Identification of a Spatially-Dependent Variable Order in One-Dimensional Subdiffusion

TL;DR

This paper addresses the inverse problem of identifying a spatially varying order in a one-dimensional subdiffusion model using boundary flux data, with proofs of unique recovery for certain cases.

Contribution

It introduces a method to uniquely recover a monotone piecewise constant variable order in subdiffusion models, expanding understanding of inverse problems in anomalous diffusion.

Findings

Unique recovery of monotone piecewise constant variable order.

Asymptotic expansion of Laplace transform data as p→0.

Applicable to known and unknown media.

Abstract

In this work we investigate an inverse problem of identifying a spatially variable order in the one-dimensional subdiffusion model from the boundary flux measurement. The model involves a generalized Caputo derivative in time, and arises in the mathematical modeling of anomalous diffusion in heterogeneous media. We prove the unique recovery of a monotone piecewise constant variable order and its range for known and unknown media, respectively. The analysis is based on a delicate asymptotic expansion of the Laplace transform of the data as $p\to0$, which is of independent interest.