Traveling Wave Solutions of the Benjamin-Bona-Mahony Water Wave Equations

and Applied Analysis 3 3. An Analysis of the Methods The following is given nonlinear partial differential equations (BBM and KdV equations) with two variables x and t as F (u, u t , u x , u xx , u xxx ) = 0 (15) can be converted to ordinary deferential equations: F (u, u 󸀠 , u 󸀠󸀠 , u 󸀠󸀠󸀠 ) = 0, (16) by using a wave variable ξ = x−ct. The equation is integrated as all terms contain derivatives where integration constants are considered zeros. 3.1. Extended Direct Algebraic Methods. We introduce an independent variable, where u = φ(ξ) is a solution of the following third-order ODE: φ 󸀠2 = ±αφ 2 (ξ) ± βφ 4 (ξ) , (17) where α, β are constants. We expand the solution of (16) as the following series: