A linearized ADI scheme for two-dimensional time-space fractional nonlinear vibration equations

Abstract

A nonlinear initial boundary value problem with a time Caputo derivative of fractional-order $\beta \in (1,2)$ and a space Riesz derivative of fractional-order $\alpha \in (1,2)$ is considered. A linearized alternating direction implicit (ADI) scheme for this problem is constructed and analysed. First, we use the classical central difference formula and a new $3-\beta$-order formula to approximate the second-order derivative and the Caputo derivative in temporal direction, respectively. Meanwhile, the central difference quotient and fractional central difference formula are applied to deal with the spatial discretizations. Then, the proposed ADI scheme is proved unconditionally stable and convergent with the $3-\beta$-order accuracy in time and the second-order accuracy in space. Furthermore, the effectiveness of the ADI scheme and theoretical findings are illustrated by numerical experiments.

Keywords:

• Time-space fractional nonlinear vibration equations
• two dimensions
• linearized ADI scheme
• stability
• convergence

2010 Mathematics Subject Classifications:

• 65M06
• 65M12
• 35R11

Acknowledgments

This research is supported by National Natural Science Foundation of China (Grant Nos. 11701502 and 11871065), and by Natural Science Foundation of Jiangsu Province of China (Grant No. BK20201427).

Disclosure statement

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

Additional information

Funding

This research is supported by National Natural Science Foundation of China [grant numbers 11701502 and 11871065], and by Natural Science Foundation of Jiangsu Province of China [grant number BK20201427].