Studies On Two Kinds Of Inverse Problems For Fractional Diffusion Equations

Fractional diffusion equations have been used to model many of anomalous diffusion phenomena,and have received extensive attention in the fields of mathematics and en-gineering science in recent years.Compared with the integer-order diffusion model,the advantage of the fractional-order diffusion model is that the fractional differential can de-scribe the memory properties of different materials.Many practical applications involve the inverse problem of fractional diffusion equations,which is a hot research topic in the fields of fractional differential equations.In this paper,we considered two kinds of inverse problem of the fractional diffusion equations.The first kind is the inverse problem of time source term.We showed that if the solution of this kind of inverse problem exists,the solution is unique,a rigorous proof was given in the paper.Then,we constructed a numerical scheme for the inverse problem by using the finite element method.In practical applications,the observed data is usually perturbed,resulting in oscillation of the solution.To overcome it,we suggested to adopt Tikhonov regularization technique.Numerical experiments implied the proposed method is efficient.The second kind is the inverse problem with the order of fractional derivative in a fractional diffusion equation.We also proved the uniqueness theorem of the second kind of inverse problem by using Laplace transform and the properties related to Mittag-Leffler function.