一类分数阶Langevin方程预估校正算法的数值分析

Abstract

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

CLC Number:

O211.63

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.

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Figures/Tables 8

References 22

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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

[1]	Yuji Liu. Solvability of BVPs for Impulsive Fractional Langevin Type Equations Involving Three Riemann-Liouville Fractional Derivatives [J]. Acta mathematica scientia,Series A, 2020, 40(1): 103-131.
[2]	Yadi Zhao, Lifei Wu, Xiaozhong Yang, Shuzhen Sun. A Kind of Efficient Difference Method for the Time Fractional Sub-Diffusion Equation [J]. Acta mathematica scientia,Series A, 2018, 38(6): 1122-1134.
[3]	WEN Juan, HE Yin-Nian, WANG Xue-Min, HUO Mi-Hui. TWO-LEVEL MULTISCALE FINITE ELEMENT METHODS FOR THE STEADY NAVIER-STOKES PROBLEM [J]. Acta mathematica scientia,Series A, 2014, 34(3): 960-972.

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

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