Inverse problems for a generalized fractional diffusion equation with unknown history

TL;DR

This paper investigates inverse problems for a generalized fractional diffusion equation, demonstrating conditions under which the kernel and history of the source function can be uniquely recovered from partial data.

Contribution

It establishes new uniqueness results for recovering the kernel and source history in fractional diffusion equations with unknown history, under various restrictions.

Findings

Unique recovery of kernel and source history under certain conditions.

Uniqueness of kernel with less restrictions on source function.

Recovery of kernel from functional data of the solution.

Abstract

Inverse problems for a diffusion equation containing a generalized fractional derivative are studied. The equation holds in a time interval $(0,T)$ and it is assumed that a state $u$ (solution of diffusion equation) and a source $f$ are known for $t\in (t_0,T)$ where $t_0$ is some number in $(0,T)$. Provided that $f$ satisfies certain restrictions, it is proved that product of a kernel of the derivative with an elliptic operator as well as the history of $f$ for $t\in (0,t_0)$ are uniquely recovered. In case of less restrictions on $f$ the uniqueness of the kernel and the history of $f$ is shown. Moreover, in a case when a functional of $u$ for $t\in (t_0,T)$ is given the uniqueness of the kernel is proved under unknown history of $f$.