  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 $\Omega$ 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.
