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A backward problem for the time-fractional pseudo-parabolic equation with a variable coefficient

Arshyn Altybay

Abstract

This work addresses an inverse reconstruction task for a time-fractional pseudo-parabolic model with a temporally varying coefficient. By imposing Dirichlet boundary conditions, we aim to recover the unknown initial state from observations collected at the final time. From a theoretical perspective, we derive existence and uniqueness results by proving that, under suitable hypotheses, the problem admits a unique solution. Computationally, we introduce a finite-difference discretisation based on a time-stepping strategy and provide a detailed stability and convergence analysis. Leveraging the resulting forward solver, we then formulate an initial-data identification procedure using Tikhonov regularisation. The proposed approach is validated with numerical simulations, and its resilience is assessed via experiments that incorporate perturbations in the final-time measurements.

A backward problem for the time-fractional pseudo-parabolic equation with a variable coefficient

Abstract

This work addresses an inverse reconstruction task for a time-fractional pseudo-parabolic model with a temporally varying coefficient. By imposing Dirichlet boundary conditions, we aim to recover the unknown initial state from observations collected at the final time. From a theoretical perspective, we derive existence and uniqueness results by proving that, under suitable hypotheses, the problem admits a unique solution. Computationally, we introduce a finite-difference discretisation based on a time-stepping strategy and provide a detailed stability and convergence analysis. Leveraging the resulting forward solver, we then formulate an initial-data identification procedure using Tikhonov regularisation. The proposed approach is validated with numerical simulations, and its resilience is assessed via experiments that incorporate perturbations in the final-time measurements.
Paper Structure (28 sections, 7 theorems, 144 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 28 sections, 7 theorems, 144 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Existence and Uniqueness of the problem
  3. Definition and assumptions
  4. Existence and uniqueness
  5. Continuous dependence on the data
  6. Discretisation of the direct model
  7. Stability–convergence analysis
  8. Preliminary properties of the graded-mesh $L1$ operator
  9. Solvability at each time level
  10. Stability estimate on a graded mesh
  11. Convergence
  12. Error decomposition.
  13. Assumption (graded-mesh regularity).
  14. Error equation.
  15. Truncation error bound.
  16. ...and 13 more sections

Key Result

Theorem 2.2

Let F1--F4 hold. Then the inverse problem 1.1--finalT admits a unique classical solution $(u,u_0)$ in the sense of Definition def:frac. Moreover, $(u,u_0)$ is given by where $\psi_k$ and $f_k$ are defined in coeff:psi:f, and $\mathcal{A}_k,\mathcal{B}_k$ are defined by def:AkBk_frac below. In particular, Lemma lem:Ak_positive yields $\mathcal{A}_k(T)>0$, so the coefficients in u_frac_compact_thm

Figures (4)

  • Figure 1: Reconstruction results for different fractional orders. Left: exact initial state $u_0(x)$ and reconstructed state $\widehat{u_0}(x)$. Right: exact final-time measurement $\psi(x)$ and the final-time state $u(x,T;\widehat{u_0})$ generated by evolving the reconstructed initial state with the direct solver.
  • Figure 2: Exact solution (left), reconstructed solution (centre), and absolute error (right) for $u(x,t)$ for several values of $\alpha$.
  • Figure 3: Recovered initial state $u_0^{\lambda,\delta}(x)$ for increasing noise levels $\delta$ in the final-time measurement, shown against the exact initial state $u_0(x)$.
  • Figure 4: Numerical reconstructions of $u(x,t)$ over the full time interval using noisy data:1% noise (left), 3% noise (middle), and 5% noise (right), for $\alpha=0.5$.

Theorems & Definitions (15)

  • Definition 2.1
  • Theorem 2.2: Existence, uniqueness, and explicit reconstruction
  • proof
  • Lemma 2.3: Positivity of $\mathcal{A}_k$
  • proof
  • Proposition 2.4: Stability estimate for the reconstructed initial state
  • proof
  • Lemma 3.1: Coercivity of the graded-mesh $L1$ operator
  • proof
  • Lemma 3.2: Unique solvability
  • ...and 5 more