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Zbl 1181.35322
Cheng, Jin; Nakagawa, Junichi; Yamamoto, Masahiro; Yamazaki, Tomohiro
Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation.
(English)
[J] Inverse Probl. 25, No. 11, Article ID 115002, 16 p. (2009). ISSN 0266-5611

Summary: We consider a one-dimensional fractional diffusion equation: $\partial_t^\alpha u(x,t)= \frac{\partial}{\partial x}(p(x)\frac{\partial u}{\partial x}(x,t))$, $0<x<\ell$, 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 $p(x)$, $0<x<\ell$, are uniquely determined by data $u(0,t)$, $0<t<T$. 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 2000:
*35R30 Inverse problems for PDE
35R11
35D30
33E12 Mittag-Leffler functions and generalizations

Keywords: fractional diffusion equation; Caputo derivative; Neumann boundary condition; Gel'fand-Levitan theory; weak solution

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