A Modified Variational Iteration Method for Nonlinear Oscillators

被引：1
作者：
Yang, Q. W. [1 ]
Chen, Y. M. [2 ]
Liu, J. K. [2 ]
Zhao, W. [3 ]
机构：
[1] Shaoxing Univ, Dept Civil Engn, Shaoxing 312000, Peoples R China
[2] Sun Yat Sen Univ, Dept Mech, Guangzhou, Guangdong, Peoples R China
[3] Jinan Univ, Key Lab Disaster Forecast & Control Engn, Minist Educ China, Guangzhou, Guangdong, Peoples R China
来源：
INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION | 2013年 / 14卷 / 7-8期
基金：
中国国家自然科学基金;
关键词：
variational iteration method; Mickens iteration procedure; Duffing equation; van der Pol equation; HOMOTOPY PERTURBATION METHOD; APPROXIMATE SOLUTION; PERIODIC-SOLUTIONS; PARTICLE; EQUATION; MOTION; FORCE; FLOW;
D O I：
10.1515/ijnsns-2011-045
中图分类号：
T [工业技术];
学科分类号：
08 ;
摘要：
A modified variational iteration method (MVIM) is proposed for nonlinear oscillators. Different from the existing VIM, the algebraic equation deduced at each iteration stage is always linear when the proposed approach is applied to conservative oscillators. Therefore, highly accurate approximations can be easily obtained. The Duffing and the van der Pol equations are presented to validate the MVIM. Numerical examples show that the presented method is valid for both weakly and strongly nonlinear oscillators. In addition, it reveals that the MVIM is equivalent to a modified Mickens iteration scheme. Furthermore, the MVIM is extended by means of an auxiliary parameter. This parameter provides us with a convenient way to control and adjust the convergence of VIM approximations when necessary.
引用
收藏
页码：453 / 462
页数：10
相关论文
共 40 条
[1]  
[Anonymous], 2010, NONLINEAR SCI LETT A
[2]   A modified Mickens iteration procedure for nonlinear oscillators [J].
Chen, Y. M. ;
Liu, J. K. .
JOURNAL OF SOUND AND VIBRATION, 2008, 314 (3-5) :465-473
[3]   Construction of exact periodic wave and solitary wave solutions for the long-short wave resonance equations by VIM [J].
Deng, Chaofa ;
Shang, Yadong .
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, 2009, 14 (04) :1186-1195
[4]   Efficiency of differential transformation method for nonlinear oscillation: Comparison with HPM and VIM [J].
Ghafoori, S. ;
Motevalli, M. ;
Nejad, M. G. ;
Shakeri, F. ;
Ganji, D. D. ;
Jalaal, M. .
CURRENT APPLIED PHYSICS, 2011, 11 (04) :965-971
[5]   Large amplitude free vibrations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia [J].
Hamdan, MN ;
Dado, MHF .
JOURNAL OF SOUND AND VIBRATION, 1997, 206 (02) :151-168
[6]   Approximate solution of nonlinear differential equations with convolution product nonlinearities [J].
He, JH .
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 1998, 167 (1-2) :69-73
[7]   Variational iteration method for autonomous ordinary differential systems [J].
He, JH .
APPLIED MATHEMATICS AND COMPUTATION, 2000, 114 (2-3) :115-123
[8]   Variational iteration method - a kind of non-linear analytical technique: Some examples [J].
He, JH .
INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS, 1999, 34 (04) :699-708
[9]   Approximate analytical solution for seepage flow with fractional derivatives in porous media [J].
He, JH .
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 1998, 167 (1-2) :57-68
[10]   Homotopy perturbation method: a new nonlinear analytical technique [J].
He, JH .
APPLIED MATHEMATICS AND COMPUTATION, 2003, 135 (01) :73-79
1234