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Space- and Time-Dependent Source Identification Problem with Integral Overdetermination Condition

R. R. Ashurov, O. T. Mukhiddinova

TL;DR

This work tackles an inverse problem for a fractional subdiffusion equation with Caputo time derivative $D_t^{\alpha}$, aiming to identify a space–time dependent source coefficient via an integral overdetermination condition $\int_\Omega u(t,x,y)\omega(y)\,dy=\psi(t,x)$. The authors employ a spectral (Fourier) expansion in the transverse variable, leading to coupled equations for the Fourier coefficients $u_k(t,x)$ and an explicit formula for the unknown coefficient $h(t,x)$ in terms of the data and the solution. Existence and uniqueness of a weak solution are established using a Galerkin-type approach in $y$ and a successive-approximation scheme for the coefficient system, accompanied by detailed a priori estimates. The results are presented as new for fractional (subdiffusion) equations and extendable to parabolic equations, providing a rigorous foundation and quantitative bounds for the inverse problem under integral overdetermination.

Abstract

This paper is devoted to the study of the inverse problem of determining the right-hand side of the subdiffusion equation with the Caputo derivative with respect to time. In our case, the inverse problem consists in restoring the coefficient of the right-hand side, which depends on both the time and the spatial variable, when measured in integral form. Previously, similar inverse problems were studied for hyperbolic and parabolic equations with a different overdetermination condition, and in some works the existence and uniqueness of generalized solutions was established, while in others, the uniqueness of classical solutions was established. However, similar inverse problems for fractional equations with an integral overdetermination condition have not been considered before this work. The existence and uniqueness of a weak solution to the inverse problem under consideration is established. It is noteworthy that the results obtained are new for parabolic equations as well.

Space- and Time-Dependent Source Identification Problem with Integral Overdetermination Condition

TL;DR

This work tackles an inverse problem for a fractional subdiffusion equation with Caputo time derivative , aiming to identify a space–time dependent source coefficient via an integral overdetermination condition . The authors employ a spectral (Fourier) expansion in the transverse variable, leading to coupled equations for the Fourier coefficients and an explicit formula for the unknown coefficient in terms of the data and the solution. Existence and uniqueness of a weak solution are established using a Galerkin-type approach in and a successive-approximation scheme for the coefficient system, accompanied by detailed a priori estimates. The results are presented as new for fractional (subdiffusion) equations and extendable to parabolic equations, providing a rigorous foundation and quantitative bounds for the inverse problem under integral overdetermination.

Abstract

This paper is devoted to the study of the inverse problem of determining the right-hand side of the subdiffusion equation with the Caputo derivative with respect to time. In our case, the inverse problem consists in restoring the coefficient of the right-hand side, which depends on both the time and the spatial variable, when measured in integral form. Previously, similar inverse problems were studied for hyperbolic and parabolic equations with a different overdetermination condition, and in some works the existence and uniqueness of generalized solutions was established, while in others, the uniqueness of classical solutions was established. However, similar inverse problems for fractional equations with an integral overdetermination condition have not been considered before this work. The existence and uniqueness of a weak solution to the inverse problem under consideration is established. It is noteworthy that the results obtained are new for parabolic equations as well.
Paper Structure (8 sections, 8 theorems, 135 equations)

This paper contains 8 sections, 8 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eigenvalues of the Laplace Operator with Dirichlet Boundary Conditions
  4. Definition of the Weak Solution and Formulation of the Main Result
  5. A priori estimates
  6. Convergence
  7. Proof of Theorem
  8. 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 (15)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 4.1
  • Theorem 4.1
  • Remark 4.2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 5 more