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Some Unexplored Topics in The Reconstruction of Scalar Parameters of Subdiffusion

Sergii V. Siryk, Lidiia Tereshchenko, Nataliya Vasylyeva

Abstract

In the paper, we discuss the reconstruction of scalar parameters in a linear diffusion equation with fractional in time differential operators and with additional nonlocal (convolution) terms, which incorporate memory effects in models. Although, under suitable assumptions on the data, inverse problems associated with recovery of these parameters are nowadays well understood, several important questions related with numerical reconstructions of these parameters via a nonlocal observation in a small time interval have not yet been analyzed. This paper aims to provide some answers.

Some Unexplored Topics in The Reconstruction of Scalar Parameters of Subdiffusion

Abstract

In the paper, we discuss the reconstruction of scalar parameters in a linear diffusion equation with fractional in time differential operators and with additional nonlocal (convolution) terms, which incorporate memory effects in models. Although, under suitable assumptions on the data, inverse problems associated with recovery of these parameters are nowadays well understood, several important questions related with numerical reconstructions of these parameters via a nonlocal observation in a small time interval have not yet been analyzed. This paper aims to provide some answers.

Paper Structure

This paper contains 20 sections, 23 theorems, 207 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Notations and General Hypothesis in the Model
  3. Functional setting
  4. General assumptions in the model
  5. Statement of the Main Results
  6. Auxiliary Results
  7. Proof of Theorem \ref{['t2.3']}
  8. Theorem \ref{['t2.3']} in a simpler case: the reconstruction of $\nu_1$ only
  9. Completion of the proof of Theorem \ref{['t2.3']}
  10. Proof of Theorem \ref{['t2.4']}
  11. Estimates \ref{['5.2.1']}-\ref{['5.2.2']}
  12. Estimate \ref{['5.2.3']}
  13. Proof of Theorem \ref{['t2.5']}
  14. Estimates \ref{['6.3.1']}-\ref{['6.3.2']}
  15. Estimate \ref{['6.3.3']}
  16. ...and 5 more sections

Key Result

Theorem 3.1

Let positive $T$ be arbitrary but finite, and let assumptions h1-h6 hold. If $\mathfrak{C}_{\nu,0}\neq 0$ and $\rho_{i^*}(t)\neq 0$ for all $t\in[0,t^{*}]$, then the first inverse problem i.1-i.4 has a unique solution $(\nu_{1},\nu_{i^*},u)$ with $\nu_{1}$ and $\nu_{i^*}$ being computed via formulas where the positive quantity $C_0$ depends only on $T$, the Lebesgue measure of $\Omega$, $\|\mathca

Theorems & Definitions (46)

  • Definition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.1
  • Theorem 3.3
  • Remark 3.1
  • Remark 3.2
  • Corollary 3.2
  • Lemma 3.1
  • Theorem 3.4
  • ...and 36 more