Homotopy analysis method to obtain numerical solutions of the Painlevé equations

In this paper, the homotopy analysis method (HAM) is presented to obtain the numerical solutions for the two kinds of the Painlevé equations with a number of initial conditions. Then, a numerical evaluation and comparison with the results obtained via the HAM are included. It illustrates the validity and the great potential of the HAM in solving Painlevé equations. Although the HAM contains the auxiliary parameter, the convergence region of the series solution can be controlled in a simple way. Copyright © 2012 John Wiley & Sons, Ltd.