch Antenna --- p.56 / Chapter 3.3.2 --- Analysis of EMC Patch Antenna --- p.60 / Chapter 3.3.3 --- Analysis of Aperture-Coupled Patch Antenna --- p.63 / Chapter 3.4 --- Verification of FD-TD: Radiation Pattern Analysis --- p.66 / Chapter 3.4.1 --- The Absolute and Relative Approaches --- p.67 / Chapter 3.4.2 --- The Inset Fed Patch Antenna --- p.69 / Chapter 3.5 --- Summary --- p.71 / Chapter Chapter 4: --- Space Diversity Design --- p.73 / Chapter 4.1 --- Introduction --- p.73 / Chapter 4.2 --- How Space Diversity Antenna Works --- p.74 / Chapter 4.3 --- Criteria for Evaluation and Optimization of Diversity Performance --- p.77 / Chapter 4.4 --- Simple Approach for Two-Patch Diversity Array --- p.82 / Chapter 4.4.1 --- Performance as a Function of Antenna Separation --- p.83 / Chapter 4.5 --- Novel Designs for Performance Improvement --- p.89 / Chapter 4.5.1 --- Shorting Post Isolation --- p.90 / Chapter 4.5.2 --- Offset-positioned Configuration --- p.101 / Chapter 4.6 --- Three-Patch Diversity Array --- p.106 / Chapter 4.6.1 --- Co-aligned Configurations --- p.107 / Chapter 4.6.2 --- Offset-Positioned Configurations --- p.112 / Chapter 4.7 --- Summary --- p.117 / Chapter Chapter 5: --- Conclusion --- p.118 / Appendix A: Publication --- p.121 / Appendix B: References List --- p.122

We have constructed a new finite-difference time-domain (FDTD) method in this project. Our new algorithm focuses on the most important and more challenging transverse electric (TE) case. In this case, the electric field is discontinuous across the interface between different dielectric media. We use an electric permittivity that stays as a constant in each medium, and magnetic permittivity that is constant in the whole domain. To handle the interface between different media, we introduce new effective permittivities that incorporates electromagnetic fields boundary conditions. That is, across the interface between two different media, the tangential component, Er(x,y), of the electric field and the normal component, Dn(x,y), of the electric displacement are continuous. Meanwhile, the magnetic field, H(x,y), stays as continuous in the whole domain. Our new algorithm is built based upon the integral version of the Maxwell's equations as well as the above continuity conditions. The theoretical analysis shows that the new algorithm can reach second-order convergence O(∆x2)with mesh size ∆x. The subsequent numerical results demonstrate this algorithm is very stable and its convergence order can reach very close to second order, considering accumulation of some unexpected numerical approximation and truncation errors. In fact, our algorithm has clearly demonstrated significant improvement over all related FDTD methods using effective permittivities reported in the literature. Therefore, our new algorithm turns out to be the most effective and stable FDTD method to solve Maxwell's equations involving multiple media.

Includes bibliographical references (leaves 158-163). Concerns the development of high-order finite-difference methods on a uniform rectangular grid for advection and diffuse problems with smooth variable coefficients. This technique has been successfully applied to variable-coefficient advection and diffusion problems. Demonstrates that the new schemes may readily be incorporated into multi-dimensional problems by using locally one-dimensional techniques, or that they may be used in process splitting algorithms to solve complicatef time-dependent partial differential equations.

Bibliography: leaves 264-267.

Includes bibliographical references (leaves 158-163). / vi, 166 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Concerns the development of high-order finite-difference methods on a uniform rectangular grid for advection and diffuse problems with smooth variable coefficients. This technique has been successfully applied to variable-coefficient advection and diffusion problems. Demonstrates that the new schemes may readily be incorporated into multi-dimensional problems by using locally one-dimensional techniques, or that they may be used in process splitting algorithms to solve complicatef time-dependent partial differential equations. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 2001

<p>The famous Black-Scholes (BS) model used in the option pricing theory</p><p>contains two parameters - a volatility and an interest rate. Both</p><p>parameters should be determined before the price evaluation procedure</p><p>starts. Usually one use the historical data to guess the value of these</p><p>parameters. For short lifetime options the interest rate can be estimated</p><p>in proper way, but the volatility estimation is, as well in this case,</p><p>more demanding. It turns out that the volatility should be considered</p><p>as a function of the asset prices and time to make the valuation self</p><p>consistent. One of the approaches to this problem is the method of</p><p>uncertain volatility and the static hedging. In this case the envelopes</p><p>for the maximal and minimal estimated option price will be introduced.</p><p>The envelopes will be described by the Black - Scholes - Barenblatt</p><p>(BSB) equations. The existence of the upper and lower bounds for the</p><p>option price makes it possible to develop the worse and the best cases</p><p>scenario for the given portfolio. These estimations will be financially</p><p>relevant if the upper and lower envelopes lie relatively narrow to each</p><p>other. One of the ideas to converge envelopes to an unknown solution</p><p>is the possibility to introduce an optimal static hedged portfolio.</p>

Graduation date: 1996