Identification problems for anisotropic time-fractional subdiffusion equations

TL;DR

This paper addresses the inverse problem of identifying constant coefficients in anisotropic time-fractional subdiffusion equations, proving uniqueness for fractional orders in (0,1), and analyzing the limit as the order approaches 1, with applications to heat and elasticity.

Contribution

It provides the first uniqueness proof for inverse coefficient problems in anisotropic time-fractional PDEs with fractional order in (0,1), along with existence results and numerical insights.

Findings

Uniqueness of solutions for fractional order in (0,1)

Solution convergence to classical case as fractional order approaches 1

Numerical methods supporting theoretical results

Abstract

We investigate the inverse problem consisting in the identification of constant coefficients for a fractional-in-time partial differential equation governed by a finite sum of positive self-adjoint operators on a Hilbert space under energy-type overdeterminating conditions. We prove the uniqueness of the solution to the inverse problem when the fractional order $\alpha$ of the derivative is in $(0,1)$. A conditioned existence result is also provided, complemented with a suitable selection of numerical calculations. In addition, we prove that, as $\alpha\to 1^{-}$, the solution corresponding to $\alpha$ tends to the classical one ($\alpha=1$). Applications to examples of heat diffusion and elasticity are presented.