A note on the convergence of the Zakharov-Kuznetsov equation by homotopy analysis method

محورهای موضوعی : Partial differential equations

A. Fallahzadeh ¹ , M. A. Fariborzi Araghi ²

1 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 13185.768, Tehran, Iran
2 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 13185.768, Tehran, Iran

تاریخ دریافت : 1393/02/31 تاریخ پذیرش : 1393/05/13 تاریخ انتشار : 1392/12/10

کلید واژه: Homotopy Analysis Method, Zakharov-Kuznetsov equation, convergence, recursive method, Partial Differential Equation,

چکیده مقاله :

In this paper, the convergence of Zakharov-Kuznetsov (ZK) equation by homotopy analysis method (HAM) is investigated. A theorem is proved to guarantee the convergence of HAM and to find the series solution of this equation via a reliable algorithm.

منابع و مأخذ:

[1] S. Abbasbandy, Y. Tan, S. J. Liao, Newton-homotopy analysis method for nonlinear equations, Appl. Math. Comput., 188 (2007) 1794-1800.

 

[2] S. Abbasbandy, Homotopy analysis method for the Kawahara equation, Nonlinear Analysis: Real World Applications, 11 (2010) 307-312.

 

[3] J. Biazar, F. Badpeimaa, F. Azimi, Application of the homotopy perturbation method to Zakharov-Kuznetsov equations, Computers and Mathematics with Applications 58 (2009) 2391-2394.

 

[4] W. Huang, A polynomial expansion method and its application in the coupled Zakharov-Kuznetsov equations, Chaos Solitons Fractals, 29 (2006) 365-371.

 

[5] S. Hesam, A. Nazemi, A. Haghbin, Analytical solution for the Zakharov-Kuznetsov equations by differential transform method, International Journal of Engineering and Natural Sciences 4 (4) (2010).

 

[6] M. Inc, Exact solutions with solitary patterns for the Zakharov-Kuznetsov equations with fully nonlinear dispersion, Chaos Solitons Fractals, 33 (15) (2007) 1783-1790.

 

[7] S. J. Liao, Beyond pertubation: Introduction to the homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, (2003).

 

[8] S.J. Liao, Notes on the homotopy analysis method: some definitions and theorems, Communication in Nonlinear Science and Numnerical Simulation, 14 (2009) 983-997.

 

[9] M. A. Fariborzi Araghi, A. Fallahzadeh, On the convergence of the Homotopy Analysis method for solving the Schrodinger Equation, Journal of Basic and Applied Scientific Research, 2(6) (2012) 6076-6083.

 

[10] M. A. Fariborzi Araghi, A. Fallahzadeh, Explicit series solution of Boussinesq equation by homotopy analysis method, Journal of American Science, 8(11) 2012.

 

[11] M. A. Fariborzi Araghi, S. Naghshband, On convergence of homotopy analysis method to solve the Schrodinger equation with a power law nonlinearity, Int. J. Industrial Mathematics, 5 (4) (2013) 367-374.

 

[12] S. Monro, E. J. Parkes, The derivation of a modified Zakharov-Kuznetsov equation and the stability of its solutions, Journal of Plasma Physics, 62 (3) (1999) 305-317.

 

[13] S. Monro, E. J. Parkes, Stability of solitary-wave solutions to a modified ZakharovKuznetsov equation, Journal of Plasma Physics, 64 (3) (2000) 411-426.

 

[14] M. Usman, I. Rashid, T. Zubair, A. Waheed, S. T. Mohyuddin, Homotopy analysis method for Zakharov-Kuznetsov (ZK) equation with fully nonlinear dispersion, Scientific Research and Essays, 8 (23) (2013) 1065-1072.

 

[15] A. M. Wazwaz, The extended tanh method for the Zakharov-Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms, Communications in Nonlinear Science and Numerical Simulation, 13 (2008) 1039-1047.

 

[16] X. Zhao, H. Zhou, Y. Tang, H. Jia, Travelling wave solutions for modified Zakharov-Kuznetsov equation, Applied Mathematics and Computation, 181 (2006) 634-648.