An undetermined coefficient problem for a fractional diffusion equation

We consider a fractional diffusion equation (FDE) = a D u a t u C t xx ( ) with an undetermined time-dependent diffusion coefficient a(t). Firstly, for the direct problem part, we establish the existence, uniqueness and some regularity properties of the weak solution for this FDE with a fixed a t . ( ) Secondly, for the inverse problem part, in order to recover a t , ( ) we introduce an operator and show its monotonicity. With this property, we establish the uniqueness of a(t) and create an efficient reconstruction algorithm to recover this coefficient.