Unique Determination of Variable Order in Subdiffusion from a Single Measurement

TL;DR

This paper proves the unique recoverability of spatially varying fractional orders in subdiffusion models from a single boundary measurement, advancing inverse problem theory for anomalous diffusion.

Contribution

It introduces new uniqueness results for piecewise constant variable orders without monotonicity, using innovative analytical techniques and extending to higher dimensions.

Findings

Established uniqueness for variable order recovery from boundary data.

Extended analysis to higher-dimensional settings.

Weakened regularity assumptions on data.

Abstract

We study the inverse problem of recovering a spatially dependent variable order in a time-fractional diffusion model from the boundary flux measurement generated by a single boundary excitation. It arises in the identification of heterogeneous media in anomalous diffusion processes. In this work, we establish several new uniqueness results for the inverse problem in the case of piecewise constant variable orders, without any monotonicity condition. The analysis follows a new approach that combines properties of harmonic functions, a linearization technique in the Laplace domain, and tools from complex, asymptotic, and geometrical analysis. In addition, we weaken the regularity assumptions on the problem data and extend the analysis of previous contributions to higher-dimensional settings.