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A Regularization for Time-Fractional Backward Heat Conduction Problem with Inhomogeneous Source Function

Vighnesh V. Alavani, P. Danumjaya, M. Thamban Nair

Abstract

Recently, Nair and Danumjaya (2023) introduced a new regularization method for the homogeneous time-fractional backward heat conduction problem (TFBHCP) in a one-dimensional space variable, for determining the initial value function. In this paper, the authors extend the analysis done in the above referred paper to a more general setting of an inhomogeneous time-fractional heat equation involving the higher dimensional state variables and a general elliptic operator. We carry out the analysis for the newly introduced regularization method for the TFBHCP providing optimal order error estimates under a source condition by choosing the regularization parameter appropriately, and also carry out numerical experiments illustrating the theoretical results.

A Regularization for Time-Fractional Backward Heat Conduction Problem with Inhomogeneous Source Function

Abstract

Recently, Nair and Danumjaya (2023) introduced a new regularization method for the homogeneous time-fractional backward heat conduction problem (TFBHCP) in a one-dimensional space variable, for determining the initial value function. In this paper, the authors extend the analysis done in the above referred paper to a more general setting of an inhomogeneous time-fractional heat equation involving the higher dimensional state variables and a general elliptic operator. We carry out the analysis for the newly introduced regularization method for the TFBHCP providing optimal order error estimates under a source condition by choosing the regularization parameter appropriately, and also carry out numerical experiments illustrating the theoretical results.
Paper Structure (11 sections, 12 theorems, 111 equations, 4 figures, 3 tables)

This paper contains 11 sections, 12 theorems, 111 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Time-Fractional Backward Heat Conduction Problem
  4. Issues of well-posedness and ill-posedness of TFBHCP
  5. The Regularization
  6. Error Estimates Under Noisy Data
  7. General error estimates and convergence
  8. Error estimates under source conditions
  9. Numerical Illustrations
  10. Numerical Procedure
  11. Results and Discussions

Key Result

Proposition 2.1

Let ${\mathcal{H}}$ be an infinite dimensional real Hilbert space with inner product $\langle\cdot, \cdot\rangle_{\mathcal{H}}$ and let ${\mathcal{H}}_0$ be a subspace of ${\mathcal{H}}$ which is a Hilbert space with a stronger inner product $\langle\cdot, \cdot\rangle_{{\mathcal{H}}_0}$ and such th Then, there exists a sequence $(\lambda_n)$ of positive real numbers with $\lambda_n\to \infty$ as

Figures (4)

  • Figure 1: Solution profiles of $u_0(x,y)$ and $u_{\alpha}(x, y, t_{i})$ for $\alpha = 0.8$
  • Figure 2: Solution profiles of $u_0(x, y)$ and $u^\varepsilon_{\alpha}(x,y, t_\varepsilon)$ due to noisy in $f$ for $\alpha = 0.8$
  • Figure 3: Solution profiles of $u_0(x,y)$ and $u^{\delta,\varepsilon}_{\alpha}(x,y, t_\eta)$ due to noisy in $f$ and $g$ for $\alpha = 0.8$
  • Figure 4: $\eta$ versus error $\|u_{\alpha}^{\delta ,\varepsilon}(\cdot,t_{\eta}) - u_0\|$ for $\alpha = 0.8$.

Theorems & Definitions (21)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • ...and 11 more