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

TL;DR

This paper introduces an alternating iterative method with regularization for reconstructing initial values and source terms in a time-fractional diffusion-wave equation, supported by convergence analysis and numerical validation.

Contribution

The paper develops a novel alternating iterative reconstruction method with regularization for a coupled inverse problem in fractional diffusion equations, including convergence analysis and error estimates.

Findings

The method effectively reconstructs initial and source terms in fractional diffusion equations.

Numerical experiments confirm the accuracy and stability of the proposed approach.

Error estimates relate reconstruction accuracy to noise, iterations, and discretization.

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.