In this work, we consider a FDE (fractional diffusion equation) $${}^CD_t^\alpha u(x,t)-a(t)\mathcal{L} u(x,t)=F(x,t)$$ with a time-dependentdiffusion coefficient $a(t)$. For the direct problem, given an $a(t),$ weestablish the existence, uniqueness and some regularity properties with a moregeneral domain $\Omega$ and right-hand side $F(x,t)$. For the inverseproblem--recovering $a(t),$ we introduce an operator $K$ one of whose fixedpoints is $a(t)$ and show its monotonicity, uniqueness and existence of itsfixed points. With these properties, a reconstruction algorithm for $a(t)$ iscreated and some numerical results are provided to illustrate the theories.
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