Traveling Wave Solutions for the (3+1)-Dimensional Breaking Soliton Equation by (G'/G)- Expansion Method and Modified F-Expansion Method
In this paper, using (G/G )-expansion method and modified F-expansion method, we give some explicit formulas of exact traveling wave solutions for the (3+1)-dimensional breaking soliton equation. A modified F-expansion method is proposed by taking full advantages of F-expansion method and Riccati equation in seeking exact solutions of the equation.
[1] L. Debtnath, Nonlinear partial differential equations for scientists and
engineers, Birkhauser, Boston, 1997.
[2] A.M. Wazwaz, Partial differential equations methods and applications,
Rotterdam, Balkema, 2002.
[3] J.H. He, New interpretation of homotopy perturbation method, Int. J.
Mod. Phys. B 20(18) (2006) 2561-2568.
[4] J.H. He, Application of homotopy perturbation method to nonlinear wave
equations, Chaos, Solitons and Fractals 26(3) (2005) 695-700.
[5] J.H. He, Homotopy perturbation method for bifurcation of nonlinear
problems, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005) 207-208.
[6] M.T. Darvishi, F. Khani, Application of He-s homotopy perturbation
method to stiff systems of ordinary differential equations, Zeitschrift fur
Naturforschung A, 63a (1-2) (2008) 19-23.
[7] M.T. Darvishi, F. Khani, S. Hamedi-Nezhad, S.-W. Ryu, New modification
of the HPM for numerical solutions of the sine-Gordon and coupled sine-
Gordon equations, Int. J. Comput. Math. 87(4) (2010) 908-919.
[8] J.H. He, Variational iteration method-a kind of non-linear analytical
technique: some examples, Int. J. Non-linear Mech., 34(4) (1999) 699-
708.
[9] M.T. Darvishi, F. Khani, A.A. Soliman, The numerical simulation for stiff
systems of ordinary differential equations, Comput. Math. Appl., 54(7-8)
(2007) 1055-1063.
[10] M.T. Darvishi, F. Khani, Numerical and explicit solutions of the fifthorder
Korteweg-de Vries equations, Chaos, Solitons and Fractals, 39
(2009) 2484-2490.
[11] J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlin.
Sci. Numer. Simul., 2 (2001) 257-264.
[12] M.T. Darvishi, A. Karami, B.-C. Shin, Application of He-s parameterexpansion
method for oscillators with smooth odd nonlinearities, Phys.
Lett. A, 372(33) (2008) 5381-5384.
[13] B.-C. Shin, M.T. Darvishi, A. Karami, Application of He-s parameterexpansion
method to a nonlinear self-excited oscillator system, Int. J.
Nonlin. Sci. Num. Simul., 10(1) (2009) 137-143.
[14] M.T. Darvishi, S. Kheybari, A. Yildirim, Application of He-s parameterpxpansion
method to a system of two van der Pol oscillators coupled via
a bath, Nonlin. Science Lett. A., 1(4) (2010) 399-405.
[15] M.T. Darvishi, Preconditioning and domain decomposition schemes to
solve PDEs, Int-l J. of Pure and Applied Math., 1(4) (2004) 419-439.
[16] M.T. Darvishi, S. Kheybari and F. Khani, A numerical solution of the
Korteweg-de Vries equation by pseudospectral method using Darvishi-s
preconditionings, Appl. Math. Comput., 182(1) (2006) 98-105.
[17] M.T. Darvishi, M. Javidi, A numerical solution of Burgers- equation
by pseudospectral method and Darvishi-s preconditioning, Appl. Math.
Comput., 173(1) (2006) 421-429.
[18] M.T. Darvishi, F. Khani and S. Kheybari, Spectral collocation solution
of a generalized Hirota-Satsuma KdV equation, Int. J. Comput. Math.,
84(4) (2007) 541-551.
[19] M.T. Darvishi, F. Khani, S. Kheybari, Spectral collocation method
and Darvishi-s preconditionings to solve the generalized Burgers-Huxley
equation, Commun., Nonlinear Sci. Numer. Simul., 13(10) (2008) 2091-
2103.
[20] S.J. Liao, An explicit, totally analytic approximate solution for Blasius
viscous flow problems, Int. J. Non-Linear Mech., 34 (1999) 759-778.
[21] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis
Method, Chapman & Hall/CRC Press, Boca Raton, 2003.
[22] S.J. Liao, On the homotopy analysis method for nonlinear problems,
Appl. Math. Comput., 147 (2004) 499-513.
[23] S.J. Liao, A new branch of solutions of boundary-layer flows over an
impermeable stretched plate, Int. J. Heat Mass Transfer, 48 (2005) 2529-
2539.
[24] S.J. Liao, A general approach to get series solution of non-similarity
boundary-layer flows, Commun. Nonlinear Sci. Numer. Simul., 14(5)
(2009) 2144-2159.
[25] M.T. Darvishi, F. Khani, A series solution of the foam drainage equation,
Comput. Math. Appl., 58 (2009) 360-368.
[26] J.H. He, X.H. Wu, Construction of solitary solution and compacton-like
solution by variational iteration method, Chaos, Solitons and Fractals, 29
(2006) 108-113.
[27] F. Khani, S. Hamedi-Nezhad, M.T. Darvishi, S.-W. Ryu, New solitary
wave and periodic solutions of the foam drainage equation using the Expfunction
method, Nonlin. Anal.: Real World Appl., 10 (2009) 1904-1911.
[28] B.-C. Shin, M.T. Darvishi, A. Barati, Some exact and new solutions
of the Nizhnik-Novikov-Vesselov equation using the Exp-function method,
Comput. Math. Appl. 58(11/12) (2009) 2147-2151.
[29] X.H. Wu, J.H. He, Exp-function method and its application to nonlinear
equations, Chaos, Solitons and Fractals, 38(3) (2008) 903-910.
[30] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations,
Chaos Solitons Fractals, 30(3) (2006) 700-708.
[31] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution
equations using Exp-function method, Chaos Solitons Fractals, 34 (2007)
1421-1429.
[32] Ma S-H, Peng J and Zhang C, New exact solutions of the (2+1)-
dimensional breaking soliton system via an extended mapping method,
Chaos Solitons Fractals, 46 (2009) 210-214.
[33] A.M. Wazwaz, Integrable (2+1)-dimensional and (3+1)-dimensional
breaking soliton equations, Phys. Scr., 81 (2010) 1-5.
[34] M.-L. Wang., X.-Z. Li, J.-L. Zhang, The (G
G )-expansion method and
travelling wave solutions of nonlinear evolution equations in mathematical
physics,, Physics Letters A, 372 (2008) 417-423.
[35] M.T. Darvishi, M. Najafi, M. Najafi, New exact solutions for the
(3+1)-dimensional breaking soliton equation, International Journal of
Engineering and Mathematical Sciences, 6(2) (2010) 137-140.
[36] J. Wang, Construction of new exact traveling wave solutions to (2+1)-
dimensional mVN equation, International Journal of Nonlinear Science,
9(3) (2010) 325-329.
[1] L. Debtnath, Nonlinear partial differential equations for scientists and
engineers, Birkhauser, Boston, 1997.
[2] A.M. Wazwaz, Partial differential equations methods and applications,
Rotterdam, Balkema, 2002.
[3] J.H. He, New interpretation of homotopy perturbation method, Int. J.
Mod. Phys. B 20(18) (2006) 2561-2568.
[4] J.H. He, Application of homotopy perturbation method to nonlinear wave
equations, Chaos, Solitons and Fractals 26(3) (2005) 695-700.
[5] J.H. He, Homotopy perturbation method for bifurcation of nonlinear
problems, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005) 207-208.
[6] M.T. Darvishi, F. Khani, Application of He-s homotopy perturbation
method to stiff systems of ordinary differential equations, Zeitschrift fur
Naturforschung A, 63a (1-2) (2008) 19-23.
[7] M.T. Darvishi, F. Khani, S. Hamedi-Nezhad, S.-W. Ryu, New modification
of the HPM for numerical solutions of the sine-Gordon and coupled sine-
Gordon equations, Int. J. Comput. Math. 87(4) (2010) 908-919.
[8] J.H. He, Variational iteration method-a kind of non-linear analytical
technique: some examples, Int. J. Non-linear Mech., 34(4) (1999) 699-
708.
[9] M.T. Darvishi, F. Khani, A.A. Soliman, The numerical simulation for stiff
systems of ordinary differential equations, Comput. Math. Appl., 54(7-8)
(2007) 1055-1063.
[10] M.T. Darvishi, F. Khani, Numerical and explicit solutions of the fifthorder
Korteweg-de Vries equations, Chaos, Solitons and Fractals, 39
(2009) 2484-2490.
[11] J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlin.
Sci. Numer. Simul., 2 (2001) 257-264.
[12] M.T. Darvishi, A. Karami, B.-C. Shin, Application of He-s parameterexpansion
method for oscillators with smooth odd nonlinearities, Phys.
Lett. A, 372(33) (2008) 5381-5384.
[13] B.-C. Shin, M.T. Darvishi, A. Karami, Application of He-s parameterexpansion
method to a nonlinear self-excited oscillator system, Int. J.
Nonlin. Sci. Num. Simul., 10(1) (2009) 137-143.
[14] M.T. Darvishi, S. Kheybari, A. Yildirim, Application of He-s parameterpxpansion
method to a system of two van der Pol oscillators coupled via
a bath, Nonlin. Science Lett. A., 1(4) (2010) 399-405.
[15] M.T. Darvishi, Preconditioning and domain decomposition schemes to
solve PDEs, Int-l J. of Pure and Applied Math., 1(4) (2004) 419-439.
[16] M.T. Darvishi, S. Kheybari and F. Khani, A numerical solution of the
Korteweg-de Vries equation by pseudospectral method using Darvishi-s
preconditionings, Appl. Math. Comput., 182(1) (2006) 98-105.
[17] M.T. Darvishi, M. Javidi, A numerical solution of Burgers- equation
by pseudospectral method and Darvishi-s preconditioning, Appl. Math.
Comput., 173(1) (2006) 421-429.
[18] M.T. Darvishi, F. Khani and S. Kheybari, Spectral collocation solution
of a generalized Hirota-Satsuma KdV equation, Int. J. Comput. Math.,
84(4) (2007) 541-551.
[19] M.T. Darvishi, F. Khani, S. Kheybari, Spectral collocation method
and Darvishi-s preconditionings to solve the generalized Burgers-Huxley
equation, Commun., Nonlinear Sci. Numer. Simul., 13(10) (2008) 2091-
2103.
[20] S.J. Liao, An explicit, totally analytic approximate solution for Blasius
viscous flow problems, Int. J. Non-Linear Mech., 34 (1999) 759-778.
[21] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis
Method, Chapman & Hall/CRC Press, Boca Raton, 2003.
[22] S.J. Liao, On the homotopy analysis method for nonlinear problems,
Appl. Math. Comput., 147 (2004) 499-513.
[23] S.J. Liao, A new branch of solutions of boundary-layer flows over an
impermeable stretched plate, Int. J. Heat Mass Transfer, 48 (2005) 2529-
2539.
[24] S.J. Liao, A general approach to get series solution of non-similarity
boundary-layer flows, Commun. Nonlinear Sci. Numer. Simul., 14(5)
(2009) 2144-2159.
[25] M.T. Darvishi, F. Khani, A series solution of the foam drainage equation,
Comput. Math. Appl., 58 (2009) 360-368.
[26] J.H. He, X.H. Wu, Construction of solitary solution and compacton-like
solution by variational iteration method, Chaos, Solitons and Fractals, 29
(2006) 108-113.
[27] F. Khani, S. Hamedi-Nezhad, M.T. Darvishi, S.-W. Ryu, New solitary
wave and periodic solutions of the foam drainage equation using the Expfunction
method, Nonlin. Anal.: Real World Appl., 10 (2009) 1904-1911.
[28] B.-C. Shin, M.T. Darvishi, A. Barati, Some exact and new solutions
of the Nizhnik-Novikov-Vesselov equation using the Exp-function method,
Comput. Math. Appl. 58(11/12) (2009) 2147-2151.
[29] X.H. Wu, J.H. He, Exp-function method and its application to nonlinear
equations, Chaos, Solitons and Fractals, 38(3) (2008) 903-910.
[30] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations,
Chaos Solitons Fractals, 30(3) (2006) 700-708.
[31] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution
equations using Exp-function method, Chaos Solitons Fractals, 34 (2007)
1421-1429.
[32] Ma S-H, Peng J and Zhang C, New exact solutions of the (2+1)-
dimensional breaking soliton system via an extended mapping method,
Chaos Solitons Fractals, 46 (2009) 210-214.
[33] A.M. Wazwaz, Integrable (2+1)-dimensional and (3+1)-dimensional
breaking soliton equations, Phys. Scr., 81 (2010) 1-5.
[34] M.-L. Wang., X.-Z. Li, J.-L. Zhang, The (G
G )-expansion method and
travelling wave solutions of nonlinear evolution equations in mathematical
physics,, Physics Letters A, 372 (2008) 417-423.
[35] M.T. Darvishi, M. Najafi, M. Najafi, New exact solutions for the
(3+1)-dimensional breaking soliton equation, International Journal of
Engineering and Mathematical Sciences, 6(2) (2010) 137-140.
[36] J. Wang, Construction of new exact traveling wave solutions to (2+1)-
dimensional mVN equation, International Journal of Nonlinear Science,
9(3) (2010) 325-329.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:61219", author = "Mohammad Taghi Darvishi and Maliheh Najafi and Mohammad Najafi", title = "Traveling Wave Solutions for the (3+1)-Dimensional Breaking Soliton Equation by (G'/G)- Expansion Method and Modified F-Expansion Method", abstract = "In this paper, using (G/G )-expansion method and modified F-expansion method, we give some explicit formulas of exact traveling wave solutions for the (3+1)-dimensional breaking soliton equation. A modified F-expansion method is proposed by taking full advantages of F-expansion method and Riccati equation in seeking exact solutions of the equation.
", keywords = "Exact solution, The (3+1)-dimensional breaking soliton equation, ( G G )-expansion method, Riccati equation, Modified Fexpansion method.", volume = "5", number = "7", pages = "1039-6", }
