Uniqueness in determination of the fractional order in the TFDE using one measurement

TL;DR

This paper proves that the fractional order in a 1D time fractional diffusion equation can be uniquely determined from a single space-time measurement, using properties of the Mittag-Leffler function.

Contribution

It establishes the unique solvability of the inverse problem for identifying the fractional order from one measurement, leveraging the monotonicity of the Mittag-Leffler function.

Findings

Unique determination of fractional order from one measurement

Use of Mittag-Leffler function monotonicity for solution

Theoretical proof of inverse problem solvability

Abstract

This article deals with an inverse problem of identifying the fractional order in the 1D time fractional diffusion equation (TFDE in short) using the measurement at one space-time point. Based on the expression of the solution to the forward problem, the inverse problem is transformed to a nonlinear algebraic equation. By choosing suitable initial values and the measured point, the nonlinear equation has a unique solution by the monotonicity of the Mittag-Lellfer function. Theoretical testifications are presented to demonstrate the unique solvability of the inverse problem.