Monotonicity in the parameter of the Mittag-Leffler function and determining the fractional exponent of the subdiffusion equation
Ravshan Ashurov, Ilyoskhuja Sulaymonov
TL;DR
The paper addresses identifying the fractional order $\rho$ in subdiffusion models from data at a single point by proving strict monotonicity of Mittag-Leffler functions in $\rho$. It shows that $E_{\rho}(-t^{\rho})$ is increasing in $\rho$ while $t^{\rho-1}E_{\rho,\rho}(-t^{\rho})$ is decreasing, enabling a monotone mapping for the inverse problem and yielding unique recovery under mild data conditions; forward problem solutions are given by Fourier eigenfunction expansions. The authors extend previous results by removing dimensional/coefficients restrictions, provide a concrete Alimov example with guaranteed uniqueness, and apply the monotonicity approach to other inverse problems, including a new proof related to Pskhu's classical result. Overall, the work enhances the robustness of fractional-order identification in subdiffusion and broadens applicability to a wider class of fractional PDE inverse problems.
Abstract
In this paper, we prove the strict monotonicity in the parameter $ρ$ of the Mittag-Leffler functions $E_ρ(-t^ρ)$ and $t^{ρ-1}E_{ρ,ρ}(-t^ρ)$. 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.