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Solving fractional van der Pol system based on Adams discretization and Newmark-
β method- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 61, Issue 5, Pages: 126-132(2022)
- 作者机构：
中山大学航空航天学院，广东 广州 510006
- 作者简介：
- 基金信息：
DOI：10.13471/j.cnki.acta.snus.2021B034
CLC： V21Published：25 September 2022，
Published Online：26 January 2022，
Received：17 April 2021，
Accepted：21 May 2021
扫 描 看 全 文
姜宏杰,刘祚秋,吕中荣等.基于Adams离散和Newmark-,β法求解分数阶van der Pol系统[J].中山大学学报(自然科学版),2022,61(05):126-132.
JIANG Hongjie,LIU Zuoqiu,LU Zhongrong,et al.Solving fractional van der Pol system based on Adams discretization and Newmark-,β method[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2022,61(05):126-132.
姜宏杰,刘祚秋,吕中荣等.基于Adams离散和Newmark-,β法求解分数阶van der Pol系统[J].中山大学学报(自然科学版),2022,61(05):126-132. DOI： 10.13471/j.cnki.acta.snus.2021B034.
JIANG Hongjie,LIU Zuoqiu,LU Zhongrong,et al.Solving fractional van der Pol system based on Adams discretization and Newmark-,β method[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2022,61(05):126-132. DOI： 10.13471/j.cnki.acta.snus.2021B034.
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
<math id="M1"><mn mathvariant="normal">0</mn><mo><</mo><mi>α</mi><mo><</mo><mn mathvariant="normal">1</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438686&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438632&type=
12.78466606
2.28600001
和
<math id="M2"><mn mathvariant="normal">1</mn><mo><</mo><mi>α</mi><mo><</mo><mn mathvariant="normal">2</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438754&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438723&type=
12.78466606
2.28600001
的van der Pol系统的数值响应。当
<math id="M3"><mi>α</mi><mo>→</mo><mn mathvariant="normal">1</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438801&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438783&type=
8.72066593
2.28600001
和
<math id="M4"><mi>α</mi><mo>→</mo><mn mathvariant="normal">2</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438866&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49438845&type=
8.72066593
2.28600001
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。
Abstract
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 proposed for Caputo fractional derivative. Then， a complete iterative scheme is constructed based on the Newmark-
β
method. 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 systems with fractional order
<math id="M5"><mn mathvariant="normal">0</mn><mo><</mo><mi>α</mi><mo><</mo><mn mathvariant="normal">1</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49441798&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49441781&type=
14.90133286
2.70933342
and
<math id="M6"><mn mathvariant="normal">1</mn><mo><</mo><mi>α</mi><mo><</mo><mn mathvariant="normal">2</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49441816&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49441797&type=
14.90133286
2.62466669
are discussed respectively. Moreover， the comparison between the proposed method and the fourth-order Runge-Kutta method is also discussed. The results 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
Keywords
fractional-order derivativevan der Pol systemAdams discretizationNewmark-β methodnumerical solution
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