Recovery of a Distributed Order Fractional Derivative in an Unknown Medium

In this work, we study an inverse problem of recovering information about the weight distribution in a distributed-order time-fractional diffusion model from the observation at one single point on the domain boundary. In the absence of an explicit knowledge of the involved problem data, we prove that the one-point observation can uniquely determine the support of the weight distribution. The proof is based on asymptotic behavior of the measurement as well as analytic continuation and Titchmarch convolution theorem. In addition, when the problem data are fully known, we give an alternative proof of an existing result, which states that the over-posed one-point boundary observation can uniquely determine the weight. Several numerical experiments are presented to complement the theoretical analysis.