基于Gr&uuml;nwald&ndash;Letnikov定义改进的短记忆原理方法

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振动与冲击 ›› 2022, Vol. 41 ›› Issue (10) : 215-221.

论文

基于Grünwald–Letnikov定义改进的短记忆原理方法

马瑞群，张波，员海玮，韩景龙

作者信息 +

Improved short memory principle method based on the Grünwald-Letnikov definition

MA Ruiqun,ZHANG Bo,YUAN Haiwei,HAN Jinglong

Author information +

文章历史 +

摘要

本文提出一种改进的短记忆原理方法，即将传统的短记忆原理(short memory principle ,SMP)对时间的截断替换为对二项式系数的截断，然后有限数量的二项式系数反复应用于不断倍增的步长，直至覆盖所有先前的时间点。该方法的目的是用小步长保证计算精度，同时用逐渐增大的步长减少计算量。文章中用带有分数阶阻尼的受迫振动、分数阶非线性Duffing方程和分数阶Lorenz混沌系统为算例说明了该方法的准确性和有效性。

Abstract

In this paper, an improved method of short memory principle is proposed, that is, the truncation of time by the traditional short memory principle (SMP) is replaced by the truncation of binomial coefficients, and then a finite number of binomial coefficients are repeatedly applied to the step size of multiplication until all previous time points are covered. The purpose of this method is to ensure the accuracy of calculation with small step length, and reduce the amount of calculation with gradually increasing step size. In this paper, the forced vibration with fractional damping , the fractional nonlinear Duffing equation and fractional Lorenz Chaos System are used as examples to illustrate the accuracy and effectiveness of the method.

导出引用

马瑞群，张波，员海玮，韩景龙. 基于Grünwald–Letnikov定义改进的短记忆原理方法[J]. 振动与冲击, 2022, 41(10): 215-221

MA Ruiqun,ZHANG Bo,YUAN Haiwei,HAN Jinglong. Improved short memory principle method based on the Grünwald-Letnikov definition[J]. Journal of Vibration and Shock, 2022, 41(10): 215-221

参考文献

1.LEIBNIZ G W. Leibnizens mathematische schriften. 7vols [M]. Hildesheim: Georg Olms, 1971.
2.COMPTE A. Continuous time random walks on moving fluids[J]. Physical Review E, 1997, 55 (6): 6821–6831.
3. FELLAH Z, DEPOLLIER C, FELLAH M, Application of fractional calculus to the sound waves propagation in rigid porous materials: validation via ultrasonic measurements[J]. Acta Acustica united with Acustica. 2002, 88 (1): 34–39.
4.METZLER R, BARKAI E, KLAFTER J. Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker–Planck equation approach[J]. Physical Review Letters. 1999, 82 (18): 3563–3567.
5.SOCZKIEWICZ E, KRZYWOUSTEGO B, GLIWICE P. Application of fractional calculus in the theory of viscoelasticity[J]. Molecular and Quantum Acoustics, 2002, 23: 397-404.
6.GORENFLO R, MAINARDI F, MORETTI D, PARADISI P. Time fractional diffusion: a discrete random walk approach[J]. Nonlinear Dynamics, 2002, 29 (1): 129–143.
7.SEKI K, WOJCIK M, TACHIYA M. Fractional reaction–diffusion equation[J]. The Journal of Chemical Physics， 2003, 119 (4): 2165-2170.
8.CUSHMAN J H, GINN T R. Fractional advection–dispersion equation: a classical mass balance with convolution-Fickian flux[J]. Water Resources Research, 2000, 36 (12): 3763–3766.
9.BAEUMER B, BENSON D A, MEERSCHAERT M M, WHEATCRAFT S W. Subordinated advection–dispersion equation for contaminant transport[J]. Water Resources Research, 2001, 37 (6): 1543–1550.
10.MATHIEU B, MELCHIOR P, OUSTALOUP A, CEYRAL C. Fractional differentiation for edge detection[J]. Signal Process, 2003, 83 (11): 2421–2432.
11.CHEN C, LIU F, BURRAGE K. Finite difference methods and a Fourier analysis for the fractional reaction–subdiffusion equation[J] Applied Mathematics and Computation, 2008, 198(2): 754–769.
12.LANGLANDS T A M, HENRY B I. The accuracy and stability of an implicit solution method for the fractional diffusion equation[J]. Journal of Computational Physics, 2005, 205 (2): 719–736.
13.LIU F, ZHUANG P, ANH V, TURNER I, BURRAGE K. Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation[J]. Applied Mathematics and Computation, 2007, 191 (1): 12–20.
14.MAINARDI F, MURA A, PAGNINI G, GORENFLO R. Sub-diffusion equations of fractional order and their fundamental solutions[J], Mathematical Methods in Engineering, 2007: 23–55.
15.MCLEAN W, MUSTAPHA K. Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation[J]. Numerical Algorithms, 2009, 52 (1): 69–88.
16.Podlubny I, Chechkin A, Skovranek T, Chen Y Q, B.M. Vinagre Jara, Matrix approach to discrete fractional calculus II: partial fractional differential equations[J]. Journal of Computational Physics, 2009, 228 (8): 3137–3153.
17.YUSTE S B, ACEDO L. On an explicit finite difference method for fractional diffusion equations[J]. SIAM Journal on Numerical Analysis, 2005, 42(5): 1862–1874.
18.CHEN L C, ZHAO T L, LI W. Bifurcation control of bounded noise excited Duffing oscillator by a weakly fractional-order PI^λ D^μfeedback controller[J]. Nonlinear Dynamics, 2016, 83(1-2):529-539.
19. PODLUBNY I. Fractional Differential Equations, San Diego：Lightning Source Inc, 1999.
20.ABDELOUAHAB M S, HAMRI N E. The Grünwald–Letnikov Fractional-Order Derivative with Fixed Memory Length [J]. Mediterranean Journal of Mathematics, 2015,13(2): 557–572.
21.ABEDINI M, NOJOUMIAN M A, SALARIEH H, et al. Model reference adaptive control in fractional order systems using discrete-time approximation methods[J]. Commun Nonlinear Sci Numer Simulat, 2015, 25(1-3):27–40.
22.WEI Y H, CHEN Y Q, CHENG S S, et al. A note on short memory principle of fractional calculus[J]. Fractional Calculus and Applied Analysis, 2017, 20(6): 1383–1404.
23.DIETHELM K, GARRAPPA R, STYNES M. Good (and not so good) practices in computational methods for fractional calculus[J]. Mathematics, 2020, 8(3): 324.
24.FORD N J, SIMPSON A C. The numerical solution of fractional differential equations: Speed versus accuracy[J]. Numer. Algorithms, 2001, 26(1), 333–346.
25.DIETHELM K, FREED A D. An efficient algorithm for the evaluation of convolution integrals[J]. Computers and Mathematics with Applications. 2006, 51(1): 51–72.