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The variational iteration method for characteristic problem of strong damping generalized sine-Gordon equation

Xu Yong-Hong Shi Lan-Fang Mo Jia-Qi

The variational iteration method for characteristic problem of strong damping generalized sine-Gordon equation

Xu Yong-Hong, Shi Lan-Fang, Mo Jia-Qi
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  • A class of nonlinear strong damping sine-Gordon disturbed evolution differential equation is studied which appears widely in mathematics and mechanics. Firstly, we introduce a traveling wave transformation, and obtain the exact solution of degenerate equation. Then a functional calculating method for variational iteration is constructed, thus an iterative expansion is found. Finally, the approximate traveling wave analytic solutions for the original strong damping generalized sine-Gordon disturbed evolution equation are found. The arbitrary order approximate solutions, and the simple variational iteration method are obtained with higher accuracy. The approximate analytic solution can make up for the imperfection of the simple numerical simulation solution.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11202106), the Fundamental Research Funds for the Central Universities, China (Grant No. 2232012D3-34), the Natural Science Foundation of the Education Department of Anhui Province, China (Grant No. KJ2014A151) and the Natural Sciences Foundation from the Universities of Jiangsu Province, China (Grant No. 13KJB170016).
    [1]

    Parkes E J 2008 Chaos Solitons Fractals 38 154

    [2]

    Sirendaoreji J S 2003 Phys. Lett. A 309 387

    [3]

    McPhaden M J, Zhang D 2002 Nature 415 603

    [4]

    Gu D F 1997 Science 275 805

    [5]

    Wu J P 2011 Chin. Phys. Lett. 28 060207

    [6]

    Zuo J M, Zhang Y M 2011 Chin. Phys. B 20 010205

    [7]

    Pang J, Jin L H, Zhao Q 2012 Acta Phys. Sin. 61 140201 (in Chinese) [庞晶, 靳玲花, 赵强 2012 物理学报 61 140201]

    [8]

    Xin P, Liu X Q, Zhang L L 2011 Chin. Phys. Lett. 28 020201

    [9]

    Li Ning, Liu Xi 2013 Acta Phys. Sin. 62 160203 (in Chinese) [李宁, 刘希 2013 物理学报 62 160203]

    [10]

    Xu Y H, Lin W T, Xu H, Yao J S, Mo J Q 2012 J. Lanzhou Univ. 48 100 (in Chinese) [许永红, 林万涛, 徐惠, 姚静荪, 莫嘉琪 2012 兰州大学学报 48 100]

    [11]

    Shi L F, Lin W T, Lin Y H, Mo J Q 2013 Acta. Phys. Sin. 62 010201 (in Chinese) [石兰芳, 林万涛, 林一骅, 莫嘉琪 2013 物理学报 62 010201]

    [12]

    Mo Jiaqi 2009 Science in China, Ser. G 52 1007

    [13]

    Mo J Q, Chen X F 2010 Chin. Phys. B 19 100203

    [14]

    Mo J Q 2010 Chin. Phys. 19 010203

    [15]

    Mo J Q, Lin Y H, Lin W T 2010 Chin. Phys. 19 030202

    [16]

    2011 Commun Theor. Phy. 55 387

    [17]

    Mo J Q, Lin W T 2011 J. Sys. Sci. Complexity 24 271

    [18]

    Mo J Q, Lin W T, Lin Y H 2011 Chin. Phys. B 20 070205

    [19]

    Mo J Q, Lin W T, Lin Y H 2011 Chin. Phys. B 20 010208

    [20]

    He J H 2002 Approximate Nonlinear Analytical Methods in Engineering and Sciences (Zhengzhou: Henan Science and Technology Press) (in Chinese) [何吉欢 2002 工程和科学计算中的近似非线性分析方法 郑州: 河南科学技术出版社]

    [21]

    Lebedev L P, Cloud M J 2003 The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics (New York: World Scientific)

    [22]

    de Jager E M, Jiang F R 1996 The Theory of Singular Perturbation (Amsterdam: North-Holland Publishing Co)

    [23]

    Barbu L, Morosanu G 2007 Singularly Perturbed Boundary-Value Problem, (Basel: Birkhauserm Verlag AG)

    [24]

    He J H 1999 J. Non-Linear Mech 34 699

    [25]

    He J H 2000 Appl. Math. Comput 114 115

    [26]

    He J H 2004 Chaos, Solitons & Fractals 19 847

    [27]

    He J H, Wu X H 2006 Chaos, Solitons & Fractals 29 108

    [28]

    He J H 2007 Chaos, Solitons & Fractals 34 1430

    [29]

    He J H, Wu G C 2010 Nonlinear Sci. Lett. A 1 1

    [30]

    He J H 2010 J. Nonlinear Scoences and Nimweical Simulation 11 555

    [31]

    Wang Q, Fu F H 2012 Int. J. Engineering and Manufacturing 2 36

  • [1]

    Parkes E J 2008 Chaos Solitons Fractals 38 154

    [2]

    Sirendaoreji J S 2003 Phys. Lett. A 309 387

    [3]

    McPhaden M J, Zhang D 2002 Nature 415 603

    [4]

    Gu D F 1997 Science 275 805

    [5]

    Wu J P 2011 Chin. Phys. Lett. 28 060207

    [6]

    Zuo J M, Zhang Y M 2011 Chin. Phys. B 20 010205

    [7]

    Pang J, Jin L H, Zhao Q 2012 Acta Phys. Sin. 61 140201 (in Chinese) [庞晶, 靳玲花, 赵强 2012 物理学报 61 140201]

    [8]

    Xin P, Liu X Q, Zhang L L 2011 Chin. Phys. Lett. 28 020201

    [9]

    Li Ning, Liu Xi 2013 Acta Phys. Sin. 62 160203 (in Chinese) [李宁, 刘希 2013 物理学报 62 160203]

    [10]

    Xu Y H, Lin W T, Xu H, Yao J S, Mo J Q 2012 J. Lanzhou Univ. 48 100 (in Chinese) [许永红, 林万涛, 徐惠, 姚静荪, 莫嘉琪 2012 兰州大学学报 48 100]

    [11]

    Shi L F, Lin W T, Lin Y H, Mo J Q 2013 Acta. Phys. Sin. 62 010201 (in Chinese) [石兰芳, 林万涛, 林一骅, 莫嘉琪 2013 物理学报 62 010201]

    [12]

    Mo Jiaqi 2009 Science in China, Ser. G 52 1007

    [13]

    Mo J Q, Chen X F 2010 Chin. Phys. B 19 100203

    [14]

    Mo J Q 2010 Chin. Phys. 19 010203

    [15]

    Mo J Q, Lin Y H, Lin W T 2010 Chin. Phys. 19 030202

    [16]

    2011 Commun Theor. Phy. 55 387

    [17]

    Mo J Q, Lin W T 2011 J. Sys. Sci. Complexity 24 271

    [18]

    Mo J Q, Lin W T, Lin Y H 2011 Chin. Phys. B 20 070205

    [19]

    Mo J Q, Lin W T, Lin Y H 2011 Chin. Phys. B 20 010208

    [20]

    He J H 2002 Approximate Nonlinear Analytical Methods in Engineering and Sciences (Zhengzhou: Henan Science and Technology Press) (in Chinese) [何吉欢 2002 工程和科学计算中的近似非线性分析方法 郑州: 河南科学技术出版社]

    [21]

    Lebedev L P, Cloud M J 2003 The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics (New York: World Scientific)

    [22]

    de Jager E M, Jiang F R 1996 The Theory of Singular Perturbation (Amsterdam: North-Holland Publishing Co)

    [23]

    Barbu L, Morosanu G 2007 Singularly Perturbed Boundary-Value Problem, (Basel: Birkhauserm Verlag AG)

    [24]

    He J H 1999 J. Non-Linear Mech 34 699

    [25]

    He J H 2000 Appl. Math. Comput 114 115

    [26]

    He J H 2004 Chaos, Solitons & Fractals 19 847

    [27]

    He J H, Wu X H 2006 Chaos, Solitons & Fractals 29 108

    [28]

    He J H 2007 Chaos, Solitons & Fractals 34 1430

    [29]

    He J H, Wu G C 2010 Nonlinear Sci. Lett. A 1 1

    [30]

    He J H 2010 J. Nonlinear Scoences and Nimweical Simulation 11 555

    [31]

    Wang Q, Fu F H 2012 Int. J. Engineering and Manufacturing 2 36

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  • Received Date:  24 July 2014
  • Accepted Date:  14 August 2014
  • Published Online:  05 January 2015

The variational iteration method for characteristic problem of strong damping generalized sine-Gordon equation

  • 1. Department of Mathematics & Physics, Bengbu College, Bengbu 233030, China;
  • 2. College of Mathematics and Statistics, Nanjing University of Information Science & Technology, Nanjing 210044, China;
  • 3. Department of Mathematics, Anhui Normal University, Wuhu 241003, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 11202106), the Fundamental Research Funds for the Central Universities, China (Grant No. 2232012D3-34), the Natural Science Foundation of the Education Department of Anhui Province, China (Grant No. KJ2014A151) and the Natural Sciences Foundation from the Universities of Jiangsu Province, China (Grant No. 13KJB170016).

Abstract: A class of nonlinear strong damping sine-Gordon disturbed evolution differential equation is studied which appears widely in mathematics and mechanics. Firstly, we introduce a traveling wave transformation, and obtain the exact solution of degenerate equation. Then a functional calculating method for variational iteration is constructed, thus an iterative expansion is found. Finally, the approximate traveling wave analytic solutions for the original strong damping generalized sine-Gordon disturbed evolution equation are found. The arbitrary order approximate solutions, and the simple variational iteration method are obtained with higher accuracy. The approximate analytic solution can make up for the imperfection of the simple numerical simulation solution.

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