Analytic Approximate Solution for the KdV Equation with the Homotopy Analysis Method

Abstract

The homotopy analysis method (HAM) is applied to obtain the analytic approximate solution of the well-known Korteweg-de Vries (KdV) equation. The HAM is an analytic technique which provides us with a new way to obtain series solutions of such nonlinear problems. HAM contains the auxiliary parameter $\hbar,$, which provides us with a straightforward way to adjust and control the convergence region of the series solution. The resulted HAM solution at eighth order approximation is then compared with that of the exact soliton solution of KdV equation, and shown to be in excellent agreement. Keywords: KdV equation; homotopy analysis method; approximate analytic solution; soliton solution; $\hbar$-curve. 2010 Mathematics Subject Classification: 74J30, 78M99