Analytical Solution for the Zakharov-Kuznetsov Equations by Differential Transform Method
This paper presents the approximate analytical solution of a Zakharov-Kuznetsov ZK(m, n, k) equation with the help of the differential transform method (DTM). The DTM method is a powerful and efficient technique for finding solutions of nonlinear equations without the need of a linearization process. In this approach the solution is found in the form of a rapidly convergent series with easily computed components. The two special cases, ZK(2,2,2) and ZK(3,3,3), are chosen to illustrate the concrete scheme of the DTM method in ZK(m, n, k) equations. The results demonstrate reliability and efficiency of the proposed method.
[1] S. Monro, E. J. Parkes, The derivation of a modified ZakharovKuznetsov
equation and the stability of its solutions, Journal of Plasma Physics, 62
(3) (1999) 305-317.
[2] S. Monro, E. J. Parkes, Stability of solitary-wave solutions to a modified
ZakharovKuznetsov equation, Journal of Plasma Physics, 64 (3) (2000)
411-426.
[3] V. E. Zakharov, E. A. Kuznetsov, On three-dimensional solitons, Soviet
Physics, 39 (1974) 285-288.
[4] 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.
[5] W. Huang, A polynomial expansion method and its application in
the coupled Zakharov-Kuznetsov equations, Chaos Solitons Fractals 29
(2006) 365-371.
[6] X. Zhao, H. Zhou, Y. Tang, H. Jia, Travelling wave solutions for modified
Zakharov-Kuznetsov equation, Applied Mathematics and Computation,
181 (2006) 634-648.
[7] M. Inc, Exact solutions with solitary patterns for the Zakharov-Kuznetsov
equations with fully nonlinear dispersion, Chaos Solitons Fractals 33 (15)
(2007) 1783-1790.
[8] 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.
[9] X. Zhou, Differential Transformation and its Applications for Electrical
Circuits. Huazhong University Press, Wuhan, China, 1986 (in Chinese).
[10] L. Zou, Z. Zong, Z. Wang, S. Tian, Differential transform method for
solving solitary wave with discontinuity, Physics Letters A, 374 (2010)
3451-3454.
[11] D. Nazari, S. Shahmorad, Application of the fractional differential
transform method to fractional-order integro-differential equations with
nonlocal boundary conditions, Journal of Computational and Applied
Mathematics, 234 (2010) 883-891.
[12] M. Thongmoon, S. Pusjuso, The numerical solutions of differential
transform method and the Laplace transform method for a system of
differential equations, Nonlinear Analysis: Hybrid Systems, 4 (2010) 425-
431.
[13] J. Biazar, M. Eslami, Analytic solution for Telegraph equation by
differential transform method, Physics Letters A, 374 (2010) 2904-2906.
[14] V. S. Ert¨urk, S. Momani, Z. Odibat, Application of generalized differential
transform method to multi-order fractional differential equations,
Communications in Nonlinear Science and Numerical Simulation, 13,
(2008) 1642-1654.
[15] A. Al-rabtah, V. S. Ert¨urk, S. Momani, Solutions of a fractional oscillator
by using differential transform method, Computers & Mathematics with
Applications, 59 (2010) 1356-1362.
[16] M. Kurulay, M. Bayram, Approximate analytical solution for the fractional
modified KdV by differential transform method, Communications
in Nonlinear Science and Numerical Simulation, 15 (2010) 1777-1782.
[17] C. K. Chen, S. H. Ho, Solving partial differential equations by two dimensional
differential transform, Applied Mathematics and Computation,
106 (1999) 171-179.
[18] M. J. Jang, C. L. Chen, Y.C. Liu, Two-dimensional differential transform
for partial differential equations, Applied Mathematics and Computation,
121 (2001) 261-270.
[19] F. Ayaz, On the two-dimensional differential transform method, Applied
Mathematics and Computation, 143 (2003) 361-374.
[20] F. Ayaz, Solutions of the system of differential equations by differential
transform method, Applied Mathematics and Computation, 147 (2004)
547-567.
[1] S. Monro, E. J. Parkes, The derivation of a modified ZakharovKuznetsov
equation and the stability of its solutions, Journal of Plasma Physics, 62
(3) (1999) 305-317.
[2] S. Monro, E. J. Parkes, Stability of solitary-wave solutions to a modified
ZakharovKuznetsov equation, Journal of Plasma Physics, 64 (3) (2000)
411-426.
[3] V. E. Zakharov, E. A. Kuznetsov, On three-dimensional solitons, Soviet
Physics, 39 (1974) 285-288.
[4] 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.
[5] W. Huang, A polynomial expansion method and its application in
the coupled Zakharov-Kuznetsov equations, Chaos Solitons Fractals 29
(2006) 365-371.
[6] X. Zhao, H. Zhou, Y. Tang, H. Jia, Travelling wave solutions for modified
Zakharov-Kuznetsov equation, Applied Mathematics and Computation,
181 (2006) 634-648.
[7] M. Inc, Exact solutions with solitary patterns for the Zakharov-Kuznetsov
equations with fully nonlinear dispersion, Chaos Solitons Fractals 33 (15)
(2007) 1783-1790.
[8] 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.
[9] X. Zhou, Differential Transformation and its Applications for Electrical
Circuits. Huazhong University Press, Wuhan, China, 1986 (in Chinese).
[10] L. Zou, Z. Zong, Z. Wang, S. Tian, Differential transform method for
solving solitary wave with discontinuity, Physics Letters A, 374 (2010)
3451-3454.
[11] D. Nazari, S. Shahmorad, Application of the fractional differential
transform method to fractional-order integro-differential equations with
nonlocal boundary conditions, Journal of Computational and Applied
Mathematics, 234 (2010) 883-891.
[12] M. Thongmoon, S. Pusjuso, The numerical solutions of differential
transform method and the Laplace transform method for a system of
differential equations, Nonlinear Analysis: Hybrid Systems, 4 (2010) 425-
431.
[13] J. Biazar, M. Eslami, Analytic solution for Telegraph equation by
differential transform method, Physics Letters A, 374 (2010) 2904-2906.
[14] V. S. Ert¨urk, S. Momani, Z. Odibat, Application of generalized differential
transform method to multi-order fractional differential equations,
Communications in Nonlinear Science and Numerical Simulation, 13,
(2008) 1642-1654.
[15] A. Al-rabtah, V. S. Ert¨urk, S. Momani, Solutions of a fractional oscillator
by using differential transform method, Computers & Mathematics with
Applications, 59 (2010) 1356-1362.
[16] M. Kurulay, M. Bayram, Approximate analytical solution for the fractional
modified KdV by differential transform method, Communications
in Nonlinear Science and Numerical Simulation, 15 (2010) 1777-1782.
[17] C. K. Chen, S. H. Ho, Solving partial differential equations by two dimensional
differential transform, Applied Mathematics and Computation,
106 (1999) 171-179.
[18] M. J. Jang, C. L. Chen, Y.C. Liu, Two-dimensional differential transform
for partial differential equations, Applied Mathematics and Computation,
121 (2001) 261-270.
[19] F. Ayaz, On the two-dimensional differential transform method, Applied
Mathematics and Computation, 143 (2003) 361-374.
[20] F. Ayaz, Solutions of the system of differential equations by differential
transform method, Applied Mathematics and Computation, 147 (2004)
547-567.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51305", author = "Saeideh Hesam and Alireza Nazemi and Ahmad Haghbin", title = "Analytical Solution for the Zakharov-Kuznetsov Equations by Differential Transform Method", abstract = "This paper presents the approximate analytical solution of a Zakharov-Kuznetsov ZK(m, n, k) equation with the help of the differential transform method (DTM). The DTM method is a powerful and efficient technique for finding solutions of nonlinear equations without the need of a linearization process. In this approach the solution is found in the form of a rapidly convergent series with easily computed components. The two special cases, ZK(2,2,2) and ZK(3,3,3), are chosen to illustrate the concrete scheme of the DTM method in ZK(m, n, k) equations. The results demonstrate reliability and efficiency of the proposed method.
", keywords = "Zakharov-Kuznetsov equation, differential transform method, closed form solution.", volume = "5", number = "3", pages = "276-6", }
