Some Inverse Problems On Fractional Diffusion Equation In One Dimension

In this thesis, two inverse problems for time-fractional diffusion equations are considered. Specifically, one is an inverse source problem in one dimension and the other is an inverse coefficient problem on conditional stability based upon Carleman estimates.First of all, a brief introduction of research background about fractional diffu-sion equations is given in Chapter1, including some definitions, applications and modeling.In Chapter2, some previous works are presented briefly, including Maximum Principle which guarantees the uniqueness and the stability for forward problems in classical sense, some classic uniqueness results on inverse problems, and some basic numerical methods for forward problems.An inverse source problem for a fractional diffusion equation with initial bound-ary condition and source term in one dimension is studied in Chapter3. Firstly Duhamel’s Principle for nonhomogeneous linear fractional equations with zero bound-ary condition is proved to deduce an eigenfunction expansion solution. We conclude that additional data can determine the initial condition, furthermore, the uniqueness of inverse source problems is obtained. Analytic continuation and Laplace transform are employed through our proof. Also numerical examples are presented to illustrate the rationality of the uniqueness result.In Chapter4, Carleman estimate is reviewed briefly. A special Carleman es-timate for a fractional diffusion equation is proved, and we mainly apply the mul-tiplier method into the transformation from fractional order equation to integral order equation. Integration by parts is the major skill when we deal with the inte-gral order equation. In the second part of this Chapter, we investigate a half-order fractional diffusion equation with zeroth-order coefficient. The difficulty also lies on the transformation from fractional order to integral order, and the key to the proof is Carleman estimate. Under some assumptions on regularity of the solutions, coefficients and source term, we obtain the conditional stability of Holder type in determining zeroth-order coefficient in a fractional diffusion equation.