Numerical methods for solving initial value problems in ordinary differential equations are studied. A notation is introduced to represent cyclic methods in terms of two matrices, A_h, and B_h, and this is developed to cover the very extensive class of m-block methods. Some stability results are obtained and convergence is analysed by means of a new consistency concept, namely optimal consistency. It is shown that optimal consistency allows one to give two-sided bounds on the global error, and examples are given to illustrate this. The form of the inverse of A_h is studied closely to give a criterion for the order of convergence to exceed that of consistency by one. Further convergence results are obtained , the first of which gives the orders of convergence for cases in which A_h, and B_h, have a special form, and the second of which gives rise to the possibility of the order of convergence exceeding that of consistency by two or more at some stages. In addition an alternative proof is given of the superconvergence result for collocation methods. In conclusion the work covered is set in the context of that done in recent years by various authors.

published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

Very accurate methods, based on boundary integral techniques, are developed for the study of multiple, interacting fluid interfaces in an Eulerian fluid. These methods are applied to the evolution of a thin, periodic layer of constant vorticity embedded in irrotational fluid. Numerical regularity experiments are conducted and suggest that the interfaces of the layer develop a curvature singularity in infinite time. This is to be contrasted with the more singular vorticity distribution of a vortex sheet developing such a singularity in a finite time.

In this thesis, a wavelet-based prediction method is developed for concealing packet-loss effects in wireless channels. The proposed method utilizes a wavelet decomposition algorithm in order to process the data and then applies the well known linear prediction technique to estimate one or more approximation coefficients as necessary at the lowest resolution level. The predicted sample stream is produced by using the predicted approximation coefficients and by exploiting certain sample value patterns in the detail coefficients. In order to test the effectiveness of the proposed scheme, a wireless channel based on a three-state Markov model is developed and simulated. Simulation results for transmission of image and speech packet streams over a wireless channel are reported for both the wavelet-based prediction and direct linear prediction. In all the simulations run in this work, the wavelet-based method outperformed the direct linear prediction method. / Hellenic Navy author.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Master of Science, May 14, 2013. / In this dissertation, we investigate the parabolic mixed derivative diffusion equation modeling the viscous and viscoelastic effects in a non-Newtonian viscoelastic fluid. The model is analytically considered using Fourier and Laplace transformations. The main focus of the dissertation, however, is the implementation of the Peaceman-Rachford Alternating Direction Implicit method. The one-dimensional parabolic mixed derivative diffusion equation is extended to a two-dimensional analog. In order to do this, the two-dimensional analog is solved using a Crank-Nicholson method and implemented according to the Peaceman- Rachford ADI method. The behaviour of the solution of the viscoelastic fluid model is analysed by investigating the effects of inertia and diffusion as well as the viscous behaviour, subject to the viscosity and viscoelasticity parameters. The two-dimensional parabolic diffusion equation is then implemented with a high-order method to unveil more accurate solutions. An error analysis is executed to show the accuracy differences between the numerical solutions of the general ADI and high-order compact methods. Each of the methods implemented in this dissertation are investigated via the von-Neumann stability analysis to prove stability under certain conditions.

by Zheng Yu-kun. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Includes bibliographies.

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

Chan Yuet Tai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 107-111). / Chapter 1 --- Introduction to Sturm-Liouville Problem --- p.1 / Chapter 1.1 --- What are inverse problems? --- p.1 / Chapter 1.2 --- Introductory background --- p.2 / Chapter 1.3 --- The Liouville transformation --- p.3 / Chapter 1.4 --- The Sturm-Liouville problem 一 A historical look --- p.4 / Chapter 1.5 --- Where Sturm-Liouville problems come from? --- p.6 / Chapter 1.6 --- Inverse problems of interest --- p.8 / Chapter 2 --- Reconstruction Method I --- p.10 / Chapter 2.1 --- Perturbative inversion --- p.10 / Chapter 2.1.1 --- Inversion problem via Fredholm integral equation --- p.10 / Chapter 2.1.2 --- Output least squares method for ill-posed integral equations --- p.15 / Chapter 2.1.3 --- Numerical experiments --- p.17 / Chapter 2.2 --- Total inversion --- p.38 / Chapter 2.3 --- Summary --- p.45 / Chapter 3 --- Reconstruction Method II --- p.46 / Chapter 3.1 --- Computation of q --- p.47 / Chapter 3.2 --- Computation of the Cauchy data --- p.48 / Chapter 3.2.1 --- Recovery of Cauchy data for K --- p.51 / Chapter 3.2.2 --- Numerical implementation for computation of the Cauchy data . --- p.51 / Chapter 3.3 --- Recovery of q from Cauchy data --- p.52 / Chapter 3.4 --- Iterative procedure --- p.53 / Chapter 3.5 --- Numerical experiments --- p.60 / Chapter 3.5.1 --- Eigenvalues without noised data --- p.64 / Chapter 3.5.2 --- Eigenvalues with noised data --- p.69 / Chapter 4 --- Appendices --- p.79 / Chapter A --- Tikhonov regularization --- p.79 / Chapter B --- Basic properties of the Sturm-Liouville operator --- p.80 / Chapter C --- Asymptotic formulas for the eigenvalues --- p.86 / Chapter C.1 --- Case 1: h ≠ ∞ and H ≠ ∞ --- p.87 / Chapter C.2 --- Case 2: h= ∞ and H ≠∞ --- p.90 / Chapter C.3 --- Case 3: h = ∞ and H = ∞ --- p.91 / Chapter D --- Completeness of the eigenvalues --- p.92 / Chapter E --- d'Alembert solution formula for the wave equation --- p.97 / Chapter E.1 --- "The homogeneous solution uH(x,t)" --- p.98 / Chapter E.2 --- "The particular solution up(x, t)" --- p.99 / Chapter E.3 --- "The standard d'Alembert solution u(x,t)" --- p.101 / Chapter E.4 --- Applications to our problem --- p.101 / Chapter F --- Runge-Kutta method for solving eigenvalue problems --- p.104 / Bibliography --- p.107

by Chan Kam Wai Clifford. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 105-108). / Text in English; abstracts in English and Chinese. / by Chan Kam Wai Clifford. / Abstract --- p.i / Acknowledgements --- p.iii / Contents --- p.iv / List of Figures --- p.vii / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- Motivations of the Project --- p.1 / Chapter 1.2 --- Historical Background --- p.1 / Chapter 1.3 --- Objective and Outline of Thesis --- p.3 / Chapter Chapter 2. --- Reviews on One-dimensional Dynamical Cavity --- p.4 / Chapter 2.1 --- Formalism --- p.4 / Chapter 2.2 --- Methods of Solution --- p.6 / Chapter 2.2.1 --- Phase Construction (R function) --- p.6 / Chapter 2.2.2 --- Instantaneous Mode Expansion --- p.12 / Chapter 2.2.3 --- Transformation Method --- p.15 / Chapter 2.3 --- Numerical Results --- p.15 / Chapter 2.3.1 --- Some Results using R function --- p.16 / Chapter 2.3.2 --- Some Results using Instantaneous Mode Decomposition --- p.24 / Chapter 2.3.3 --- Remarks on the Numerical Scheme used in Transformation Method --- p.28 / Chapter 2.3.4 --- "Comparisons of Results obtained by Phase Construction, In- stantaneous Mode Decomposition and Transformation" --- p.28 / Chapter 2.4 --- Conclusion --- p.30 / Chapter Chapter 3. --- Fixed-point Analysis for the One-dimensional Cavity --- p.31 / Chapter 3.1 --- Introduction --- p.31 / Chapter 3.2 --- What are the fixed-points? --- p.32 / Chapter 3.3 --- Characteristics of Fixed-points --- p.36 / Chapter 3.4 --- Fixed-points and Geometric Resonance --- p.39 / Chapter Chapter 4. --- Electromagnetic Field in an Undulating Spherical Cavity --- p.44 / Chapter 4.1 --- Classical Electromagnetic field theory --- p.44 / Chapter 4.2 --- Boundary Conditions --- p.46 / Chapter 4.3 --- The Motion of Cavity Surface --- p.47 / Chapter Chapter 5. --- Methods of Solution and Results to the Spherical Cavity --- p.48 / Chapter 5.1 --- Introduction --- p.48 / Chapter 5.2 --- Mode Decomposition and Transformation Method revisited --- p.49 / Chapter 5.2.1 --- Mode Decomposition --- p.49 / Chapter 5.2.2 --- Transformation Method --- p.50 / Chapter 5.2.3 --- Remarks on the use of Instantaneous Mode Expansion and Transformation Method --- p.51 / Chapter 5.3 --- The Ge(z) function --- p.52 / Chapter 5.3.1 --- The Ge(z) function as a solution of the scalar wave equation --- p.52 / Chapter 5.3.2 --- Numerical Results --- p.54 / Chapter 5.4 --- The Me(z) function --- p.60 / Chapter 5.4.1 --- Formalism --- p.60 / Chapter 5.4.2 --- Comparison of Me(z) with Ge(z) --- p.62 / Chapter 5.4.3 --- Numerical Results --- p.63 / Chapter 5.5 --- Conclusions and Discussions --- p.93 / Chapter 5.5.1 --- Geometric Resonances --- p.93 / Chapter 5.5.2 --- Harmonic Resonances --- p.94 / Chapter Chapter 6. --- Conclusion --- p.95 / Appendix A. Electromagnetic Field in Spherical Cavity --- p.97 / Chapter A.1 --- Field Strength --- p.97 / Chapter A.2 --- Field Energy --- p.98 / "Appendix B. Construction of Ψe(r,t) by G(z)" --- p.100 / Appendix C. The Arbitrary Part GH(z) of Ψe(r，t) --- p.103 / Bibliography --- p.105

Hung Kin Ting. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 126-129). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Inverse Problems --- p.1 / Chapter 1.2 --- Applications of Inverse Problems --- p.2 / Chapter 1.3 --- Least-squares Solutions --- p.4 / Chapter 1.4 --- Discrete Systems --- p.4 / Chapter 1.5 --- "Discretization, Regularization and Regularization Pa- rameters" --- p.5 / Chapter 1.6 --- Outline of the Thesis --- p.6 / Chapter 2 --- Some Basic Concepts and Mathematical Tools --- p.8 / Chapter 2.1 --- Singular Value Decomposition (SVD) --- p.8 / Chapter 2.2 --- Generalized Singular Value Decomposition (GSVD) --- p.13 / Chapter 2.3 --- White Noises --- p.16 / Chapter 3 --- Regularized Solutions --- p.18 / Chapter 3.1 --- Derivation of Regularized Solutions --- p.18 / Chapter 3.2 --- Discrete Picard Condition --- p.20 / Chapter 3.3 --- Relationship between Discrete Picard Condition and Regularized Solution --- p.21 / Chapter 3.4 --- Checking for the Discrete Picard Condition --- p.22 / Chapter 4 --- Different Discretization Approaches --- p.23 / Chapter 4.1 --- Problem 1 - Volterra Integral Equation of the First Kind --- p.25 / Chapter 4.2 --- Examples of Problem 1 --- p.30 / Chapter 4.3 --- Problem 2 - Fredholm Integral Equation of the First Kind --- p.49 / Chapter 4.4 --- Examples of Problem 2 --- p.53 / Chapter 4.5 --- Conclusion --- p.57 / Chapter 5 --- Effect of Different Kinds of Observation Data and Differential Operators on Accuracy --- p.59 / Chapter 5.1 --- Pointwise Observation Data --- p.60 / Chapter 5.2 --- Pointwise Observation Data of Heat Fluxes at the Boundary --- p.69 / Chapter 5.3 --- Observation Data with Heat Fluxes --- p.80 / Chapter 5.4 --- Conclusion --- p.89 / Chapter 6 --- L-curve --- p.90 / Chapter 6.1 --- Properties of L-curve --- p.93 / Chapter 6.2 --- L-curve in Log-Log Scale --- p.100 / Chapter 6.3 --- Disadvantages of the L-curve Method --- p.100 / Chapter 7 --- Algorithms of Finding the Corner of L-curve --- p.105 / Chapter 7.1 --- Cubic Spline Curve Fitting --- p.105 / Chapter 7.2 --- Conic Section Fitting --- p.106 / Chapter 7.3 --- Triangle Method --- p.109 / Chapter 8 --- Implementation of the L-curve Method --- p.111 / Chapter 8.1 --- Our Algorithm --- p.111 / Chapter 8.2 --- Numerical Experiments --- p.112 / Chapter 8.3 --- Conclusion --- p.124 / Bibliography --- p.126