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Regularization operators for identifying the unknown source in the time-fractional convection-diffusion-reaction equation

Guillermo Federico Umbricht, Diana Rubio

Abstract

This article presents a mathematical study of the problem of identifying a time-dependent source term in transport processes described by a timefractional parabolic equation, based on noisy time-dependent measurements taken at an arbitrary position. The problem is analytically solved using Fourier techniques, and it is shown that the solution is unstable. To address this instability, three one-parameter families of regularization operators are proposed, each designed to counteract the factors responsible for the instability of the inverse operator. Additionally, a new rule for selecting the regularization parameter is introduced, and an error bound is derived for each estimate. Numerical examples with varying characteristics are provided to illustrate the advantages of the proposed strategies.

Regularization operators for identifying the unknown source in the time-fractional convection-diffusion-reaction equation

Abstract

This article presents a mathematical study of the problem of identifying a time-dependent source term in transport processes described by a timefractional parabolic equation, based on noisy time-dependent measurements taken at an arbitrary position. The problem is analytically solved using Fourier techniques, and it is shown that the solution is unstable. To address this instability, three one-parameter families of regularization operators are proposed, each designed to counteract the factors responsible for the instability of the inverse operator. Additionally, a new rule for selecting the regularization parameter is introduced, and an error bound is derived for each estimate. Numerical examples with varying characteristics are provided to illustrate the advantages of the proposed strategies.

Paper Structure

This paper contains 15 sections, 10 theorems, 87 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Source identification
  3. Presentation of the problem
  4. Problem solution
  5. Ill-posed problem
  6. Regularization operators
  7. Regularization solutions
  8. Error Analysis
  9. Auxiliary results
  10. Analytical bound of error
  11. Numerical examples and discussion of results
  12. Numerical examples
  13. Discussion of results
  14. Conclusions
  15. Additional Results

Key Result

Theorem 1

Let $0<\alpha\leq1$ and the parameters $\omega,\beta, \nu ,x_0, \delta, \delta_M \in \mathbb R^+$ with $\delta<\delta_M$. Consider functions $u(x, \cdot), f(\cdot), y(\cdot), y_\delta(\cdot) \in L^2(\mathbb R)$ such that $||y-y_{\delta}||_{L^2(\mathbb R)}\le \delta$ and $y_{\delta}(t)=y(t) + \textit Then, the expression for the source in the Fourier variable is given by where $\blacktriangleleft

Figures (2)

  • Figure 1: Example \ref{['example1']}: Non-regularized source with $x_0=0.5$ (top-left); Regularized sources with $x_0=0.5$ and $p=1$, using $R^{1}_\mu$ (top-right), $R^{2}_\mu$ (bottom-left), $R ^{3}_\mu$ (bottom-right) for different noise levels.
  • Figure 2: Example \ref{['example1']}: Non-regularized source with $x_0=10$ (top-left); Regularized sources with $x_0=10$ and $p=2$, using $R^{1}_\mu$ (top-right), $R^{2}_\mu$ (bottom-left), $R ^{3}_\mu$ (bottom-right) for different noise levels.

Theorems & Definitions (25)

  • Definition 1
  • Theorem 1: Source identification
  • proof
  • Theorem 2: The problem is ill-posed
  • proof
  • Proposition 1
  • proof
  • Definition 2
  • Theorem 3: Convergent regularization operators
  • proof
  • ...and 15 more