Title from PDF of title page (University of Missouri--Columbia, viewed on Feb 26, 2010). The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Dissertation advisor: Dr. Peter Pfeifer. Vita. Includes bibliographical references.

For a sign-changing function a(x) E C^αloc(Rn) with bounded Ω+ = {x E R^n |a(x) > O}, we study non-negative entire solutions u(x) ≥ 0 of the semilinear elliptic equation -Δu = a(x)u^q + b(x)u^p in R^n with n ≥ 3.0 < q < 1, p > q, and λ > 0. We consider two types of coefficient b(x) E C^αloc(R^n), either b(x) ≤ 0 in (R^n) or b(x) ≡ 1. In each case, we give sufficient conditions on a(x) for which all solutions must have compact support. In case Ω+ has several connected components, we also give conditions under which there exist "dead core'' solutions which vanish identically in one or more of these components. In the "logistic" case b(x) ≤ 0, we prove that there can be only one solution with given dead core components. In the case b(x) ≡ 1, the question of existence is more delicate, and we introduce a parametrized family of equations by replacing a(x) by ay = ya^+(x) - a^- (x). We show that there exists a maximal interval y E (0, f] for which there exists a stable (locally minimizing) solution. Under some hypotheses on a^- near infinity, we prove that there are two solutions for each y E (0, f). Some care must be taken to ensure the compactness of Palais-Smale sequences, and we present an example which illustrates how the Palais-Smale condition could fail for certain a(x). The analysis is based on a combination of comparison arguments, a priori estimates, and variational methods. / Thesis / Doctor of Philosophy (PhD)

Predation and harvesting play critical roles in maintaining biodiversity in ecological communities. Too much harvesting may drive a species to extinction, while too little harvesting may allow a population to drive out competing species. The spatial features of a habitat can also significantly affect population dynamics within these communities. Here, we formulate and analyze three ordinary differential equation models for the population density of a single species. Each model differs in its assumptions about how the species is harvested. We then extend each of these models to analogous partial differential equation models that more explicitly describe the spatial habitat and the movement of individuals using reaction-diffusion equations. We study the existence and stability of non-zero equilibria of these models in terms of each model's parameters. Biological interpretations for these results are discussed.

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Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2020 / In this study we present the numerical solution o fboundary value problems for the reaction-diffusion equations in 1-d and 2-d that model phenomena such as kinetics and population dynamics.These differential equations are solved nu- merically using the finite element method (FEM).The FEM was chosen due to several desirable properties it possesses and the many advantages it has over other numerical methods.Some of its advantages include its ability to handle complex geometries very well and that it is built on well established Mathemat- ical theory,and that this method solves a wider class of problems than most numerical methods.The Lax-Milgram lemma will be used to prove the existence and uniqueness of the finite element solutions.These solutions are compared with the exact solutions,whenever they exist,in order to examine the accuracy of this method.The adaptive finite element method will be used as a tool for validating the accuracy of theFEM.The convergence of the FEM will be proven only on the real line.

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (leaves 93-99).

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 50-51). Also available in electronic version. Access restricted to campus users.

In recent decades, significant interest, based on physics and engineering applications, has developed on so-called anomalous diffusion processes that possess different spread functions with classical ones. The resulting differential equation whose fundamental solution matches this decay process is best modeled by an equation containing a fractional order derivative. This dissertation mainly focuses on some inverse problems for fractional diffusion equations. After some background introductions and preliminaries in Section 1 and 2, in the third section we consider our first inverse boundary problem. This is where an unknown boundary condition is to be determined from overposed data in a time- fractional diffusion equation. Based upon the fundamental solution in free space, we derive a representation for the unknown parameters as the solution of a nonlinear Volterra integral equation of second kind with a weakly singular kernel. We are able to make physically reasonable assumptions on our constraining functions (initial and given boundary values) to be able to prove a uniqueness and reconstruction result. This is achieved by an iterative process and is an immediate result of applying a certain fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. In the fourth section a reaction-diffusion problem with an unknown nonlinear source function, which has to be determined from overposed data, is considered. A uniqueness result is proved and a numerical algorithm including convergence analysis under some physically reasonable assumptions is presented in the one-dimensional case. To show effectiveness of the proposed method, some results of numerical simulations are presented. In Section 5, we also attempted to reconstruct a nonlinear source in a heat equation from a number of known input sources. This represents a new research even for the case of classical diffusion and would be the first step in a solution method for the fractional diffusion case. While analytic work is still in progress on this problem, Newton and Quasi-Newton method are applied to show the feasibility of numerical reconstructions. In conclusion, the fractional diffusion equations have some different properties with the classical ones but there are some similarities between them. The classical tools like integral equations and fixed point theory still hold under slightly different assumptions. Inverse problems for fractional diffusion equations have applications in many engineering and physics areas such as material design, porous media. They are trickier than classical ones but there are also some advantages due to the mildly ill-conditioned singularity caused by the new kernel functions.

Thesis (M. Sc.)--University of Stellenbosch, 2008. / Includes bibliographical references. Also available via the Internet.

Thesis (Ph. D.)--Ohio State University, 2006. / Available online via OhioLINK's ETD Center; full text release delayed at author's request until 2007 Mar 21