the extended Korteweg de-Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water wave equations and the Lagrangian for the water wave equations. For the special case which only the higher order nonlinear term is retained, the extended Korteweg de-Vries equation is transformed into the Korteweg de-Vries equation. A Bubnov- Galerkin finite element method with quintic B-spline functions taken as element shape and weight functions is presented for the solution of the extended Korteweg de-Vries equation. Modulation equations for this equation are then derived from the modulation equation for the Korteweg de-Vries equation and the solution for the extended Korteweg de-Vries equation is found as a simple wave solution of these modulation equations. This solution is compared with the numerical solution with different time displacement. Simulations undertaken proved that the scheme can model faithfully the physics of the Extended Korteweg de-Vries equation.