基于Adams离散和Newmark-β法求解分数阶van der Pol系统  

Solving fractional van der Pol system based on Adams discretization and Newmark-β method

作　　者：姜宏杰 刘祚秋 吕中荣 刘广 JIANG Hongjie;LIU Zuoqiu;LU Zhongrong;LIU Guang(School of Aeronautics and Astronautics,Sun Yat-sen University,Guangzhou 510006,China)

机构地区：[1]中山大学航空航天学院,广东广州510006

出　　处：《中山大学学报（自然科学版）（中英文）》2022年第5期126-132,共7页Acta Scientiarum Naturalium Universitatis Sunyatseni

基　　金：国家自然科学基金(11272361,11972380)。

摘　　要：文章着重研究了含分数阶微分算子的van der Pol方程的数值解法。首先,基于Adams离散提出了一种针对Caputo分数阶导数的离散格式;然后,进一步基于Newmark-β法构造了完整的逐步迭代格式;最后,通过Newton-Raphson迭代求得了非线性系统的响应。在算例分析部分,讨论了分数阶次为0<α<1和1<α<2的van der Pol系统的数值响应。当α→1和α→2时,将所提算法和四阶Runge-Kutta法进行了对比。结果表明,所提数值方法对整数阶微分系统也同样适用。This paper focuses on the numerical solution method of the van der Pol equation with fractional differential operators.In this paper,a discretization scheme based on the Adams discretization is pro⁃posed for Caputo fractional derivative.Then,a complete iterative scheme is constructed based on the Newmark-βmethod.Finally,the numerical solution of the nonlinear discretization equation is obtained by Newton-Raphson iteration.In the numerical examples,the numerical responses of van der Pol sys⁃tems with fractional order 0<α<1 and 1<α<2 are discussed respectively.Moreover,the compari⁃son between the proposed method and the fourth-order Runge-Kutta method is also discussed.The re⁃sults proved that the proposed numerical scheme is also suitable for integer-order differential systems.

关 键 词：分数阶导数 van der Pol系统 Adams离散 NEWMARK-Β法 数值解 

分 类 号：V21[航空宇航科学与技术—航空宇航推进理论与工程]