Analytical solution for the Zakharov-Kuznetsov equations by differential transform method Saeideh Hesam, Alireza Nazemi and Ahmad Haghbin 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. I. I NTRODUCTION I N this paper the applied DTM is used to solve the Zakharov-Kuznetsov ZK(m, n, k) equations of the form u t + a(u m ) x + b(u n ) xxx + c(u k ) yyx =0, m, n, k =0, (1) where a, b, c are arbitrary constants and m, n, k are inte- gers. This equation governs the behavior of weakly nonlinear ion-acoustic waves in plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field [1]-[2]. The ZK equation was first derived for describing weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two dimensions [3]. Wazwaz [4] used extended tanh method for analytic treat- ment of the ZK equation, the modified ZK equation, and the generalized forms of these equations. Huang [5] applied the polynomial expansion method to solve the coupled ZK equations. Zhao et al. [6] obtained numbers of solitary waves, periodic waves and kink waves using the theory of bifurcations of dynamical systems for the modified ZK equation. Inc [7] solved nonlinear dispersive ZK equations using the Adomian decomposition method, and Biazar et al. [8] applied the ho- motopy perturbation method to solve the Zakharov-Kuznetsov ZK(m, n, k) equations. In the present work, we are concerned with the application of the DTM for the ZK equations. The DTM is a numerical method based on a Taylor expansion. This method constructs an analytical solution in the form of a polynomial. The concept of DTM was first proposed and applied to solve linear and nonlinear initial value problems in electric circuit analysis by Saeideh Hesam and Alireza Nazemi (Corresponding author) are with the Department of Mathematics, School of Mathematical Sciences, Shahrood University of Technology, P. O. Box 3619995161-316, Tel-Fax No:+98273-3392012, Shahrood, Iran. email: taranome2009@yahoo.com, e- mail: nazemi20042003@yahoo.com. Ahmad Haghbin is with the Department of Mathematics, Islamic Azad University of Ghorghan, Ghorghan, Iran. email: Ahmadbin@yahoo.com. [9]. Unlike the traditional high order Taylor series method which requires a lot of symbolic computations, the DTM is an iterative procedure for obtaining Taylor series solutions. This method will not consume too much computer time when applying to nonlinear or parameter varying systems. This method gives an analytical solution in the form of a polynomial. But, it is different from Taylor series method that requires computation of the high order derivatives. The DTM is an iterative procedure that is described by the transformed equations of original functions for solution of differential equations. Recently, the application of DTM is successfully extended to obtain analytical approximate solutions to various linear and nonlinear problems. For instance see [10]-[16]. The paper is organized as follows. In Section 2, theoretical aspects of the method are discussed. In Section 3, several examples with analytical solutions will be given to show the impressiveness of the suggested method. A proof of solution is exhibited in section 4. Finally, conclusions are given in Section 5. II. DIFFERENTIAL TRANSFORM METHOD 2.1 Two-dimensional differential transform The basic definition and the fundamental theorems of the DTM and its applicability for various kinds of differential equations are given in [17]-[20]. For convenience of the reader, we present a review of the DTM. The differential transform function of the function w(x, y) is the following form: W (k,h)= 1 k!h! [ ∂ (k+h) w(x, y) ∂x k ∂y h ] (x=x0,y=y0) , (2) where w(x, y) is the original function and W (k,h) is the transformed function. The inverse differential transform of W (k,h) is defined as w(x, y)= ∞  k=0 ∞  h=0 W (k,h)(x − x 0 ) k (y − y 0 ) h . (3) Combining Eq. (2) and Eq. (3), it can be obtained that W (k,h)= ∞  k=0 ∞  h=0 1 k!h! [ ∂ (k+h) w(x, y) ∂x k ∂y h ] (x=x0,y=y0) (x − x 0 ) k (y − y 0 ) h . (4) When (x 0 ,y 0 ) are taken as (0, 0), the function w(x, y) in Eq. (4) is expressed as the following W (k,h)= ∞  k=0 ∞  h=0 1 k!h! [ ∂ (k+h) w(x, y) ∂x k ∂y h ] (x=x0,y=y0) x k y h , (5) World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:5, No:3, 2011 496 International Scholarly and Scientific Research & Innovation 5(3) 2011 scholar.waset.org/1307-6892/2172 International Science Index, Mathematical and Computational Sciences Vol:5, No:3, 2011 waset.org/Publication/2172