一类分数阶Langevin方程预估校正算法的数值分析

数学物理学报 ›› 2020, Vol. 40 ›› Issue (4): 1018-1028.

一类分数阶Langevin方程预估校正算法的数值分析

蒲琳涓,杨晓忠*(),孙淑珍

^{华北电力大学数理学院信息与计算研究所北京 102206}

收稿日期:2019-05-24 出版日期:2020-08-26 发布日期:2020-08-20
通讯作者: 杨晓忠 E-mail:yxiaozh@necpu.edu.cn
基金资助:
国家科技重大专项子课题(2017ZX07101001-01);国家自然科学基金(11371135)

Numerical Analysis of a Class of Fractional Langevin Equation by Predictor-Corrector Method

Linjuan Pu,Xiaozhong Yang*(),Shuzhen Sun

^{Institution of Information and Computation, School of Mathematics and Physics, North China Electric Power University, Beijing 102206}

Received:2019-05-24 Online:2020-08-26 Published:2020-08-20
Contact: Xiaozhong Yang E-mail:yxiaozh@necpu.edu.cn
Supported by:
the Subproject of National Science and Technology Major Project of China(2017ZX07101001-01);the NSFC(11371135)

摘要/Abstract

摘要：

该文将经典Langevin方程在分数阶上进行拓展，使其具有时间记忆性，采用预估校正算法数值求解一类分数阶Langevin方程.先用R0算法求出预估值，再将预估值代入R2算法中，对数值解进行校正，最终得到一类分数阶Langevin方程预估校正算法的数值解.误差分析证明在该方程的$0 < \alpha < 1$条件下，预估校正算法是$(1+\alpha)$阶收敛的.数值试验也表明不同$\alpha$，步长$h$取值下，预估校正算法的数值解都是收敛的.

关键词: 分数阶Langevin方程, 预估校正算法, 误差分析, 数值试验

Abstract:

This paper extends the Langevin equation on the fractional order to make it time-memory, and a class of fractional-order Langevin equation is solved numerically using the predictor-corrector method. Firstly, the estimated value is obtained by R0 algorithm, then the estimated value is substituted into the R2 algorithm to correct the numerical solution. Finally, the numerical solution of a class of fractional-order Langevin equation by predictor-corrector method is obtained. The error analysis proves that under the condition of $0 < \alpha < 1$ of the equation, the result of predictor-corrector method is $(1+\alpha)$ order convergence. Numerical experiments also show that the numerical solution of the predictor-corrector method is convergent under different values of $\alpha$ and step size $h$.

Key words: Fractional Langevin equation, Predictor-Corrector method, Error analysis, Numerical experiments

中图分类号:

O211.63

蒲琳涓,杨晓忠,孙淑珍. 一类分数阶Langevin方程预估校正算法的数值分析[J]. 数学物理学报, 2020, 40(4): 1018-1028.

Linjuan Pu,Xiaozhong Yang,Shuzhen Sun. Numerical Analysis of a Class of Fractional Langevin Equation by Predictor-Corrector Method[J]. Acta mathematica scientia,Series A, 2020, 40(4): 1018-1028.

图/表 8

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图 7

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$\alpha$	$h$	$e(h)$	$\gamma$	$\alpha$	$h$	$e(h)$	$\gamma$
	0.005	0.0034			0.005	0.0010
	0.008	0.0105			0.008	0.0028
0.1	0.01	0.0117	1.0732	0.3	0.01	0.0026	1.2005
	0.05	0.0638			0.05	0.0635
	0.1	0.0941			0.1	0.018
	0.005	0.0013			0.005	0.0012
	0.008	0.0013			0.008	0.0022
0.5	0.01	0.0032	1.3476	0.7	0.01	0.0041	1.5506
	0.05	0.0364			0.05	0.0578
	0.1	0.0462			0.1	0.1037
	0.005	0.0006
	0.008	0.0007
0.9	0.01	0.0015	1.8435
	0.05	0.0478
	0.1	0.0850

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