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Numerical Identification of a Time-Dependent Coefficient in a Time-Fractional Diffusion Equation with Integral Constraints

Arshyn Altybay

TL;DR

The work addresses the inverse problem of identifying a time-dependent coefficient $p(t)$ in a time-fractional diffusion equation with Caputo derivative $\partial_t^\alpha$, using an integral overdetermination condition. It develops a fully implicit finite-difference scheme on graded meshes for the direct problem, proves unconditional stability and convergence, and derives a data-driven integral formulation to recover $p(t)$ alongside the state $u(x,t)$. An a priori estimate and existence/uniqueness results for the inverse problem are established, and the method is validated through numerical experiments that demonstrate high accuracy and robustness to noise. The proposed framework is scalable and practical for applications where integral measurements are available in fractional diffusion contexts.

Abstract

In this paper, we numerically address the inverse problem of identifying a time-dependent coefficient in the time-fractional diffusion equation. An a priori estimate is established to ensure uniqueness and stability of the solution. A fully implicit finite-difference scheme is proposed and rigorously analysed for stability and convergence. An efficient algorithm based on an integral formulation is implemented and verified through numerical experiments, demonstrating accuracy and robustness under noisy data.

Numerical Identification of a Time-Dependent Coefficient in a Time-Fractional Diffusion Equation with Integral Constraints

TL;DR

The work addresses the inverse problem of identifying a time-dependent coefficient in a time-fractional diffusion equation with Caputo derivative , using an integral overdetermination condition. It develops a fully implicit finite-difference scheme on graded meshes for the direct problem, proves unconditional stability and convergence, and derives a data-driven integral formulation to recover alongside the state . An a priori estimate and existence/uniqueness results for the inverse problem are established, and the method is validated through numerical experiments that demonstrate high accuracy and robustness to noise. The proposed framework is scalable and practical for applications where integral measurements are available in fractional diffusion contexts.

Abstract

In this paper, we numerically address the inverse problem of identifying a time-dependent coefficient in the time-fractional diffusion equation. An a priori estimate is established to ensure uniqueness and stability of the solution. A fully implicit finite-difference scheme is proposed and rigorously analysed for stability and convergence. An efficient algorithm based on an integral formulation is implemented and verified through numerical experiments, demonstrating accuracy and robustness under noisy data.

Paper Structure

This paper contains 12 sections, 9 theorems, 71 equations, 8 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A priori estimate
  4. Existence and Uniqueness of the Inverse Problem
  5. Numerical Methods for the direct problem
  6. Stability and convergence of the implicit difference schemes
  7. Assumption.
  8. Determination of the unknown coefficient $p(t)$
  9. Numerical experiments
  10. Numerical example
  11. Effect of noisy data on the reconstruction of $p(t)$
  12. Conclusion

Key Result

Lemma 2.1

(Alikhanov Alikhanov10) For an arbitrary absolutely continuous function $u(t)$ defined on the interval $[0,T]$, the following inequality holds:

Figures (8)

  • Figure 1: The analytical and numerical solutions of $\{p(t), u(x,t)\}$ when $\alpha=0.25$.
  • Figure 2: The analytical and numerical solutions of $\{p(t), u(x,t)\}$ when $\alpha=0.75$.
  • Figure 3: The analytical and numerical solutions of ${p(t),u(x,t)}$ when $\alpha=0.95$.
  • Figure 4: The numerical solutions of ${p(t),u(x,t)}$ when $T=1$ and at different $\alpha$.
  • Figure 5: Comparison of the exact (left) and numerical (right) solutions for $u(x,t)$ at different values of the fractional order $\alpha$.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Theorem 3.1
  • proof
  • Theorem 3.2: Existence, DurdRah25
  • Theorem 3.3: Uniqueness, DurdRah25
  • Theorem 4.1: Stability
  • ...and 3 more