Inverse problem of determining an order of the Riemann-Liuville time-fractional derivative

TL;DR

This paper addresses the inverse problem of uniquely determining the order of a Riemann-Liouville time-fractional derivative in a subdiffusion equation, using Fourier methods to recover the derivative order from solution norms.

Contribution

It proves the unique recoverability of the fractional order from solution norms for a broad class of equations using classical Fourier analysis.

Findings

Solution norm at fixed time uniquely determines the fractional order.

Applicable to various models including fractional Sturm-Liouville and systems with involution.

Provides a theoretical foundation for inverse fractional order problems.

Abstract

The inverse problem of determining the order of the fractional Riemann- Liouville derivative with respect to time in the subdi_usion equation with an arbitrary positive self-adjoint operator having a discrete spectrum is considered. Using the classical Fourier method it is proved, that the value of the norm jju(t)jj of the solution at a_xed time instance recovers uniquely the order of derivative. A list of examples is discussed, including a linear system of fractional di_erential equations, di_erential models with involution, fractional Sturm-Liouville operators, and many others.