Exact Travelling Solutions for the Sixth-order Boussinesq Equation
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Authors
M. Hosseini
- Department of Mechanical Engineering, Islamic Azad University, Ghaemshahr, Iran
H. Abdollahzadeh
- University of Kurdistan, Department of Industrial Engineering, Sanandaj, Iran
M. Abdollahzadeh
- Babol University of Technology, Department of Mechanical Engineering, P. O. Box: 484, Babol, Iran
Abstract
In this paper, we establish some distinct exact solutions for a nonlinear evolution equation. The sin-cosine method and the rational Exp-Function and the rational hyperbolic function method are used to construct the solitary travelling wave solutions of the sixth-order Boussinesq equation . These solutions may be important of significance for the explanation of some practical physical problem.
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ISRP Style
M. Hosseini, H. Abdollahzadeh, M. Abdollahzadeh, Exact Travelling Solutions for the Sixth-order Boussinesq Equation, Journal of Mathematics and Computer Science, 2 (2011), no. 2, 376--387
AMA Style
Hosseini M., Abdollahzadeh H., Abdollahzadeh M., Exact Travelling Solutions for the Sixth-order Boussinesq Equation. J Math Comput SCI-JM. (2011); 2(2):376--387
Chicago/Turabian Style
Hosseini, M., Abdollahzadeh, H., Abdollahzadeh, M.. "Exact Travelling Solutions for the Sixth-order Boussinesq Equation." Journal of Mathematics and Computer Science, 2, no. 2 (2011): 376--387
Keywords
- Traveling wave solutions
- sin-cosine method
- Exponential rational function method
- the rational hyperbolic functions methods
- the sixth-order Boussinesq equation
MSC
References
-
[1]
G. Eilenberger, Solitons: mathematical methods for physicists, Springer-Verlag, Berlin (1981)
-
[2]
G. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York (1974)
-
[3]
P. Gray, S. Scott, Chemical Oscillations and Instabilities, Clarendon Press, Oxford (1990)
-
[4]
A. Hasegawa, Plasma Instabilities and Nonlinear Effects, Springer-Verlag, Berlin (1975)
-
[5]
C. S. Gardner, J. M. Green, D. Kruskal, R. M. Miura, Method for solving the Korteweg-deVries equation, Phys. Rev. Lett., 19 (1967), 1095--1105
-
[6]
M. J. Ablowitz, H. Segur, Solitons and inverse scattering transform, SIAM, Philadelphia (1981)
-
[7]
R .Hirota, Direct method of finding exact solutions of nonlinear evolution equations, in: Backlund transformations, 1980 (1980), 1157--1175
-
[8]
M. Wadati, H. Sanuki, K. Konno, Relationships among inverse method, Backlund transformation and an infinite number of conservation laws, Prog. Theor. Phys., 53 (1975), 419--436
-
[9]
M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge (1991)
-
[10]
A. Coely, Bäcklund and Darboux Transformations, American Mathematical Society, Providence (2001)
-
[11]
V. B. Matveev, M. A. Salle, Darboux Transformations and Solitons, Springer-Verlag, Berlin (1991)
-
[12]
W. Malfliet, Solitary wave solutions of nonlinear wave equations, Amer. J. Phys., 60 (1992), 650--654
-
[13]
W. Malfliet, W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 54 (1996), 563--568
-
[14]
D. D. Ganji, M. Abdollahzadeh, Exact travelling solutions for the Lax’s seventh-order KdV equation by sech method and rational exp-function method, Applied Mathematics and Computation, 206 (2008), 438--444
-
[15]
E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 227 (2000), 212--218
-
[16]
J.-H. He , X.-H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 30 (2006), 700--708
-
[17]
J.-H. He, M. A. Abdou, New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos Solitons Fractals, 34 (2007), 1421--1429
-
[18]
X.-H. Wu , J.-H. He, EXP-function method and its application to nonlinear equations, Chaos Solitons Fractals, 38 (2008), 903--910
-
[19]
Z. Z. Ganji, D. D. Ganji, H. Bararnia, Approximate general and explicit solutions of nonlinear BBMB equations by Exp-Function method, Applied Mathematical Modelling, 33 (2009), 1836--1841
-
[20]
A.-M. Wazwaz, The sine–cosine method for obtaining solutions with compact and noncompact structures, Applied Mathematics and Computation, 159 (2004), 559--576
-
[21]
A.-M. Wazwaz, M. A. Helal, Nonlinear variants of the BBM equation with compact and noncompact physical structures, Chaos Solitons Fractals, 26 (2005), 767--776
-
[22]
A.-M. Wazwaz, A sine-cosine method for handling nonlinear wave equations, Mathematical And Computer Modeling, 40 (2004), 499--508
-
[23]
S. Liu, Z. Fu, S. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001), 69--74
-
[24]
Z. Fu, S. Liu, S. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A, 290 (2001), 72--76
-
[25]
E. J. Parkes , B. R. Duffy, P. C. Abbott, The Jacobi elliptic-function method for finding periodic- wave solutions to nonlinear evolution equations, Phys. Lett. A, 295 (2002), 280--286
-
[26]
D. D. Ganji, M. Abdollahzadeh, Exact traveling solutions of some nonlinear evolution equation by (G’/G)-expansion method, Journal of Mathematical Physics, 50 (2009), 11 pages
-
[27]
M. Abdollahzadeh, M. Ghanbarpour, M. Hosseini, H. Ebrahimnejad Shirvani, Exact travelling solutions for fifth order Caudrey-Dodd-Gibbon equation, International Journal of Applied Mathematics and Computation, 4 (2010), 81--90
-
[28]
P. J. Olver, Application of Lie Group to Differential Equation, Springer, New York (1986)
-
[29]
A. Sadighi, D. D. Ganji, Exact solutions of nonlinear diffusion equations by variational iteration method, Computers and Mathematics with Applications, 54 (2007), 1112--1121
-
[30]
D. D. Ganji, M. Rafei, Solitary wave solutions for a generalized Hirota–Satsuma coupled KdV equation by homotopy perturbation method, Phys. Lett. A, 356 (2006), 131--137
-
[31]
D. D. Ganji, The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer, Phys. Lett. A, 355 (2006), 337--341
-
[32]
D. D. Ganji, G. A. Afrouzi, R. A. Talarposhti, Application of variational iteration method and homotopy–perturbation method for nonlinear heat diffusion and heat transfer equations, Phys. Lett. A, 368 (2007), 450--457
-
[33]
E. Fan, H. Q. Zhang, A note on the homogeneous balance method, Phys. Lett. A, 246 (1998), 403--406
-
[34]
L. Wang, J. B. Zhou, L. H. Ren, The Exact Solitary Wave Solutions for a Family of BBM Equation, International Journal of Nonlinear Science, 1 (2006), 58--64
-
[35]
Z. Yan, Constructing exact solutions for two-dimensional nonlinear dispersion Boussinesq equation. II: Solitary pattern solutions, Chaos Solitons Fractals, 18 (2003), 869--880
-
[36]
A.-M. Wazwaz, Two reliable methods for solving variants of the KdV equation with compact and noncompact structures, Chaos Solitons Fractals, 28 (2006), 454--462
-
[37]
A.-M. Wazwaz, New compactons, solitons and periodic solutions for nonlinear variants of the KdV and the KP equations, Chaos Solitons Fractals, 22 (2004), 249--260
-
[38]
A.-M. Wazwaz, Exact and explicit travelling wave solutions for the nonlinear Drinfeld–Sokolov system, Commun. Nonlin. Sci. Numer. Simul., 11 (2006), 311--325
-
[39]
A.-M. Wazwaz, Multiple-soliton solutions for the ninth-order KdV equation and sixth-order Boussinesq equation, Applied Mathematics and Computation , 203 (2008), 277--283
-
[40]
S. Zhang, Application of Exp-function method to a KdV equation with variable coefficients, Phys. Lett. A, 365 (2007), 448--453
-
[41]
S. Zhang, Application of Exp-function method to high-dimensional nonlinear evolution equations, Chaos Solitons Fractals, 38 (2008), 270--276
-
[42]
A. Ebaid, Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method, Phys. Lett. A, 365 (2007), 213--219
-
[43]
E. Yusufoglu, A. Bekir, Symbolic computation and new families of exact travelling solutions for the Kawahara and modified Kawahara equations, Computers and Mathematics with Applications, 55 (2008), 1113--1121
-
[44]
A.-M. Wazwaz, New solitary wave solutions to the modified forms of Degasperis–Procesi. and Camassa–Holm equations, Applied Mathematics and Computation, 186 (2007), 130--141
-
[45]
A.-M. Wazwaz, New solitons and kink solutions for the Gardner equation, Communications in Nonlinear Science and Numerical Simulation, 12 (2007), 1395--1404