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Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. (English) Zbl 1181.35322
Summary: We consider a one-dimensional fractional diffusion equation: ${\partial }_{t}^{\alpha }u\left(x,t\right)=\frac{\partial }{\partial x}\left(p\left(x\right)\frac{\partial u}{\partial x}\left(x,t\right)\right)$, $0, where $0<\alpha <1$ and ${\partial }_{t}^{\alpha }$ denotes the Caputo derivative in time of order $\alpha$. We attach the homogeneous Neumann boundary condition at $x=0$, $\ell$ and the initial value given by the Dirac delta function. We prove that $\alpha$ and $p\left(x\right)$, $0, are uniquely determined by data $u\left(0,t\right)$, $0. The uniqueness result is a theoretical background in experimentally determining the order $\alpha$ of many anomalous diffusion phenomena which are important, for example, in environmental engineering. The proof is based on the eigenfunction expansion of the weak solution to the initial value/boundary value problem and the Gel’fand-Levitan theory.
##### MSC:
 35R30 Inverse problems for PDE 35R11 Fractional partial differential equations 35D30 Weak solutions of PDE 33E12 Mittag-Leffler functions and generalizations