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.

Author's `Characterization of subgradients: 1` (31 leaves) in pocket / Bibliography: leaves 229-231 / 231 leaves : 1 port ; 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, 1986

Thesis (Ph.D.)--Baylor University, 2007. / Includes bibliographical references (p. 64-66).

Thesis (Ph.D.)--McMaster University, 2005. / Advisor: Ian Hambleton. Includes bibliographical references (leaves 74-75). Also available via World Wide Web.

Thesis (M.S.)--University of Akron, Dept. of Mathematics, 2007. / "May, 2007." Title from electronic thesis title page (viewed 4/28/2009) Advisor, Ali Hajjafar; Faculty readers, Linda Marie Saliga, Lala Krishna; Department Chair, Joseph W. Wilder; Dean of the College, Ronald F. Levant; Dean of the Graduate School, George R. Newkome. Includes bibliographical references.

Thesis (M.S. in Electrical Engineering)--Naval Postgraduate School, September 2002. / Thesis advisor(s): Tri Ha, Roberto Cristi. Includes bibliographical references (p. 69). Also available online.

Thesis (M. A.)--Marshall University, 2009. / Title from document title page. Includes abstract. Document formatted into pages: contains 43 p. Includes bibliographical references (p.42-43)

Thesis (Ph. D.)--University of Chicago, Dept. of Physics, March 2003. / Includes bibliographical references. Also available on the Internet.

Neste trabalho tem por objetivo a construção de conjugações suaves de pontos fixos hiperbólicos com condições de não ressonância. Por tanto, inicialmente são apresentados alguns conceitos básicos sobre espaços de Banach e alguns resultados de equações diferenciais ordinárias em espaços de Banach e sistemas dinâmicos, apresentamos o teorema de Hartman Grobman como motivação inicial de Linearização. Apresentamos também vários exemplos como motivação para estudar o Teorema de Sternberg para contrações hiperbólicas, o principal resultado estudado nesta dissertação para contrações hiperbólicas / This work has the objetive of building smooth conjugations of hyperbolic fixed points with non-resonance conditions. So, first we present some basics of Banach spaces and some results of ordinary differential equations in Banach spaces and dynamical systems, we present the theorem of Hartman Grobman as original motivation for linearization . We also present several examples as motivation to study the Sternberg theorem for hyperbolic contractions, as main result studied in this dissertation

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