An undetermined coefficient problem for a fractional diffusion equation

We consider a fractional diffusion equation (FDE) 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 Secondly, for the inverse problem part, in order to recover 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.