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

Lingyun Qiu, Jiwoon Sim

TL;DR

This work investigates the inverse source problem for the fractional diffusion-wave equation with a separable source $F(x,t)=f(x)\mu(t)$ in a bounded domain, using data taken after an incident and without assuming vanishing of either component. The authors derive an asymptotic expansion for the solution in terms of Mittag-Leffler functions and eigenfunctions, and prove uniqueness results that recover either $f$ or $\mu$ (or the whole source) from late-time observations, with the outcome depending on the fractional order $\alpha$ (irrational $\alpha$ allows full recovery, while rational $\alpha$ yields component-wise uniqueness). The proofs hinge on two key propositions and a detailed analysis of the asymptotic expansion $\psi_n(t)$, enabling translation of asymptotic data into spectral information and constraints on the source components. The results extend previous vanishing-condition–based uniqueness results by exploiting asymptotic data and show significant practical implications for applications such as pollutant source localization, where observations may begin after a delay. Limitations and open directions include handling finite-time data, relaxing regularity assumptions, and exploring the role of $\alpha$ further, as well as extending to more general observation regions and source structures.

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.

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

TL;DR

This work investigates the inverse source problem for the fractional diffusion-wave equation with a separable source in a bounded domain, using data taken after an incident and without assuming vanishing of either component. The authors derive an asymptotic expansion for the solution in terms of Mittag-Leffler functions and eigenfunctions, and prove uniqueness results that recover either or (or the whole source) from late-time observations, with the outcome depending on the fractional order (irrational allows full recovery, while rational yields component-wise uniqueness). The proofs hinge on two key propositions and a detailed analysis of the asymptotic expansion , enabling translation of asymptotic data into spectral information and constraints on the source components. The results extend previous vanishing-condition–based uniqueness results by exploiting asymptotic data and show significant practical implications for applications such as pollutant source localization, where observations may begin after a delay. Limitations and open directions include handling finite-time data, relaxing regularity assumptions, and exploring the role of further, as well as extending to more general observation regions and source structures.

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.
Paper Structure (8 sections, 10 theorems, 81 equations, 1 table)

This paper contains 8 sections, 10 theorems, 81 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and main theorems
  3. Proof of the main theorems
  4. The key propositions
  5. The asymptotic expansion of $\psi_n(t)$
  6. Proof of the key propositions
  7. Proof of the theorems
  8. Conclusions

Key Result

Proposition 2.1

The Mittag-Leffler function $E_{\alpha, \beta}$ satisfies the following properties: (a) $E_{\alpha, \beta}$ is entire in $\mathbb{C}$. (b) $E_{\alpha, \beta}$ admits an asymptotic expansion (p99): Here $\frac{1}{\Gamma(z)} = 0$ for $z = 0, -1, -2, \ldots$. (c) $E_{\alpha, \beta}(-x) \leq C(1 + x)^{-1}$ for any $x \geq 0$ (p99).

Theorems & Definitions (15)

  • Proposition 2.1
  • Theorem 2.2
  • remark 1
  • Theorem 2.3
  • remark 2
  • Theorem 2.4
  • remark 3
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3
  • ...and 5 more