New Application of EHTA for the Generalized(2+1)-Dimensional Nonlinear Evolution Equations
In this paper, the generalized (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (shortly CBS) equations are investigated. We employ the Hirota-s bilinear method to obtain the bilinear form of CBS equations. Then by the idea of extended homoclinic test approach (shortly EHTA), some exact soliton solutions including breather type solutions are presented.
[1] 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.
[2] 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.
[3] M.T. Darvishi, F. Khani, Numerical and explicit solutions of the fifth-order
Korteweg-de Vries equations, Chaos, Solitons and Fractals 39 (2009)
2484-2490.
[4] J.H. He, New interpretation of homotopy perturbation method, Int. J.
Mod. Phys. B 20(18) (2006) 2561-2568.
[5] J.H. He, Application of homotopy perturbation method to nonlinear wave
equations, Chaos, Solitons and Fractals 26(3) (2005) 695-700.
[6] J.H. He, Homotopy perturbation method for bifurcation of nonlinear
problems, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005) 207-208.
[7] 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.
[8] 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.
[9] J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlin.
Sci. Numer. Simul. 2 (2001) 257-264.
[10] 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.
[11] 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.
[12] M.T. Darvishi, Preconditioning and domain decomposition schemes to
solve PDEs, Int-l J. of Pure and Applied Math. 1(4) (2004) 419-439.
[13] 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.
[14] 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.
[15] 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.
[16] 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.
[17] S.J. Liao, An explicit, totally analytic approximate solution for Blasius
viscous flow problems, Int. J. Non-Linear Mech. 34 (1999) 759-778.
[18] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis
Method, Chapman & Hall/CRC Press, Boca Raton, 2003.
[19] S.J. Liao, On the homotopy analysis method for nonlinear problems,
Appl. Math. Comput. 147 (2004) 499-513.
[20] 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.
[21] 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.
[22] M.T. Darvishi, F. Khani, A series solution of the foam drainage equation,
Comput. Math. Appl. 58 (2009) 360-368.
[23] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution
equations using Exp-function method, Chaos, Solitons and Fractals 34
(2007) 1421-1429.
[24] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations,
Chaos, Solitons and Fractals, 30(3) (2006) 700-708.
[25] 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.
[26] 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.
[27] 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.
[28] X.H. Wu, J.H. He, Exp-function method and its application to nonlinear
equations, Chaos, Solitons and Fractals 38(3) (2008) 903-910.
[29] A.M. Wazwaz, New solutions of distinct physical structures to highdimensional
nonlinear evolution equations, Appl. Math. Comput., 196
(2008) 363-370.
[30] A.M. Wazwaz, The (2+1) and (3+1)-Dimensional CBS Equations: Multiple
Soliton Solutions and Multiple Singular Soliton Solutions, Zeitschrift
fur Naturforschung A 65a, (2010) 173-181.
[31] A.M. Wazwaz, Multiple-soliton solutions for the Calogero-
Bogoyavlenskii-Schiff, Jimbo-Miwa and YTSF equations, Appl. Math.
Comput., 203 (2008) 592-597.
[32] Z.-H. Zhao, Z. Dai, S. Han, The EHTA for nonlinear evolution equations,
Appl. Math. Comput., (in press), doi:10.1016/j.amc.2010.09.069.
[1] 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.
[2] 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.
[3] M.T. Darvishi, F. Khani, Numerical and explicit solutions of the fifth-order
Korteweg-de Vries equations, Chaos, Solitons and Fractals 39 (2009)
2484-2490.
[4] J.H. He, New interpretation of homotopy perturbation method, Int. J.
Mod. Phys. B 20(18) (2006) 2561-2568.
[5] J.H. He, Application of homotopy perturbation method to nonlinear wave
equations, Chaos, Solitons and Fractals 26(3) (2005) 695-700.
[6] J.H. He, Homotopy perturbation method for bifurcation of nonlinear
problems, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005) 207-208.
[7] 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.
[8] 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.
[9] J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlin.
Sci. Numer. Simul. 2 (2001) 257-264.
[10] 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.
[11] 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.
[12] M.T. Darvishi, Preconditioning and domain decomposition schemes to
solve PDEs, Int-l J. of Pure and Applied Math. 1(4) (2004) 419-439.
[13] 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.
[14] 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.
[15] 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.
[16] 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.
[17] S.J. Liao, An explicit, totally analytic approximate solution for Blasius
viscous flow problems, Int. J. Non-Linear Mech. 34 (1999) 759-778.
[18] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis
Method, Chapman & Hall/CRC Press, Boca Raton, 2003.
[19] S.J. Liao, On the homotopy analysis method for nonlinear problems,
Appl. Math. Comput. 147 (2004) 499-513.
[20] 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.
[21] 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.
[22] M.T. Darvishi, F. Khani, A series solution of the foam drainage equation,
Comput. Math. Appl. 58 (2009) 360-368.
[23] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution
equations using Exp-function method, Chaos, Solitons and Fractals 34
(2007) 1421-1429.
[24] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations,
Chaos, Solitons and Fractals, 30(3) (2006) 700-708.
[25] 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.
[26] 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.
[27] 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.
[28] X.H. Wu, J.H. He, Exp-function method and its application to nonlinear
equations, Chaos, Solitons and Fractals 38(3) (2008) 903-910.
[29] A.M. Wazwaz, New solutions of distinct physical structures to highdimensional
nonlinear evolution equations, Appl. Math. Comput., 196
(2008) 363-370.
[30] A.M. Wazwaz, The (2+1) and (3+1)-Dimensional CBS Equations: Multiple
Soliton Solutions and Multiple Singular Soliton Solutions, Zeitschrift
fur Naturforschung A 65a, (2010) 173-181.
[31] A.M. Wazwaz, Multiple-soliton solutions for the Calogero-
Bogoyavlenskii-Schiff, Jimbo-Miwa and YTSF equations, Appl. Math.
Comput., 203 (2008) 592-597.
[32] Z.-H. Zhao, Z. Dai, S. Han, The EHTA for nonlinear evolution equations,
Appl. Math. Comput., (in press), doi:10.1016/j.amc.2010.09.069.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:62986", author = "Mohammad Taghi Darvishi and Maliheh Najafi and Mohammad Najafi", title = "New Application of EHTA for the Generalized(2+1)-Dimensional Nonlinear Evolution Equations", abstract = "In this paper, the generalized (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (shortly CBS) equations are investigated. We employ the Hirota-s bilinear method to obtain the bilinear form of CBS equations. Then by the idea of extended homoclinic test approach (shortly EHTA), some exact soliton solutions including breather type solutions are presented.
", keywords = "EHTA, (2+1)-dimensional CBS equations, (2+1)-dimensional breaking solution equation, Hirota's bilinear form.", volume = "4", number = "1", pages = "159-7", }
