Logarithmic stable recovery of the source and the initial state of time fractional diffusion equations

TL;DR

This paper establishes logarithmic stability estimates for inverse problems in time-fractional diffusion equations, revealing how ill-posedness varies with fractional order and source regularity, using Laplace inversion and unique continuation techniques.

Contribution

It provides the first logarithmic stability estimates for recovering sources and initial states in time-fractional diffusion equations, highlighting the effects of fractional order and regularity.

Findings

Stability deteriorates exponentially as fractional order approaches zero.

Higher regularity of source or initial state improves stability.

Derived a global time regularity result for the fractional diffusion equation.

Abstract

In this paper we study the inverse problem of identifying a source or an initial state in a time-fractional diffusion equation from the knowledge of a single boundary measurement. We derive logarithmic stability estimates for both inversions. These results show that the ill-posedness increases exponentially when the fractional derivative order tends to zero, while it exponentially decreases when the regularity of the source or the initial state becomes larger. The stability estimate concerning the problem of recovering the initial state can be considered as a weak observability inequality in control theory. The analysis is mainly based on Laplace inversion techniques and a precise quantification of the unique continuation property for the resolvent of the time-fractional diffusion operator as a function of the frequency in the complex plane. We also determine a global time regularity for the time-fractional diffusion equation which is of interest itself.