# Inverse moving source problem for fractional diffusion(-wave) equations: Determination of orbits

## TL;DR

This paper addresses the inverse problem of determining the orbit of a moving source in fractional diffusion(-wave) equations, establishing stability and uniqueness results using interior observations and a new fractional Duhamel's principle.

## Contribution

It introduces a fractional Duhamel's principle and provides stability and uniqueness results for the inverse orbit determination problem in fractional diffusion(-wave) equations.

## Key findings

- Lipschitz stability estimate for localized sources
- Uniqueness of orbit determination with additional interior data
- Development of a fractional Duhamel's principle

## Abstract

This paper is concerned with the inverse problem on determining an orbit of the moving source in a fractional diffusion(-wave) equations in a connected bounded domain of $\mathbb R^d$ or in the whole space $\mathbb R^d$. Based on a newly established fractional Duhamel's principle, we derive a Lipschitz stability estimate in the case of a localized moving source by the observation data at $d$ interior points. The uniqueness for the general non-localized moving source is verified with additional data of more interior observations.