Recovering the Potential and Order in One-Dimensional Time-Fractional Diffusion with Unknown Initial Condition and Source

TL;DR

This paper addresses an inverse problem in one-dimensional time-fractional diffusion, demonstrating unique recovery of potential, fractional order, and some initial/source data from limited boundary measurements, supported by numerical validation.

Contribution

It introduces a novel method for simultaneously recovering potential, fractional order, and some initial/source data without full prior knowledge, using boundary data and analytical techniques.

Findings

Unique recovery of potential and fractional order from boundary data.

Feasibility of numerical reconstruction demonstrated through experiments.

Partial initial/source data can be recovered under certain conditions.

Abstract

This paper is concerned with an inverse problem of recovering a potential term and fractional order in a one-dimensional subdiffusion problem, which involves a Djrbashian-Caputo fractional derivative of order $\alpha\in(0,1)$ in time, from the lateral Cauchy data. In the model, we do not assume a full knowledge of the initial data and the source term, since they might be unavailable in some practical applications. We prove the unique recovery of the spatially-dependent potential coefficient and the order $\alpha$ of the derivation simultaneously from the measured trace data at one end point, when the model is equipped with a boundary excitation with a compact support away from $t=0$. One of the initial data and the source can also be uniquely determined, provided that the other is known. The analysis employs a representation of the solution and the time analyticity of the associated function. Further, we discuss a two-stage procedure, directly inspired by the analysis, for the numerical identification of the order and potential coefficient, and illustrate the feasibility of the recovery with several numerical experiments.