An undetermined timedependent coefficient in a fractional diffusion equation
Zhidong Zhang  Department of Mathematics, Texas A&M University, College Station, TX 778433368, United States (email) Abstract: In this work, we consider a FDE (fractional diffusion equation) $$C_{D^\alpha_t} u(x,t)a(t)\mathcal{L}u(x,t)=F(x,t)$$ with a timedependent diffusion coefficient $a(t)$. This is an extension of [13], which deals with this FDE in onedimensional space. For the direct problem, given an $a(t),$ we establish the existence, uniqueness and some regularity properties with a more general domain $\Omega$ and righthand side $F(x,t)$. For the inverse problemrecovering $a(t),$ 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.
Keywords: Fractional diffusion, fractional inverse problem, uniqueness, existence, monotonicity, iteration algorithm.
Received: July 2016; Revised: October 2016; Available Online: July 2017. 
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