Recovering Multiple Fractional Orders in Time-Fractional Diffusion in an Unknown Medium

TL;DR

This paper addresses the inverse problem of uniquely recovering multiple fractional orders and their weights in a time-fractional diffusion model from boundary data, without full knowledge of the medium, using analytical and numerical methods.

Contribution

It proves the unique recovery of fractional orders and weights without requiring detailed medium information and proposes a numerical fitting procedure for practical parameter estimation.

Findings

Successful numerical recovery of fractional orders from boundary data.

Proof of uniqueness for the inverse problem.

Effective numerical method demonstrated with experiments.

Abstract

In this work, we investigate an inverse problem of recovering multiple orders in a time-fractional diffusion model from the data observed at one single point on the boundary. We prove the unique recovery of the orders together with their weights, which does not require a full knowledge of the domain or medium properties, e.g., diffusion and potential coefficients, initial condition and source in the model. The proof is based on Laplace transform and asymptotic expansion. Further, inspired by the analysis, we propose a numerical procedure for recovering these parameters based on a nonlinear least-squares fitting with either fractional polynomials or rational approximations as the model function, and provide numerical experiments to illustrate the approach for small time $t$.