Initial-boundary value and inverse problems for subdiffusion equations in $\mathbb{R}^N$

TL;DR

This paper studies initial-boundary value and inverse problems for subdiffusion equations with fractional derivatives, proving existence, uniqueness, and parameter recovery results using Fourier methods and analyzing the influence of parameters on solutions.

Contribution

It introduces new inverse problem results for determining fractional derivative orders and operator powers in subdiffusion equations, extending existing theory.

Findings

Unique recovery of fractional order from Fourier transform at fixed time

Existence and uniqueness of solutions for inverse problems involving fractional derivatives

Parameter  of Mittag-Leffler functions decreases in inverse problem solutions

Abstract

An initial-boundary value problem for a subdiffusion equation with an elliptic operator $A(D)$ in $\mathbb{R}^N$ is considered. The existence and uniqueness theorems for a solution of this problem are proved by the Fourier method. Considering the order of the Caputo time-fractional derivative as an unknown parameter, the corresponding inverse problem of determining this order is studied. It is proved, that the Fourier transform of the solution $\hat{u}(\xi, t)$ at a fixed time instance recovers uniquely the unknown parameter. Further, a similar initial-boundary value problem is investigated in the case when operator $A(D)$ is replaced by its power $A^\sigma$. Finally, the existence and uniqueness theorems for a solution of the inverse problem of determining both the orders of fractional derivatives with respect to time and the degree $ \sigma $ are proved. We also note that when solving the inverse problems, a decrease in the parameter $\rho$ of the Mettag-Leffler functions $E_\rho$ has been proved.