Numerical solution of the Cauchy problem for the Painlevé; I and II equations

A numerical method for solving the Cauchy problem for the first and second Painlevé; differential equations is proposed. The presence of movable poles of the solution is allowed. The positions of the poles are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to an auxiliary system of differential equations in a neighborhood of a pole. The equations in this system and its solution have no singularities in either the pole or its neighborhood. Numerical results confirming the efficiency of this method are presented.