An undetermined time-dependent coefficient in a fractional diffusion equation

In this work, we consider a FDE (fractional diffusion equation) $${}^C{D}_t^{\alpha }u(x,t)-a(t)\mathcal{L}u(x,t)=F(x,t)$$ with a time-dependent diffusion coefficient $a(t)$. For the direct problem, given an $a(t),$ we establish the existence, uniqueness and some regularity properties with a more general domain $\mathrm{\Omega }$ and right-hand side $F(x,t)$. For the inverse problem–recovering $a(t),$ we introduce an operator $K$ one of whose fixed points is $a(t)$ and show its monotonicity, uniqueness and existence of its fixed points. With these properties, a reconstruction algorithm for $a(t)$ is created and some numerical results are provided to illustrate the theories.