Inverse Problems for Fractional Diffusion Equations
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By Fick’s laws of diffusion, in the classical diffusion process, the mean square path ‹x2› is proportional to the time t as t →∞,. However, in practice, some anomalous diffusion processes may occur, in which the relation ‹x2› t, ≠ 1 holds. To describe such processes, we need to add the fractional derivative on the time t, which forms the fractional diffusion equation, and we call it FDE for short. This dissertation contains some inverse problems in FDEs. Specifically, the recovery of unknown conditions of coefficients from additional data on the solution u will be considered. The results of fractional inverse problems are totally different from the ones of the classical case. For instance, the degree of ill-posedness. This is due to the polynomial asymptotic behavior of the Mittag-Leffler function, which consists of the fundamental solution of FDE. This difference leads to new physics and we can ask a question that do similar things always occur? The short answer is not always and the slightly longer version is the analysis is always more complex. This makes the research on inverse problems in FDEs both challenging and interesting. For each inverse problem in this dissertation, at first it was necessary to extend existing results about the direct problem, namely the situation where all parameters in the equation are known and we must recover u(x, t). This includes the existence, uniqueness and regularity estimates of the solution. Then for the inverse problem, the initial step in many of these situations is to use the equation structure to obtain an operator K one of whose fixed points is the unknown function we seek. With this K; the key step is proving the monotonicity of the operator in a suitable partially ordered space and then showing uniqueness of its fixed points. In conclusion, the monotonicity property and the domain of the operator K will lead to an iterative reconstruction algorithm and some numerical results are reproduced to verify the theoretical conclusions.
Zhang, Zhidong (2017). Inverse Problems for Fractional Diffusion Equations. Doctoral dissertation, Texas A & M University. Available electronically from