Inverse Problems and Imaging (IPI)

An undetermined time-dependent coefficient in a fractional diffusion equation

Pages: 875 - 900, Volume 11, Issue 5, October 2017      doi:10.3934/ipi.2017041

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Zhidong Zhang - Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, 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 time-dependent diffusion coefficient $a(t)$. This is an extension of [13], which deals with this FDE in one-dimensional 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 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.

Keywords:  Fractional diffusion, fractional inverse problem, uniqueness, existence, monotonicity, iteration algorithm.
Mathematics Subject Classification:  35R11, 35R30, 65M32.

Received: July 2016;      Revised: October 2016;      Available Online: July 2017.