Thesis (doctoral)--Faculté des sciences de Paris, 1913.

Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (p. 165-174).

Thesis (MTech. : Mathematical Technology.)--Tshwane University of Technology, 2008.

This thesis investigates the convergence of numerical schemes for the computation of equilibrium solutions. These are solutions of evolutionary PDEs that arise from (bounded, non-decaying) boundary forcing after the dissipation of any (initial data dependent) transients. A rigorous definition of the term 'equilibrium solution' is given. Classes of evolutionary PDEs for which equilibrium solutions exist uniquely are identified. The uniform well-posedness of equilibrium problems is also investigated. Equilibrium solutions may be approximated by evolutionary initialization: that is, by finding the solution of an initial boundary value problem, with arbitrary initial data, over a period of time t ϵ [0,T]. If T is chosen large enough, the analytic transient will be small, and the analytic solution over t ϵ [T, T + T₀] will be a good approximation to the analytic equilibrium solution. However, in numerical computations, T must be chosen so that the analytical transient is small in comparison with the numerical error E_h, which depends on the fineness of the grid h. Thus T = T_h, and, in general, T_h→∞ as h→0. Convergence is required over t ϵ [T_h,T_h + T₀]. The existing Lax-Richtmyer and GKS convergence theories cannot ensure convergence over such increasing periods of time. Furthermore, neither of these theories apply when the forcing does not decay. Consequently, these theories are of little help in predicting the convergence of finite difference methods for the computation of equilibrium solutions. For these reasons, a new definition of stability - uniform stability — is proposed. Uniformly stable, consistent, finite difference schemes, for uniformly well posed problems, converge uniformly over t ≥ 0. Uniformly convergent schemes converge for bounded and nondecaying forcing. Finite difference schemes for hyperbolic PDEs may admit waves of zero group velocity, even when the underlying analytic problem does not. Such schemes may be GKS convergent, provided that the boundary conditions exclude these waves. The deficiency of the GKS theory for equilibrium computations is traced to this fact. However, uniform stability finds schemes that admit waves of zero group velocity to be (weakly) unstable, regardless of the boundary conditions. It is also shown that weak uniform instabilities are the result of time-dependent analogues of the 'spurious modes' that occur in steady-state calculations. In addition, uniform stability theory sheds new light on the phenomenon of spurious modes.

Thesis (Ph. D.)--Rheinische Friedrich-Wilhelms-Universität zu Bonn, 1994. / Cover title. Includes bibliographical references (p. 166-168).

Wong Tak Kwong. / Thesis submitted in: August 2004. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 67-72). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Basic Notations --- p.7 / Chapter 1.2 --- Riemann Problems --- p.10 / Chapter 1.3 --- Elementary Waves --- p.10 / Chapter 1.3.1 --- Rarefaction Waves --- p.11 / Chapter 1.3.2 --- Shock Waves --- p.11 / Chapter 1.3.3 --- Composite Waves --- p.13 / Chapter 1.4 --- Remarks --- p.14 / Chapter 2 --- Non-strictly Hyperbolic Conservation Laws --- p.16 / Chapter 2.1 --- Systems with Isolated Umbilic Degeneracy --- p.16 / Chapter 2.1.1 --- Mathematical Motivations --- p.17 / Chapter 2.2 --- Complex Burgers' Equation --- p.21 / Chapter 2.2.1 --- Introduction --- p.21 / Chapter 2.2.2 --- Basic Properties --- p.22 / Chapter 2.2.3 --- Riemann Solutions --- p.24 / Chapter 2.2.4 --- Under-Compressive Shocks --- p.31 / Chapter 3 --- Relaxation Approximation --- p.34 / Chapter 3.1 --- Basic Ideas of the Relaxation Approximation --- p.34 / Chapter 3.1.1 --- General Settings --- p.35 / Chapter 3.1.2 --- Subcharacteristic Condition --- p.36 / Chapter 3.2 --- Relaxation of Scalar Conservation Laws --- p.39 / Chapter 3.2.1 --- Perturbation Problems --- p.39 / Chapter 3.3 --- Jin-Xin Relaxation Systems --- p.42 / Chapter 3.3.1 --- Basic Ideas of the Jin-Xin Systems --- p.42 / Chapter 3.4 --- Zero-Relaxation Limit --- p.45 / Chapter 3.4.1 --- 2x2 Hyperbolic Relaxation Systems --- p.45 / Chapter 3.4.2 --- Jin-Xin Relaxation Systems --- p.48 / Chapter 4 --- Jin-Xin Relaxation Limit for the Complex Burgers' Equations --- p.51 / Chapter 4.1 --- Jin-Xin Relaxation Limit for the UCUI Solutions --- p.52 / Chapter 4.1.1 --- Main Statements --- p.52 / Chapter 4.1.2 --- Analysis on UCUI Solution --- p.53 / Chapter 4.1.3 --- Shock Profiles --- p.56 / Chapter 4.1.4 --- Re-scaled Relaxation System --- p.60 / Chapter 4.1.5 --- Proof of Theorem 4.1.1.3 --- p.63 / Bibliography --- p.67

Xiao, Jingjing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 46-50). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Backgrounds and Our Main Results --- p.4 / Chapter 2.1 --- Backgrounds --- p.4 / Chapter 2.1.1 --- The Scalar Case --- p.4 / Chapter 2.1.2 --- 2x2 Systems --- p.5 / Chapter 2.1.3 --- General n x n(n ≥ 3) Systems --- p.9 / Chapter 2.2 --- Our Main Results --- p.18 / Chapter 3 --- Lifespan of Periodic Solutions to Gas Dynamics Systems --- p.21 / Chapter 3.1 --- Riemann Invariant Formulation --- p.21 / Chapter 3.2 --- Calculation along Characteristics --- p.26 / Chapter 3.3 --- Estimate of the Global Wave Interaction --- p.35 / Chapter 3.4 --- Proof of Theorem 2.2.1 --- p.38 / Chapter 4 --- Proof of Theorem 2.2.2 and a Special Case --- p.40 / Chapter 4.1 --- Proof of Theorem 2.2.2 --- p.40 / Chapter 4.2 --- A Special Case --- p.43 / Chapter 5 --- Appendix --- p.45

Thesis (Ph. D.)--University of Wisconsin--Madison, 1981. / Typescript. Vita. Description based on print version record. Includes bibliographical references (leaves 164-166).

Thesis--Wisconsin. / Vita. Includes bibliographical references (leaves 239-241).

A research project submitted in partial fulfilment of the requirements of the degree of Masters in Science Education (Mathematics Education) University of the Witwatersrand Johannesburg South Africa May 2016 / This study investigated learners’ mathematical discourse on the hyperbola using commognitive theory, with particular focus on the use of words, narratives, routines and visual mediators. Data was collected by means of task-based interviews with nine Grade 10 and 11 learners from a township school in Johannesburg, South Africa. An analytical tool, named the Discourse Profile of the Hyperbola, was adapted from the Arithmetic Discourse Profile of Ben-Yahuda et al (2005) and was used to analyse learners’ mathematical discourse. The study focused on three representations of the hyperbola, namely, the formulae (equation); the graph and the table. Learners’ views and definition of the asymptote, in relation to the graph, emerged as a central theme in the analysis. The analysis also focused on the mismatch between what is said and what is done by learners, for example most learners sketched the graph of a hyperbola showing a vertical asymptote yet talked as if there is no vertical asymptote. Most routines were ritualized, for example learners failed to link iconic and symbolic mediators they had used in responding to tasks. However, there were traces of exploratory routines from a few learners, evidenced by links between equations, and identifying the hyperbola from unfamiliar tasks. While a few learners used literate words, colloquial word use was dominant. The discourse of learners was found to be visual. For example, some reasoned that an equation with a fraction represents a hyperbola while an equation not expressed in standard form does not represent a hyperbola. Some learner narratives are not endorsed by the community of mathematicians.