Monotonicity in the parameter of the Mittag-Leffler function and determining the fractional exponent of the subdiffusion equation

TL;DR

This paper establishes the strict monotonicity of certain Mittag-Leffler functions with respect to their parameter and applies these results to uniquely determine the fractional order in subdiffusion equations from single-point measurements.

Contribution

It proves the monotonicity of Mittag-Leffler functions in their parameter and uses this to solve inverse problems for identifying the fractional derivative order in subdiffusion equations.

Findings

Proved strict monotonicity of $E_ ho(-t^ ho)$ and $t^{ ho -1}E_{ ho, ho}(-t^ ho)$ in $ ho$

Established unique solvability of the inverse problem for fractional order identification

Extended applicability to a broader class of subdiffusion equations

Abstract

In this paper, we prove the strict monotonicity in the parameter $\rho$ of the Mittag-Leffler functions $E_\rho(-t^\rho)$ and $t^{\rho -1}E_{\rho,\rho}(-t^\rho)$. Then, these results are applied to solve the inverse problem of determining the order of the fractional derivative in subdiffusion equations, where the available measurement is given at one point in space-time. In particular, we find the missing conditions in the previously known work in this area. Moreover, the obtained results are valid for a wider class of subdiffusion equations than those considered previously. An example of an initial boundary value problem constructed by Sh.A. Alimov is given, for which the inverse problem under consideration has a unique solution. We also point out the application of the monotonicity of the Mittag-Leffler functions to solving some other inverse problems of determining the order of a fractional derivative.