Abstract
Problem statement: In this research, the Variational Homotopy Perturba tion Method (VHPM) which is a combination of Variational Iterat ion Method (VIM) and Homotopy Perturbation Method (HPM) used for the Zakharove-Kuznetsov equations (ZK-equations). Approach: These two methods are proposed by chinese researcher J.H.He. M.A.Noor improved these two methods and established the VHPM. The numerical solution of ZK-equation is of great importance dut to it's ability to model of traveling wave and nuclear fusion so, f inding it's solution is very important. Results: In this study we presented an efficient and reliable t reatment of the VHPM for this nonlinear Partial Differential Equations (PDEs). This method is based on Lagrange multipliers for identification of optimal value of parameters in a functional and Homotopy Perturbation Method. By applying this method we found the solution of ZK-equations with simple and reliable method and without time consuming calculations. Comparisons were made among the Variational Iteration Method (VIM), Adomian Decomposition Method (ADM) and the proposed method. Conclusion: The results revealed that the proposed method is very effective and can be used for other nonlinear problems in applied mathematics. In following sections, first we introd uce the applied method , then we used that for finding the solution of our equations and finally t he effectiveness and usefulness of proposed method was shown in comparison with other methods.
Highlights
Traveling waves are very important because various phenomena in nature such as vibration and solitons or self-reinforcing solitary waves are described by them
To illustrate the accuracy of applied method comparison of the Variational Homotopy Perturbation Method (VHPM) with other approaches is presented in Table 1 and 2 and Fig. 1-4
This is important that one application of ZKequation is describing Ion Acoustic Wave (IAW) in inertial fusion and Tokamak
Summary
Traveling waves are very important because various phenomena in nature such as vibration and solitons or self-reinforcing solitary waves are described by them. ∂3un ∂x3 are presented by Ismail and Taha (1998) and Wazwaz (2002b) and Ganjavi et al (2008) They use a finite difference method and a finite element method to investigate the approximate solutions of K(2,2) and K(3,3) in (1). & Stat., 6 (4): 425-430, 2010 where, m, n, k are integers and a, b, c are arbitrary constants This equation governs the behavior of weakly nonlinear ion-acoustic waves in plasma comprising cold ions and hot isothermal electrons in the present of a uniform magnetic field (Zhu, 2004); Yan (2002) Osman and Beech (2004) ZK-equation was solved by the sine-cosine and the hyperbolic tangent(tanh)-function methods. We applyied the variational homotopy perturbation method and variational iteration method to ZK-equation
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