PM  >> Vol. 7 No. 5 (September 2017)

    分数阶广义KDV方程的精确解
    Travelling Wave Solution of the Generalized KDV Equation

  • 全文下载: PDF(403KB) HTML   XML   PP.378-385   DOI: 10.12677/PM.2017.75049  
  • 下载量: 164  浏览量: 507   国家自然科学基金支持

作者:  

王小娇,周贤林,韦方棋:四川师范大学数学与软件科学学院,四川 成都

关键词:
复变换椭圆函数展开法修正Riemann-Liouville函数分数阶广义KDV方程Complex-Transform-Cn Expansion Method Modified Riemann-Liouville Derivative Fractional Generalized KDV Equation

摘要:

本文将分数阶复变换方法和椭圆函数展开法相结合,给出了求解分数阶广义KDV方程的复变换椭圆函数展开法。进而得到了分数阶广义KDV方程的周期波解和孤立波解。

By combining the fractional transform with Cn-expansion method, we give the improved elliptic expansion method to solve the generalized fraction KDV equations, and obtain some new periodic solution and solitary wave solutions.

文章引用:
王小娇, 周贤林, 韦方棋. 分数阶广义KDV方程的精确解[J]. 理论数学, 2017, 7(5): 378-385. https://doi.org/10.12677/PM.2017.75049

参考文献

[1] Rosenau, P. (1997) On Nonanalytic Solitary Waves Formed by a Nonlinear Dispersion. Physical Letters A, 230, 305-318.
https://doi.org/10.1016/S0375-9601(97)00241-7
[2] Lai, S.Y. and Yin, Q. (2001) A New Study for the Modified Nonlinear Dispersive mK(m,n)Equations in Higher Dimensional Spaces. Journal of Pure and Applied Mathematical Science in China(Series A), 44, 396-401.
[3] Pelinovsky, D. and Grimshaw, R. (1996) An Asymptotic Approach to Solitary Wave Instability and Critical Collapse in Long-Wave KdV-Type Evolution Equations. Physica D, 98, 139-155.
https://doi.org/10.1016/0167-2789(96)00093-0
[4] Klaus, M., Pelinovsky, D. and Rothos, V. (2006) Evans Function for Lax Operators with Algebraically Decaying Potentials. Journal of Nonlinear Science, 16, 1-44.
https://doi.org/10.1007/s00332-005-0652-7
[5] Pelinovsky, D., Afanasjev, V. and Kivshar, V. (1996) Nonlinear Theory of Oscillating, Decaying and Collapsing Solitions in the Generalized Nonlinear Schrodinger Equation. Physical Review E, 53, 1940-1953.
https://doi.org/10.1103/PhysRevE.53.1940
[6] Wazwaz, A.M. (2006) Compactons and Soltary Wave Solutions for the Bousinesq Wave Equation and Its Generalized Form. Applied Mathematics Computation, 182, 529-535.
https://doi.org/10.1016/j.amc.2006.04.014
[7] Wazwaz, A.M. (2006) The Sine-Cosine and the Tanh Methods: Reliable Tools for Analytic Treatment of Nonlinear Dispersive Equations. Applied Mathematics Computation, 173, 150-164.
https://doi.org/10.1016/j.amc.2005.02.047
[8] Wazwaz, A.M. (2005) Nonlinear Variants of the Improved Boussinesq Equation with Compact and Noncompact Structures. Applied Mathematics Computation, 49, 565-574.
https://doi.org/10.1016/j.camwa.2004.07.016
[9] Wazwaz, A.M. (2006) Kinks and Solitons Solutions for the Generalized KdV Equation with Two Power Nonlinearities. Applied Mathematics Computation, 183, 1181-1189.
https://doi.org/10.1016/j.amc.2006.06.042
[10] Wazwaz, A.M. (2001) A Study of Nonlinear Dispersive Equations with Solitary-Wave Solutions Having Compact Support. Mathematics and Computers in Simulation, 56, 269-276.
https://doi.org/10.1016/S0378-4754(01)00291-9
[11] Ismail, M.S. and Taha, T.R. (1998) A Numerical Study of Compactons. Mathematics and Computers in Simulation, No. 47, 519-530.
[12] Jumarie, G. (2006) Modified Riemann-Liouville Derivative and Fractional Taylor Series of Non-Differentiable Functions Further Results. Computers and Mathematics with Applications, 51, 1367-1376.
[13] Sabatier, J., Agrawal, O.P. and Tenreiro Machado, J.A. (2007) Advances in Fractional Calculus: Theoretical Develop¬ments and Applications in Physics and Engineering. Springer, 50, 1648-1650.
[14] Baleanu, D., Diethelm, K., Scalas, E., et al. (2012) Fractional Calculus Models and Numerical Methods in Series on Complexity, Monlinearity and Chaos. World Scientific, Singapore, 216, 67-75.
[15] Liu, Y.Q. and Xin, B.G. (2011) Numerical Solutions of a Fractional Predator-Prey System. Advances in Difference Equations, 51, 1159-1162.
[16] Wang, M.L., Li, X.Z. and Zhang, J.L. (2008) The G/G-Expansion Method and Travelling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics. Physics Letters A, 37, 417-423.
[17] Liu, Y.Q. and Yan, L.M. (2013) Solutions of Fractional Konopelchenko-Dubrovsky and Nizhnik-Novikov-Veselov Equations using a Generalized Fractional Sub-Equation Method. Abstract and Applied Analysis, 11, 515-521.
[18] Khan, N.A., Ara, A. and Mahmood, A. (2012) Numerical Solutions of Time Fractional Burgers Equations: A Compar¬ison between Generalized Differential Transformation Technique and Homotopy Perturbation Methon. International Journal of Numerical Methods for Heat and Fluid Flow, 24, 175-193.
https://doi.org/10.1108/09615531211199818
[19] He, J.H. (2003) Homotopy Perturbation Method: A New Nonlinear Analytical Technique. Applied Mathematics and Computation, 135, 73-79.
[20] Li, Z.B. and He, J.H. (2010) Fractional Complex Transform for Fractional Differential Equations. Mathematical and Computational Applications, 15, 970-973.
https://doi.org/10.3390/mca15050970
[21] 刘式达, 傅遵涛. 一类非线性方程的新周期解[J]. 物理学报, 2002, 51(1): 10-14.