分数阶Fornberg-Whitham方程及其改进方程的变分迭代解

鲍四元; 邓子辰

doi:10.3879/j.issn.1000-0887.2013.12.002

分数阶Fornberg-Whitham方程及其改进方程的变分迭代解

doi: 10.3879/j.issn.1000-0887.2013.12.002

鲍四元¹,
邓子辰^2,3

¹苏州科技学院土木工程学院工程力学系, 江苏苏州 215011;²西北工业大学工程力学系,西安 710072;³大连理工大学工业装备结构分析国家重点实验室, 辽宁大连 116023

基金项目: 国家自然科学基金资助项目(11202146)

详细信息

作者简介:
鲍四元（1980—），男，安徽人，副教授，博士(通讯作者. E-mail: bsiyuan@126.com)；邓子辰(1964—),男,西安人，教授,博士生导师(E-mail: dweifan@nwpu.edu.cn).

中图分类号: O175.29
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出版历程
- 收稿日期: 2013-08-27
- 修回日期: 2013-10-30
- 刊出日期: 2013-12-16

Variational Iteration Solutions for Fractional FornbergWhitham Equation and Its Modified Equation

BAO Si-yuan¹,
DENG Zi-chen^2,3

¹Department of Engineering Mechanics, School of Civil Engineering,Suzhou University of Science and Technology, Suzhou, Jiangsu 215011, P.R.China;²Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, P.R.China;³State Key Laboratory of Structural Analysis of Industrial Equipment,Dalian University of Technology, Dalian, Liaoning 116023, P.R.China

Funds: The National Natural Science Foundation of China(11202146)

摘要

摘要: 给出分数阶FornbergWhitham方程(FFW)并把其中非线性项uu_x换为u²u_x后所得的改进Fornberg-Whitham方程的解．使用了分数阶变分迭代法(fractional variational iteration method，FVIM)，其中Lagrange乘子由泛函和Laplace变换确定．讨论了分数阶次的数值在两种情况下FFW方程的解，因为确定FFW方程中时间微分的阶次需要比较原方程中含时间的两个微分的阶次．最后，给出两个使用分数阶变分迭代法的算例．算例结果证明了所提方法的有效性
- 分数阶Fornberg-Whitham方程 /
- 分数阶变分迭代法 /
- Lagrange乘子 /
- 近似解 /
- 初值问题
Abstract: The solutions to the fractional FornbergWhitham (FFW) equation and the modified FFW equation generated by change of one nonlinear term uu_x with u²u_x were presented. The fractional variational iteration method (FVIM) was used, in which the Lagrange multiplier was determined with the variational function and the Laplace transformation. Two cases were discussed respectively for the FFW equation because the order of time differentiation was determined through comparison of the two derivatives’orders in the fractional differential equation. Finally, two numerical examples of the FVIM solution were given. The computational results demonstrate the high efficiency of the presented method.
- fractional FornbergWhitham equation /
- fractional variational iteration method /
- Lagrange multiplier /
- approximate solution /
- initial value problem

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参考文献(34)

[1]	孙文, 孙洪广, 李西成. 力学与工程问题的分数阶导数建模[M]. 北京:科学出版社, 2010.（SUN Wen, SUN Hong-guang, LI Xi-cheng. Modeling Using the Fractional Derivative in Mechanics and Engineering Problems [M]. Beijing: Science Press, 2010.(in Chinese)）
[2]	Miller K S, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations [M]. New York: Wiley, 1993.
[3]	Oldham K B, Spanier J. The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order [M]. New York: Academic Press, 1974.
[4]	Debnath L. Fractional integrals and fractional differential equations in fluid mechanics[J]. Fractional Calculus & Applied Analysis,2003, 6(2): 119-155.
[5]	Podlubny I. Fractional Differential Equations [M]. New York: Academic Press, 1999.
[6]	Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations [M]. Amsterdam: Elsevier, 2006.
[7]	DENG Wei-hua. Short memory principle and a predictor-corrector approach for fractional differential equations[J]. Journal of Computational and Applied Mathematics,2007,206(1): 174-188.
[8]	Liu F W, Anh V, Turner I. Numerical solution of the space fractional Fokker-Planck equation[J]. Journal of Computational and Applied Mathematics,2004,166(1): 209-219.
[9]	Odibat Z, Momani S. A generalized differential transform method for linear partial differential equations of fractional order[J].Applied Mathematics Letters,2008,21(2): 194-199.
[10]	LIAO Shi-jun. A short review on the homotopy analysis method in fluid mechanics[J]. Journal of Hydrodynamics, Series B,2010,22(5): 882-884.
[11]	LI Chang-pin, WANG Yi-hong. Numerical algorithm based on Adomian decomposition for fractional differential equations[J]. Computers & Mathematics With Applications,2009, 57(10): 1672-1681.
[12]	Duan J S, Rach R, Buleanu D, Wazwaz A M. A review of the Adomian decomposition method and its applications to fractional differential equations[J]. Communications in Fractional Calculus,2012, 3(2): 73-99.
[13]	Momani S, Odibat Z. Homotopy perturbation method for nonlinear partial differential equations of fractional order[J]. Physics Letters A,2007, 365(5/6): 345-350.
[14]	HE Ji-huan. Variational iteration method for delay differential equations[J]. Communications in Nonlinear Science and Numerical Simulation,1997, 2(4): 230-235.
[15]	GUO Shi-min, MEI Li-quan, LI Ying. Fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation[J]. Applied Mathematics and Computation,2013, 219(11): 5909-5917.
[16]	HE Ji-huan, WU Xu-hong. Variational iteration method: new development and applications[J]. Computers & Mathematics With Applications,2007, 54(7/8): 881-894.
[17]	HE Ji-huan. Asymptotic methods for solitary solutions and compactons[J]. Abstract and Applied Analysis,2012: 916793.
[18]	莫嘉琪, 张伟江, 陈贤峰. 一类强非线性发展方程孤波变分迭代解法[J]. 物理学报, 2009, 58(11): 7397-7401.（MO Jia-qi, ZHANG Wei-jiang, CHEN Xian-feng. Variational iteration method for solving a class of strongly nonlinear evolution equations[J]. Acta Physica Sinica,2009, 58(11): 7397-7401.（in Chinese））
[19]	Abbasbandy S. A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials[J]. Journal of Computational and Applied Mathematics,2007, 207(1): 59-63.
[20]	Noor M A, Mohyud-Din S T. Variational iteration method for solving higher-order nonlinear boundary value problems using He’s polynomials[J]. International Journal of Nonlinear Sciences and Numerical Simulation,2008, 9(2): 141-156.
[21]	GENG Fa-zhan. A modified variational iteration method for solving Riccati differential equations[J]. Computers & Mathematics With Applications,2010, 60 (7): 1868-1872.
[22]	Ghorbani A, Momani S. An effective variational iteration algorithm for solving Riccati differential equations[J]. Applied Mathematics Letters,2010, 23(8): 922-927.
[23]	HE Bin, MENG Qing, LI Shao-lin. Explicit peakon and solitary wave solutions for the modified Fornberg-Whitham equation[J]. Applied Mathematics and Computation,2010, 217(5): 1976-1982.
[24]	Fornberg B, Whitham G B. A numerical and theoretical study of certain nonlinear wave phenomena[J]. Phil Trans R Soc A,1978, 289: 373-404.
[25]	Abidi F, Omrani K. The homotopy analysis method for solving the Fornberg-Whitham equation and comparison with Adomian’s decomposition method[J]. Computers & Mathematics With Applications,2010, 59(8): 2743-2750.
[26]	Gupta P K, Singh M. Homotopy perturbation method for fractional Fornberg-Whitham equation[J]. Computers & Mathematics With Applications,2011, 61(2): 250-254.
[27]	Saker M G, Erdogan F, Yildirim A. Variational iteration method for the time fractional Fornberg-Whitham equation[J]. Computers & Mathematics With Applications,2012, 63(9): 1382-1388.
[28]	Merdan M, Gokdogan A, Yildirim A, Mohyud-Din S T. Numerical simulation of fractional Fornberg-Whitham equation by differential transformation method[J]. Abstract and Applied Analysis,2012, 2012: 1-8.
[29]	Lu J. An analytical approach to the Fornberg-Whitham type equations by using the variational iteration method[J]. Computers & Mathematics With Applications,2011, 61(8): 2010-2013.
[30]	Javidi M, Raji M A. Combination of Laplace transform and homotopy perturbation method to solve the parabolic partial differential equations[J]. Commun Fract Calc,2012, 3(1): 10-19.
[31]	Singha J, Vitae A, Kumarb D, Vitae A, Kumar S. New treatment of fractional Fornberg-Whitham equation via Laplace transform[J]. Ain Shams Engineering Journal,2013, 4(3): 557-562.
[32]	Zeng D Q, Qin Y M. The Laplace-Adomian-Pade technique for the seepage flows with the Riemann-Liouville derivatives[J]. Commun Fract Calc,2012, 3(1): 26-29.
[33]	Tsai P Y, Chen C K. An approximate analytic solution of the nonlinear Riccati differential equation[J]. Journal of the Franklin Institute,2010, 347(10): 1850-1862.
[34]	WU Guo-cheng, Baleanu D. Variational iteration method for the Burgers’ flow with fractional derivatives—new Lagrange multipliers[J]. Applied Mathematical Modelling,2013, 37(9): 6183-6190.

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分数阶Fornberg-Whitham方程及其改进方程的变分迭代解

doi: 10.3879/j.issn.1000-0887.2013.12.002

作者简介:
鲍四元（1980—），男，安徽人，副教授，博士(通讯作者. E-mail: bsiyuan@126.com)；邓子辰(1964—),男,西安人，教授,博士生导师(E-mail: dweifan@nwpu.edu.cn).

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Variational Iteration Solutions for Fractional FornbergWhitham Equation and Its Modified Equation

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留言板

分数阶Fornberg-Whitham方程及其改进方程的变分迭代解

doi: 10.3879/j.issn.1000-0887.2013.12.002

作者简介: 鲍四元（1980—），男，安徽人，副教授，博士(通讯作者. E-mail: bsiyuan@126.com)；邓子辰(1964—),男,西安人，教授,博士生导师(E-mail: dweifan@nwpu.edu.cn).

计量

出版历程

Variational Iteration Solutions for Fractional FornbergWhitham Equation and Its Modified Equation

计量

出版历程

目录

作者简介:
鲍四元（1980—），男，安徽人，副教授，博士(通讯作者. E-mail: bsiyuan@126.com)；邓子辰(1964—),男,西安人，教授,博士生导师(E-mail: dweifan@nwpu.edu.cn).