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: ${\partial}_{t}^{\alpha}u(x,t)=\frac{\partial}{\partial x}\left(p\left(x\right)\frac{\partial u}{\partial x}(x,t)\right)$, $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 $p\left(x\right)$, $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 PDE |

35R11 | Fractional partial differential equations |

35D30 | Weak solutions of PDE |

33E12 | Mittag-Leffler functions and generalizations |