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.

In this thesis we study the thin-film type equations in one spatial dimension. These equations arise from the lubrication approximation to the thin films of viscous fluids which is described by the Navier-Stokes equations with free boundary. From the structural point of view, they are fourth-order degenerate nonlinear parabolic equations, with principal term from diffusion and lower order term from external forces. In Chapter one we study the dynamics of the equations when the external force is given by a power law. Classification of steady states of this equation, which is important for the dynamics, was already known. Previous numerical studies show that there is a mountain pass scenario among the steady states. We shall provide a rigorous justification to these numerical results. As a result, a rather complete picture of the dynamics of the thin film is obtained when the power law is in the range (1,3). In Chapter two we turn to the special case of the equation where the external force is the gravity. This is important, but, unfortunately not a power law. We study and classify the steady states of this equation as well as compare their energy levels. Some numerical results are also present. / Zhang, Zhenyu. / Asviser: Kai Seng Chou. / Source: Dissertation Abstracts International, Volume: 73-06, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 103-107). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

by Leung Tin Chi. / Bibliography: leaf 32 / Thesis (M.Phil.)--Chinese University of Hong Kong, 1982

Consider a family of cubic parabolic polynomials given by for non-zero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their Julia sets. We have proved that the parameter set is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff dimension function defined by is real analytic. To prove this we have constructed a holomorphic family of holomorphic parabolic graph directed Markov systems whose limit sets coincide with the Julia sets of polynomials up to a countable set, and hence have the same Hausdorff dimension. Then we associate to this holomorphic family of holomorphic parabolic graph directed Markov systems an analytic family, call it , of conformal graph directed Markov systems with infinite number of edges in order to reduce the problem of real analyticity of Hausdorff dimension for the given family of polynomials to prove the corresponding statement for the family .

We deal with a free boundary problem for a nonlinear parabolic equation, which includes a parameter in the free boundary condition. This type of system has been used in models of ecological systems, in chemical reactor theory and other kinds of propagation phenomena involving reactions and diffusion. The main purpose of this dissertation is to show the global existence, uniqueness of solutions and that a Hopf bifurcation occurs at a critical value of the parameter r. The existence and uniqueness of the solution for this problem are shown by finding an equivalent regular free boundary problem to which existence results can be applied. We then show that as the bifurcation parameter r decreases and passes through a critical value rc, the stationary solution loses stability and a stable periodic solution appears. Several figures have been included, which illustrate this transistion. The pascal source program used in the numerical simulation is included in an appendix.

In this paper, we discuss the local and global existence of week solutions for some hyperbolic-parabolic systems modelling chemotaxis.

This dissertation addresses the accuracy of a-posteriori error estimators for finite element solutions of problems with high orthotropy especially for cases where rather coarse meshes are used, which are often encountered in engineering computations. We present sample computations which indicate lack of robustness of all standard residual estimators with respect to high orthotropy. The investigation shows that the main culprit behind the lack of robustness of residual estimators is the coarseness of the finite element meshes relative to the thickness of the boundary and interface layers in the solution. With the introduction of an elliptic reconstruction procedure, a new error estimator based on the solution of the elliptic reconstruction problem is invented to estimate the exact error measured in space-time C-norm for both semi-discrete and fully discrete finite element solutions to linear parabolic problem. For a fully discrete solution, a temporal error estimator is also introduced to evaluate the discretization error in the temporal field. In the meantime, the implicit Neumann subdomain residual estimator for elliptic equations, which involves the solution of the local residual problem, is combined with the elliptic reconstruction procedure to carry out a posteriori error estimation for the linear parabolic problem. Numerical examples are presented to illustrate the superconvergence properties in the elliptic reconstruction and the performance of the bounds based on the space-time C-norm. The results show that in the case of L^2 norm for smooth solution there is no superconvergence in elliptic reconstruction for linear element, and for singular solution the superconvergence does not exist for element of any order while in the case of energy norm the superconvergence always exists in elliptic reconstruction. The research also shows that the performance of the bounds based on space-time C-norm is robust, and in the case of fully discrete finite element solution the bounds for the temporal error are sharp.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves [71]-74). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves [71]-74). Also available on the Internet.

Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2010. / Committee Chair: Gangbo, Wilfrid; Committee Member: Chow, Shui-Nee; Committee Member: Harrell, Evans; Committee Member: Swiech, Andrzej; Committee Member: Yezzi, Anthony Joseph. Part of the SMARTech Electronic Thesis and Dissertation Collection.