Simultaneous inversion of a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation. (April 2022)

• Record Type: Journal Article
• Authors: Yan, Xiong-bin; Zhang, Zheng-qiang; Wei, Ting
• Is Part Of: Chaos, solitons and fractals. Volume 157(2022)
• Publication Date: 2022-04
• Subjects: Nonlinear inverse problem -- Stability and uniqueness -- Ill-posedness -- Non-stationary iterative Tikhonov regularization method; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2022.111901
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: Consider a nonlinear inverse problem which identifies multiple parameters in a time fractional diffusion-wave equation. Prove the uniqueness of the inverse problem. Use an effective and stable algorithm to solve the inverse problem numerically. We prove the existence of minimizer of minimization functional as well as the stability for solving sub-problem. We prove for the first time that the data fidelity term decrease monotonously with the iterative operation of the algorithm provided in the paper. Abstract: The main purpose of this paper is to identify simultaneously a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation from two points observed data. First of all, using the fixed point theorem, we prove the existence and uniqueness of the solution for the direct problem. Secondly, the stability of the inverse problem is proved and the uniqueness is a direct result of the stability estimate. In addition, we illustrate the ill-posedness of the inverse problem and use a non-stationary iterative Tikhonov regularization method to recover numerically the time dependent potential coefficient and source term. At the same time, we give the existence of the minimizer for the minimization functional. In order to solve the minimization problem, we apply an alternating minimization method to find the minimizer and prove the solving sub-problems are stable on noisy data as well as prove the data fidelity item decreases monotonously with the iterative running. Finally, some numerical examples are provided to shed light on the validity and robustness of the numerical algorithm.