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: , , where and denotes the Caputo derivative in time of order . We attach the homogeneous Neumann boundary condition at , and the initial value given by the Dirac delta function. We prove that and , , are uniquely determined by data , . The uniqueness result is a theoretical background in experimentally determining the order 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.
|35R30||Inverse problems for PDE|
|35R11||Fractional partial differential equations|
|35D30||Weak solutions of PDE|
|33E12||Mittag-Leffler functions and generalizations|