Communications in Mathematical Sciences

Volume 21 (2023)

Number 7

Recovery of a distributed order fractional derivative in an unknown medium

Pages: 1791 – 1813



Bangti Jin (Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong)

Yavar Kian (Université Rouen Normandie, CNRS, Laboratoire de Mathématiques Raphaël Salem, UMR 6085, France)


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.


distributed order, time-fractional diffusion, weight recovery, ultra-slow diffusion, reconstruction

2010 Mathematics Subject Classification

35R11, 35R30, 65M32

The work of Bangti Jin is supported by UK EPSRC grant EP/T000864/1 and EP/V026259/1, and a start-up fund of The Chinese University of Hong Kong.

The work of Yavar Kian was supported by the French National Research Agency ANR (project Multi-Onde) grant ANR-17-CE40-0029.

Received 4 August 2022

Received revised 3 November 2022

Accepted 5 January 2023

Published 9 October 2023