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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: \(\partial_t^\alpha u(x,t)= \frac{\partial}{\partial x}(p(x)\frac{\partial u}{\partial x}(x,t))\), \(0<x<\ell\), 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:

35R30 Inverse problems for PDEs
35R11 Fractional partial differential equations
35D30 Weak solutions to PDEs
33E12 Mittag-Leffler functions and generalizations
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