In this research, we study new two techniques that called the extended simple equation method and the novel [Formula presented]-expansion method. The extended simple equation method depend on the auxiliary equation [Formula presented]=α+λϕ+μϕ 2 which has three ways for solving depends on the specific condition on the parameters as follow: When λ=0 this auxiliary equation reduces to Riccati equation, when α=0 this auxiliary equation reduces to Bernoulli equation and when α≠0,λ≠0,μ≠0 we the general solutions of this auxiliary equation while the novel [Formula presented]-expansion method depends also on similar auxiliary equation [Formula presented] ′ =μ+λ[Formula presented]+(v-1)[Formula presented] 2 which depend also on the value of (λ 2 -4(v-1)μ) and the specific condition on the parameters as follow: When λ=0 this auxiliary equation reduces to Riccati equation, when μ=0 this auxiliary equation reduces to Bernoulli equation and when (λ 2 ≠4(v-1)μ) we the general solutions of this auxiliary equation. This show how both of these auxiliary equation are special cases of Riccati equation. We apply these methods on two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma and three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma. We obtain the exact traveling wave solutions of these important models and under special condition on the parameters, we get solitary traveling wave solutions. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions.
Khater, M. M. A., Seadawy, A. R., & Lu, D. (2018). Bifurcations of solitary wave solutions for (two and three)-dimensional nonlinear partial differential equation in quantum and magnetized plasma by using two different methods. Results in Physics, 9, 142–150. https://doi.org/10.1016/j.rinp.2018.02.010