Variational Homotopy Perturbation Method for the Zakharove-Kuznetsov Equations

Abstract

Problem statement: In this research, the Variational Homotopy Perturbation Method (VHPM) which is a combination of Variational Iteration 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, finding it’s solution is very important. Results: In this study we presented an efficient and reliable treatment 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 introduce the applied method , then we used that for finding the solution of our equations and finally the effectiveness and usefulness of proposed method was shown in comparison with other methods.