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Space-Time-Dependent Source Identification Problem for a Subdiffusion Equation

R. R. Ashurov, O. T. Mukhiddinova

TL;DR

The paper addresses the inverse problem of identifying a space-time dependent source $h(t,x)$ in a subdiffusion equation with Caputo derivative $D_t^\alpha$, where the source depends on time and a subset of spatial variables. The authors employ a Fourier expansion in the orthogonal variable, derive a coupled system for the Fourier coefficients $u_k(t,x)$, and express $h(t,x)$ via a nonlocal relation involving these coefficients under an overdetermination condition. A successive-approximation scheme is developed to solve the resulting infinite system, with rigorous a priori estimates and convergence arguments establishing the existence and uniqueness of a weak solution, as well as coercive bounds. The results also extend to parabolic limits, providing a solid framework for inverse-source identification in fractional diffusion models. This work advances the mathematical understanding of fractional inverse problems and lays groundwork for higher-dimensional generalizations and potential applications in geophysics and imaging.

Abstract

In this paper, we investigate the inverse problem of determining the right-hand side of a subdiffusion equation with a Caputo time derivative, where the right-hand side depends on both time and certain spatial variables. Similar inverse problems have been previously explored for hyperbolic and parabolic equations, with some studies establishing the existence and uniqueness of generalized solutions, while others proved the uniqueness of classical solutions. However, such inverse problems for fractional-order equations have not been addressed prior to this work. Here, we establish the existence and uniqueness of the weak solution to the considered inverse problem. To solve it, we employ the Fourier method with respect to the variable independent of the unknown right-hand side, followed by the method of successive approximations to compute the Fourier coefficients of the solution. Notably, the results obtained are also novel for parabolic equations.

Space-Time-Dependent Source Identification Problem for a Subdiffusion Equation

TL;DR

The paper addresses the inverse problem of identifying a space-time dependent source in a subdiffusion equation with Caputo derivative , where the source depends on time and a subset of spatial variables. The authors employ a Fourier expansion in the orthogonal variable, derive a coupled system for the Fourier coefficients , and express via a nonlocal relation involving these coefficients under an overdetermination condition. A successive-approximation scheme is developed to solve the resulting infinite system, with rigorous a priori estimates and convergence arguments establishing the existence and uniqueness of a weak solution, as well as coercive bounds. The results also extend to parabolic limits, providing a solid framework for inverse-source identification in fractional diffusion models. This work advances the mathematical understanding of fractional inverse problems and lays groundwork for higher-dimensional generalizations and potential applications in geophysics and imaging.

Abstract

In this paper, we investigate the inverse problem of determining the right-hand side of a subdiffusion equation with a Caputo time derivative, where the right-hand side depends on both time and certain spatial variables. Similar inverse problems have been previously explored for hyperbolic and parabolic equations, with some studies establishing the existence and uniqueness of generalized solutions, while others proved the uniqueness of classical solutions. However, such inverse problems for fractional-order equations have not been addressed prior to this work. Here, we establish the existence and uniqueness of the weak solution to the considered inverse problem. To solve it, we employ the Fourier method with respect to the variable independent of the unknown right-hand side, followed by the method of successive approximations to compute the Fourier coefficients of the solution. Notably, the results obtained are also novel for parabolic equations.
Paper Structure (7 sections, 8 theorems, 120 equations)

This paper contains 7 sections, 8 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition of the Weak Solution and Formulation of the Main Result
  4. A priori estimates
  5. Convergence
  6. Proof of Theorem
  7. Conclusions

Key Result

Lemma 2.1

Let $0<\alpha<1$ and $v(t)$ be absolutely continuous on $[0,T]$: $v\in AC\,[0,T]$. Then

Theorems & Definitions (16)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 3.1
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 6 more