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Inverse problem to determine simultaneously several scalar parameters and a time-dependent source term in a superdiffusion equation involving a multiterm fractional Laplacian

Hany Gerges, Jaan Janno

TL;DR

We study the inverse problem of simultaneously identifying a scalar coefficient $a$, the time-fractional order $\alpha$, the multiterm Laplacian parameters $(b_j,\beta_j)$, and a time-dependent source term from time-trace data of solutions to a superdiffusion equation. The forward model employs a multiterm spectral Laplacian $L=\sum_{j=1}^m b_j(-\Delta)^{\beta_j}$ with $\alpha\in(1,2)$ and a source $\chi=g(t)f(x)+z(t,x)$; observations are given by $\Phi u(t,\cdot)=h(t)$ on $(0,T+\delta)$. A novel pole-asymptotics approach analyzes the Laplace transform $H(s)$ of the observed functional to establish uniqueness, by relating poles to the spectral data via $\mu_k=\tfrac{1}{a}\sum_{j=1}^m b_j\lambda_k^{\beta_j}$ and the Mittag-Leffler structure of the solution. The main contributions include a rigorous uniqueness theorem for the inverse problem under precise regularity and spectral assumptions, and a detailed pole-based framework that accommodates nonlocal spatial operators and a time-dependent source, supported by eigenvalue asymptotics and an established direct-problem theory. The results enhance identifiability in fractional-elliptic-diffusion models and provide a principled route to reconstruct nonlocal operators from time-trace measurements.

Abstract

Inverse problem to recover simultaneously a scalar coefficient, order of a time-fractional derivative, parameters of multiterm fractional Laplacian and a time-dependent source term occurring in a superdiffusion equation from measurements over the time is considered. Uniqueness of a solution is proved. The proof uses asymptotics of poles of the Laplace transform of a measured function.

Inverse problem to determine simultaneously several scalar parameters and a time-dependent source term in a superdiffusion equation involving a multiterm fractional Laplacian

TL;DR

We study the inverse problem of simultaneously identifying a scalar coefficient , the time-fractional order , the multiterm Laplacian parameters , and a time-dependent source term from time-trace data of solutions to a superdiffusion equation. The forward model employs a multiterm spectral Laplacian with and a source ; observations are given by on . A novel pole-asymptotics approach analyzes the Laplace transform of the observed functional to establish uniqueness, by relating poles to the spectral data via and the Mittag-Leffler structure of the solution. The main contributions include a rigorous uniqueness theorem for the inverse problem under precise regularity and spectral assumptions, and a detailed pole-based framework that accommodates nonlocal spatial operators and a time-dependent source, supported by eigenvalue asymptotics and an established direct-problem theory. The results enhance identifiability in fractional-elliptic-diffusion models and provide a principled route to reconstruct nonlocal operators from time-trace measurements.

Abstract

Inverse problem to recover simultaneously a scalar coefficient, order of a time-fractional derivative, parameters of multiterm fractional Laplacian and a time-dependent source term occurring in a superdiffusion equation from measurements over the time is considered. Uniqueness of a solution is proved. The proof uses asymptotics of poles of the Laplace transform of a measured function.
Paper Structure (9 sections, 8 theorems, 84 equations)

This paper contains 9 sections, 8 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Preliminaries
  4. Multiterm fractional powers of minus Laplacian
  5. Abstract functional spaces
  6. Direct problem
  7. Laplace transform of a functional of $u$ and its poles
  8. Uniqueness for inverse problem
  9. Appendix

Key Result

Lemma 1

Let $X$ be a complex Hilbert space. Let $\gamma \in (0,1)$, $p\in (1,\infty)$. The operator $I_t^{\gamma}$ is a bijection from $L_p((0,T);X)$ onto $_{0}H_p^\gamma((0,T);X)$, the inverse of $I_t^\gamma$ is the Riemann-Liouville fractional derivative $D_t^\gamma$ and is a norm in the space $_{0}H_p^\gamma((0,T);X)$. Moreover, in case $p\in ({1\over \gamma},\infty)$ it holds $_0H_p^\gamma((0,T);X)\h

Theorems & Definitions (13)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Lemma 3
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Remark 1
  • Remark 2
  • ...and 3 more