An alternating approach for reconstructing the initial value and source term in a time-fractional diffusion-wave equation

TL;DR

This work addresses the ill-posed, jointly difficult task of simultaneously reconstructing the initial value $a$ and a space-dependent source $f$ in a time-fractional diffusion-wave equation of order $\alpha\in(1,2)$ from terminal measurements. An alternating regularization strategy decouples the problem into backward-in-time recovery of $a$ and a regularized inverse source step for $f$, with a contraction property ensuring convergence when the final times are chosen appropriately. The authors develop two semidiscrete schemes—standard Galerkin and lumped mass—for spatial discretization, and provide comprehensive error estimates that account for noise level, regularization parameter, and mesh size. Numerical experiments in 1D and 2D validate the theory and demonstrate robustness to noise, confirming the practical viability of the method. The study contributes a rigorous framework for simultaneous parameter identification in TFDEs and offers a computationally efficient approach suitable for distributed computing contexts.

Abstract

This paper is dedicated to addressing the simultaneous inversion problem involving the initial value and space-dependent source term in a time-fractional diffusion-wave equation. Firstly, we establish the uniqueness of the inverse problem by leveraging the asymptotic expansion of Mittag-Leffler functions. Subsequently, we decompose the inverse problem into two subproblems and introduce an alternating iteration reconstruction method, complemented by a regularization strategy. Additionally, a comprehensive convergence analysis for this method is provided. To solve the inverse problem numerically, we introduce two semidiscrete schemes based on standard Galerkin method and lumped mass method, respectively. Furthermore, we establish error estimates that are associated with the noise level, iteration step, regularization parameter, and spatial discretization parameter. Finally, we present several numerical experiments in both one-dimensional and two-dimensional cases to validate the theoretical results and demonstrate the effectiveness of our proposed method.

Figures (5)

• Figure 1: Approximate solutions $a_{\mu,h,\delta}^{(k)}$ (left column) and $f_{\mu,h,\delta}^{(k)}$(right column) of Example 1 with various $\epsilon$.
• Figure 2: Approximate solutions $a_{\mu,h,\delta}^{(k)}$ (left column) and $f_{\mu,h,\delta}^{(k)}$(right column) of Example 2 based on standard Galerkin method with various $\epsilon$.
• Figure 3: Approximate solutions $a_{\mu,h,\delta}^{(k)}$ (left column) and $f_{\mu,h,\delta}^{(k)}$(right column) of Example 2 based on lumped mass method with various $\epsilon$.
• Figure 4: Approximate solutions $a_{\mu,h,\delta}^{(k)}$ (left column) and the absolute errors (right column) of Example 3 with $\alpha=1.1,1.2,1.3,1.4,1.5$ (from top to bottom).
• Figure 5: Approximate solutions $f_{\mu,h,\delta}^{(k)}$ (left column) and the absolute errors (right column) of Example 3 with $\alpha=1.1,1.2,1.3,1.4,1.5$ (from top to bottom)

Theorems & Definitions (46)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 2.5
• Proposition 2.6
• Lemma 2.7
• Lemma 2.8
• proof
• Lemma 2.9