Recovery of a Distributed Order Fractional Derivative in an Unknown Medium

TL;DR

This paper investigates an inverse problem for distributed-order fractional diffusion, demonstrating that boundary observations can uniquely identify the weight distribution or support in unknown or known media, supported by theoretical proofs and numerical experiments.

Contribution

It proves the unique determination of the weight support from boundary data in unknown media and offers an alternative proof for known media cases, with numerical validation.

Findings

Boundary observation determines the support bound of the weight in unknown media.

One-point boundary data uniquely determines the weight when the medium is known.

Numerical experiments support the theoretical results.

Abstract

In this work, we study an inverse problem of recovering information about the weight in distributed-order time-fractional diffusion from the observation at one single point on the domain boundary. In the absence of an explicit knowledge of the medium, we prove that the one-point observation can uniquely determine the support bound of the weight. The proof is based on asymptotics of the data, analytic continuation and Titchmarch convolution theorem. When the medium is known, we give an alternative proof of an existing result, i.e., the one-point boundary observation uniquely determines the weight. Several numerical experiments are also presented to complement the analysis.