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The approximate homotopy symmetry reduction for far-field model equation

Jiao Xiao-Yu

The approximate homotopy symmetry reduction for far-field model equation

Jiao Xiao-Yu
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  • The far-field model equation is investigated by the approximate homotopy symmetry method. Homotopy series solutions are constructed through summarizing the relevant general formulas for similarity reduction solutions and similarity reduction equations of differernt orders. Similarity reduction equations of different orders are linear variable coefficients ordinary differential equations, and can be solved one by one from zero-order similarity reduction equations. The auxiliary parameter in the homotopy model affects convergence of homotopy series solutions.
    • Funds:
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    Olver P J 1993 Applications of Lie Group to Differential Equations (2nd ed) (New York: Springer)

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    Bluman G W, Cole J D 1974 Similarity Methods for Differential Equations (Berlin: Springer)

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    Bluman G W, Kumei S 1989 Symmetries and Differential Equations (Berlin: Springer)

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    Cole J D 1968 Perturbation Methods in Applied Mathematics (Waltham: Blaisdell)

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    Van Dyke M 1975 Perturbation Methods in Fluid Mechanics (Stanford: Parabolic Press)

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    Nayfeh A H 2000 Perturbation Methods (New York: John Wiley Sons)

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    Baikov V A, Gazizov R K, Ibragimov N H 1988 Mat. Sb. 136 435

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    Fushchich W I, Shtelen W M 1989 J. Phys. A 22 L887

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    Pakdemirli M, Yurusoy M, Dolapci I T 2004 Acta Appl. Math. 80 243

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    Wiltshire R 2006 J. Comput. Appl. Math. 197 287

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    Jiao X Y, Yao R X, Lou S Y 2008 J. Math. Phys. 49 093505

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    Jia M, Wang J Y, Lou S Y 2009 Chin. Phys. Lett. 26 020201

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    Yao R X, Jiao X Y, Lou S Y 2009 Commun. Theor. Phys. 51 785

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    Jiao X Y, Yao R X, Zhang S L, Lou S Y 2009 Z. Naturforsch. A 64 676

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    Zhao Y, Zhang S L, Lou S Y 2009 Chin. Phys. Lett. 26 100201

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    Liao S J 1992 Ph.D. Dissertation (Shanghai: Shanghai Jiaotong University) (in Chinese) [廖世俊 1992 博士学位论文 (上海: 上海交通大学)]

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    Liao S J 1999 Int. J. Non-Linear Mech. 34 759

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    Liao S J 2002 J. Fluid Mech. 453 411

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    Liao S J 2003 Beyond Perturbation: Introduction to the Homotopy Analysis Method (Boca Raton: Chapman Hall CRC Press)

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    [39]

    Liao S J 2004 Appl. Math. Comput. 147 499

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    Liao S J 2008 Adv. Mech. 38 1 (in Chinese) [廖世俊 2008 力学进展 38 1]

    [41]
    [42]
    [43]

    Lyapunov A M 1992 General Problem of the Stability of Motion (London: Taylor and Francis) p29

    [44]

    Karmishin A V, Zhukov A I, Kolosov V G 1990 Methods of Dynamics Calculation and Testing for Thin-Walled Structures (Moscow: Mashinostroyenie) p52 (in Russian)

    [45]
    [46]

    Adomian G 1994 Solving Frontier Problems of Physics: The Decomposition Method (Boston, London: Kluwer Academic Publishers)

    [47]
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    Jiao X Y, Gao Y, Lou S Y 2009 Sci. China G 39 964 (in Chinese) [焦小玉、高 原、楼森岳 2009 中国科学G 39 964]

    [49]
    [50]

    Dai H H 1998 Acta Mech. 127 293

    [51]
    [52]
    [53]

    Dai H H, Huo Y 2000 Proc. R. Soc. Lond. A 456 331

    [54]

    Camassa R, Holm D 1993 Phys. Rev. Lett. 71 1661

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    [56]

    Benjamin T B, Bona J L, Mahony J J 1972 Phil. Trans. R. Soc. Lond. A 272 47

    [57]
  • [1]

    Olver P J 1993 Applications of Lie Group to Differential Equations (2nd ed) (New York: Springer)

    [2]

    Bluman G W, Cole J D 1974 Similarity Methods for Differential Equations (Berlin: Springer)

    [3]
    [4]
    [5]

    Bluman G W, Kumei S 1989 Symmetries and Differential Equations (Berlin: Springer)

    [6]
    [7]

    Cole J D 1968 Perturbation Methods in Applied Mathematics (Waltham: Blaisdell)

    [8]
    [9]

    Van Dyke M 1975 Perturbation Methods in Fluid Mechanics (Stanford: Parabolic Press)

    [10]
    [11]

    Nayfeh A H 2000 Perturbation Methods (New York: John Wiley Sons)

    [12]
    [13]

    Baikov V A, Gazizov R K, Ibragimov N H 1988 Mat. Sb. 136 435

    [14]

    Fushchich W I, Shtelen W M 1989 J. Phys. A 22 L887

    [15]
    [16]
    [17]

    Pakdemirli M, Yurusoy M, Dolapci I T 2004 Acta Appl. Math. 80 243

    [18]
    [19]

    Wiltshire R 2006 J. Comput. Appl. Math. 197 287

    [20]
    [21]

    Jiao X Y, Yao R X, Lou S Y 2008 J. Math. Phys. 49 093505

    [22]
    [23]

    Jia M, Wang J Y, Lou S Y 2009 Chin. Phys. Lett. 26 020201

    [24]
    [25]

    Yao R X, Jiao X Y, Lou S Y 2009 Commun. Theor. Phys. 51 785

    [26]
    [27]

    Jiao X Y, Yao R X, Zhang S L, Lou S Y 2009 Z. Naturforsch. A 64 676

    [28]
    [29]

    Zhao Y, Zhang S L, Lou S Y 2009 Chin. Phys. Lett. 26 100201

    [30]
    [31]

    Liao S J 1992 Ph.D. Dissertation (Shanghai: Shanghai Jiaotong University) (in Chinese) [廖世俊 1992 博士学位论文 (上海: 上海交通大学)]

    [32]

    Liao S J 1999 Int. J. Non-Linear Mech. 34 759

    [33]
    [34]
    [35]

    Liao S J 2002 J. Fluid Mech. 453 411

    [36]
    [37]

    Liao S J 2003 Beyond Perturbation: Introduction to the Homotopy Analysis Method (Boca Raton: Chapman Hall CRC Press)

    [38]
    [39]

    Liao S J 2004 Appl. Math. Comput. 147 499

    [40]

    Liao S J 2008 Adv. Mech. 38 1 (in Chinese) [廖世俊 2008 力学进展 38 1]

    [41]
    [42]
    [43]

    Lyapunov A M 1992 General Problem of the Stability of Motion (London: Taylor and Francis) p29

    [44]

    Karmishin A V, Zhukov A I, Kolosov V G 1990 Methods of Dynamics Calculation and Testing for Thin-Walled Structures (Moscow: Mashinostroyenie) p52 (in Russian)

    [45]
    [46]

    Adomian G 1994 Solving Frontier Problems of Physics: The Decomposition Method (Boston, London: Kluwer Academic Publishers)

    [47]
    [48]

    Jiao X Y, Gao Y, Lou S Y 2009 Sci. China G 39 964 (in Chinese) [焦小玉、高 原、楼森岳 2009 中国科学G 39 964]

    [49]
    [50]

    Dai H H 1998 Acta Mech. 127 293

    [51]
    [52]
    [53]

    Dai H H, Huo Y 2000 Proc. R. Soc. Lond. A 456 331

    [54]

    Camassa R, Holm D 1993 Phys. Rev. Lett. 71 1661

    [55]
    [56]

    Benjamin T B, Bona J L, Mahony J J 1972 Phil. Trans. R. Soc. Lond. A 272 47

    [57]
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Publishing process
  • Received Date:  28 January 2011
  • Accepted Date:  17 August 2011
  • Published Online:  15 December 2011

The approximate homotopy symmetry reduction for far-field model equation

  • 1. School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210046, China; Department of Physics, Shanghai Jiaotong University, Shanghai 200240, China

Abstract: The far-field model equation is investigated by the approximate homotopy symmetry method. Homotopy series solutions are constructed through summarizing the relevant general formulas for similarity reduction solutions and similarity reduction equations of differernt orders. Similarity reduction equations of different orders are linear variable coefficients ordinary differential equations, and can be solved one by one from zero-order similarity reduction equations. The auxiliary parameter in the homotopy model affects convergence of homotopy series solutions.

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