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.

Bibliography: leaves 264-267.

Bibliography: leaves 264-267 / 267 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, 1988

This paper is concerned with finite difference methods for approximating solutions to the partial differential heat equation. The first chapel gives some introductory background into the physical problem, then motivates three finite difference methods. Chapters II through IV provide statements and proofs for the theorems used in the methods of Chapter I. The final Chapter, V, provides conclusions and an indication of future work. An appendix includes the computer codes written by the author with numerical results.

Lau Kin Wing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 80-83). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.3 / Chapter 1.1 --- Heat conduction problem --- p.3 / Chapter 1.2 --- Direct problem --- p.4 / Chapter 1.3 --- Inverse problem --- p.4 / Chapter 1.4 --- Difficulty of the inverse problems --- p.5 / Chapter 1.5 --- A simple but important example for instability --- p.5 / Chapter 1.6 --- The purpose of this thesis --- p.7 / Chapter 2 --- Stability of the inverse problem --- p.9 / Chapter 2.1 --- Conditional stability results --- p.9 / Chapter 2.2 --- Stability of the inverse problems --- p.11 / Chapter 3 --- The continuous formulation --- p.30 / Chapter 3.1 --- Constrained minimization problem --- p.30 / Chapter 3.2 --- Existence of minimizers to the minimization problem --- p.31 / Chapter 4 --- Discretization and its convergence --- p.36 / Chapter 4.1 --- Finite element space --- p.36 / Chapter 4.2 --- Two important discrete projection operators --- p.37 / Chapter 4.3 --- Finite element problem --- p.39 / Chapter 4.4 --- Existence of minimizers to the finite element problem --- p.39 / Chapter 4.5 --- Discrete minimizers and global minimizers --- p.42 / Chapter 5 --- Numerical algorithms --- p.51 / Chapter 5.1 --- Gateaux derivative --- p.51 / Chapter 5.2 --- Nonlinear single-grid gradient method --- p.53 / Chapter 5.3 --- Nonlinear multigrid gradient method --- p.55 / Chapter 6 --- Numerical experiments --- p.60 / Chapter 6.1 --- One dimensional examples --- p.60 / Chapter 6.2 --- Two dimensional examples --- p.66

Xie Jian Li. / "August 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 106-109). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.