多项分数阶非线性波动方程的数值方法及其快速实现

doi:10.3969/j.issn.1000-1565.2022.05.002

河北大学学报(自然科学版) ›› 2022, Vol. 42 ›› Issue (5): 454-462.DOI: 10.3969/j.issn.1000-1565.2022.05.002

多项分数阶非线性波动方程的数值方法及其快速实现

邵林馨,沈卓旸,马葛沁舟,闵婕,黄健飞

收稿日期:2021-12-03 出版日期:2022-09-25 发布日期:2022-10-19
通讯作者: 黄健飞(1983—)
作者简介:邵林馨(1999—),女,江苏淮安人,扬州大学在读硕士研究生.E-mail: 316675129@qq.com
〓通信作者:黄健飞(1983—),男,江苏苏州人,扬州大学副教授,特聘教授,博士,主要从事偏微分方程数值解法方向研究.
E-mail: jfhuang@yzu.edu.cn
基金资助:
国家自然科学基金资助项目(11701502);江苏省自然科学基金资助项目(BK20201427)

A numerical method and its fast implementation for multi-term fractional nonlinear wave equations

SHAO Linxin, SHEN Zhuoyang, MA Geqinzhou, MIN Jie, HUANG Jianfei

College of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China

Received:2021-12-03 Online:2022-09-25 Published:2022-10-19

摘要/Abstract

摘要： 对带有空间四阶导数的多项时间分数阶非线性波动方程构造了一个线性化数值方法. 该方法采用线性化技术离散非线性项,从而避免求非线性方程组,并严格地证明了该方法的收敛性,在时间方向具有一阶精度,在空间方向具有四阶精度.该方法同样适用于初始奇异性问题,并且还可以用指数和技术来进行快速实现. 最后,通过数值实验验证了该方法的有效性和理论分析的正确性.

关键词: 多项分数阶波动方程, 四阶导数, 数值方法, 收敛性, 快速实现

Abstract: A linearized numerical method for multi-term time fractional nonlinear wave equations with a spatial fourth-order derivative is derived. To avoid solving the nonlinear system of equations, a linearized technique is applied to discretize the nonlinear term. Then, the convergence of this presented method is rigorously proved with the first-order accuracy in time and the fourth-order accuracy in space. It is worth mentioning that the presented method can handle the initial singularity, and can be fast calculated by the sum-of-exponentials technique. Finally, numerical experiments are given to verify the effectiveness of this method and the correctness of the theoretical results.

Key words: multi-term fractional wave equations, fourth-order derivative, numerical method, convergence, fast implementation

中图分类号:

O241.82

邵林馨,沈卓旸,马葛沁舟,闵婕,黄健飞. 多项分数阶非线性波动方程的数值方法及其快速实现[J]. 河北大学学报(自然科学版), 2022, 42(5): 454-462.

SHAO Linxin, SHEN Zhuoyang, MA Geqinzhou, MIN Jie, HUANG Jianfei. A numerical method and its fast implementation for multi-term fractional nonlinear wave equations[J]. Journal of Hebei University(Natural Science Edition), 2022, 42(5): 454-462.

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多项分数阶非线性波动方程的数值方法及其快速实现

A numerical method and its fast implementation for multi-term fractional nonlinear wave equations

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