Uniqueness of the inverse source problem for fractional diffusion-wave equations

TL;DR

This paper proves the uniqueness of determining either the spatial or temporal part of a source in fractional diffusion-wave equations using asymptotic data, without requiring source vanishing conditions, and highlights the influence of fractional order.

Contribution

It establishes a novel uniqueness result for inverse source problems in fractional diffusion-wave equations based on asymptotic behavior, without vanishing source assumptions.

Findings

Uniqueness of source components depends on asymptotic data.

The approach does not require traditional vanishing conditions.

The fractional order affects the extent of uniqueness.

Abstract

This study addresses the inverse source problem for the fractional diffusion-wave equation, characterized by a source comprising spatial and temporal components. The investigation is primarily concerned with practical scenarios where data is collected subsequent to an incident. We establish the uniqueness of either the spatial or the temporal component of the source, provided that the temporal component exhibits an asymptotic expansion at infinity. Taking anomalous diffusion as a typical example, we gather the asymptotic behavior of one of the following quantities: the concentration on partial interior region or at a point inside the region, or the flux on partial boundary or at a point on the boundary. The proof is based on the asymptotic expansion of the solution to the fractional diffusion-wave equation. Notably, our approach does not rely on the conventional vanishing conditions for the source components. We also observe that the extent of uniqueness is dependent on the fractional order.