Hyperbolic Function Expansion Method And The Generalized Kdv Equation And Kdv Equations The Exact Solution

Seeking exact solutions for partial differential equations has long been a major concert of both mathematicians and physicists. One of the most effective direct, methods to construct the exact solutions of partial different equations is tanh function method. Fan proposed an extended tanh function method. Based upon the Riccati equationφ' = k +φ², this method was further extended by Elwakil et al. Very recently the extended tanh method was improved by Wazwaz and introducing generalized the Riccati equationφ' =μ(k +φ²). In this paper, we use the extended tanh function methods to solve these equations.(1).The generalized KDV equation: u_t + (auⁿ -βu²ⁿ)u_x + u_xxx = 0.Wazwaz use tanh function method to solve this equation. In this paper, we use others extended tanh function methods to solve this equation.(2). we use the generalized extended tanh function method to solve three equations. the KDV equations: u_t + 6αuu_x－2bvv_x +αu_xxx = 0, v_t + 3βuv_x +βv_xxx = 0, the MKDV equations: u_t－1/2u_xxx+3u²ï¼3/2v_xx－3(uv)_x+3αu_x=0, v_t+v_xxx+3vv_x+3u_xv_x－3u²v_x－3αv_x = 0, and the Hirota-Satsuma-KDV equations: u_t－1/2u_xxx+3uu_x－3(vw)_x=0, v_t+v_xxx－3uv_x=0,w_t + w_xxx－3uw_x = 0,...