Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references.

In this dissertation, the quantitative and qualitative aspects of modeling shallow water flow driven mainly by gravitational forces and dominated by shear stress, using an effective equation often referred to in the literature as the diffusive wave approximation of the shallow water equations (DSW) are presented. These flow conditions arise for example in overland flow and water flow in vegetated areas such as wetlands. The DSWequation arises in shallow water flow models when special assumptions are used to simplify the shallow water equations and contains as particular cases: the Porous Medium equation and the time evolution of the p-Laplacian. It has been successfully applied as a suitable model to simulate overland flow and water flow in vegetated areas such as wetlands; yet, no formal mathematical analysis has been carried out addressing, for example, conditions for which weak solutions may exist, and conditions for which a numerical scheme can be successful in approximating them. This thesis represents a first step in that direction. The outline of the thesis is as follows. First, a survey of relevant results coming from the studies of doubly nonlinear diffusion equations that can be applied to the DSWequation when topographic effects are ignored, is presented. Furthermore, an original proof of existence of weak solutions using constructive techniques that directly lead to the implementation of numerical algorithms to obtain approximate solutions is shown. Some regularity results about weak solutions are presented as well. Second, a numerical approach is proposed as a means to understand some properties of solutions to the DSW equation, when topographic effects are considered, and conditions for which the continuous and discontinuous Galerkin methods will succeed in approximating these weak solutions are established. / text

In this dissertation we prove the homogenization for two very different classes of nonlinear partial differential equations and nonlinear elliptic integro-differential equations. The first result covers the homogenization of convex and superlinear Hamilton-Jacobi equations with stationary ergodic dependence in time and space simultaneously. This corresponds to equations of the form: [mathematical equation]. The second class of equations is nonlinear integro-differential equations with periodic coefficients in space. These equations take the form, [mathematical equation]. / text

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We carry out a preliminary group classification of the following family of non-linear wave equations u_tt =f(u_x)u_xx+g(u_x)+x. We first re-obtain the principal Lie algebra obtained by Ibragimov et al[3) and then construct the equivalence Lie algebra. In order to partially classify this family of wave equations, optimal systems of one-dimensional sub-algebras of the equivalence Lie algebra are constructed and in so doing, two distinct equations are obtained. We furthermore determine some invariant solutions of these equations. / Thesis (MSc. Mathematics) North-West University, Mafikeng Campus, 2003

Thesis (M.App.Sc.) - Carleton University, 2003. / Includes bibliographical references (p. 131-133). Also available in electronic format on the Internet.

Thesis (M. App. Sc.)--Carleton University, 2003. / Includes bibliographical references (p. 118-124). Also available in electronic format on the Internet.

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references (leaves 182-195). Also available in an electronic version.