Numerical Study of Second Painlevé Equation

The Painlevé equations and their solutions arises in parts of pure and applied mathematics and theoretical physics. In this study, Sinc-collocation method and variational iteration method coupled with Padé approximation are applied for solving a second Painlevé equation. This equation can be used as a model for describing the electric field in a semiconductor. Some properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce the computation of solution of the second Painlevé equation to some algebraic equations. It is well known that Sinc procedure converges to the solution at an exponential rate. Also the variational iteration method is used to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory. Numerical examples are included to demonstrate the validity and applicability of the techniques and a comparison is made with existing results in literatures.