Identifying a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation by using the measured data at a boundary point

Abstract

In this paper, we investigate a nonlinear inverse problem of identifying a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation by using the measured data at a boundary point. We firstly prove the existence, uniqueness and regularity of the solution for the corresponding direct problem by using the contraction mapping principle. Then we try to give a conditional stability estimate for the inverse zeroth-order coefficient problem and propose a simple condition for the initial value and zeroth-order coefficient such that the uniqueness of the inverse coefficient problem is obtained. The Levenberg–Marquardt regularization method is applied to obtain a regularized solution. Based on the piecewise linear finite elements approximation, we find an approximate minimizer at each iteration by solving a linear system of algebraic equations in which the Fréchet derivative is obtained by solving a sensitive problem. Two numerical examples in one-dimensional case and two examples in two-dimensional case are provided to show the effectiveness of the proposed method.

Keywords:

• Time-fractional diffusion-wave equation
• time-dependent zeroth-order coefficient
• uniqueness
• conditional stability
• Levenberg–Marquardt regularization method

2010 Mathematics Subject Classifications:

• 35R30
• 65M32

Acknowledgments

This work is supported by the NSF of China [grant number 11771192].

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 11771192].