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

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.