Inverse Problems of Fractional Diffusion Equations

This chapter deals with the inverse problems of time fractional diffusion equationsfractional diffusion equation(s) of order \(\alpha \in (0,1)\) . In Sect. 3.1, we study a backward problembackward problem for an inhomogeneous fractional diffusion equationfractional diffusion equation(s) in a bounded domain. By applying the properties of the Mittag-Leffler functionsMittag-Leffler function(s) and the method of eigenvalue expansion, we establish some results about the existenceexistence, uniquenessuniqueness, and regularityregularity of the mild solutionsmild solution(s) and the classical solutionsclassical solution of the proposed problem in a weighted Hölder continuous function space. In Sect. 3.2, we consider a final value problemfinal value problem(s) for a diffusion equation with time-space fractional differentiation on a bounded domain D of \( \mathbb {R}^{k}\) , \(k\ge 1\) , which includes the fractional power \(\mathscr {L}^\beta \) , \(0<\beta \leq 1\) , of a symmetric uniformly elliptic operator \(\mathscr {L}\) defined on \(L^2(D)\) . A representation of solutions is given by using the Laplace transformLaplace transform and the spectrum of \(\mathscr {L}^\beta \) . We present some existenceexistence and regularityregularity results for our problem in both the linear and nonlinear cases. The materials in Sect. 3.1 are adopted from Zhou, He, Ahmad, and Tuan [245]. The contents in Sect. 3.2 are due to Tuan, Ngoc, Zhou, and O’Regan [210].