The Numerical Solution of the Second Painleve Equation

The Painleve equations were discovered by Painleve, Gambier and their colleagues during studying a nonlinear second-order ordinary differential equation. The six equations which bear Painleve's name are irreducible in the sense that their general solutions cannot be expressed in terms of known functions. Painleve has derived these equations on the sole requirement that their solutions should be free from movable singularities. Many situations in mathematical physics reduce ultimately to Painleve equations: applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics, and fiber optics. This fact has caused a significant interest to the study of these equations in recent years. In this study, the solution of the second Painleve equation is investigated by means of Adomian decomposition method, homotopy perturbation method, and Legendre tau method. Then a numerical evaluation and comparison with the results obtained by the method of continuous analytic continuation are included. (C) 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 25: 1238-1259, 2009