The numerical solution of the second Painlevé equation

Abstract

The Painlevé equations were discovered by Painlevé, Gambier and their colleagues during studying a nonlinear second-order ordinary differential equation. The six equations which bear Painlevé's name are irreducible in the sense that their general solutions cannot be expressed in terms of known functions. Painlevé 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 Painlevé 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 Painlevé 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.

A novel computational framework to approximate analytical solution of nonlinear fractional elastic beam equation

“…(1) and (2). By using the properties of the homotopy perturbation method [11,13,14,16,19], we have the following equation:…”

“…Considerable research work has recently been conducted in application of this method to fractional advection-dispersion equations, multi-order fractional di erential equations, Navier-Stokes equations, nonlinear Schr odinger equations, Volterra integro-di erential equations, nonlinear oscillators, boundary value problems, fractional KdV equations, quadratic Riccati differential equations of fractional order and many others. For more details about the homotopy perturbation method and its applications, the reader is advised to consult the results of research work presented in [14][15][16][17][18][19]. All these successful applications veri ed the e ectiveness, exibility, and validity of the homotopy perturbation method.…”

Numerical Study of Second Painlevé Equation

“…These six equations have a great variety of interesting properties and applications. As said in [20] the Painlevé equations where first found from strictly mathematical considerations, they have recently appeared in a variety of important physical applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics, and fiber optics [10]. Also the Painlevé equations have attracted much interest since they arise as reductions of the soliton equations which are solvable by inverse scattering [6] transform such as the Kortewegde Vries equation, the modified Kortewegde Vries equation, the cylindrical Kortewegde Vries equation, the Boussinesq and Kadomtsev Petviashvili type equations, the nonlinear Schrdinger equation, the sine-Gordon equation, the equations of Einstein type, and so on.…”

“…Also the Painlevé equations have attracted much interest since they arise as reductions of the soliton equations which are solvable by inverse scattering [6] transform such as the Kortewegde Vries equation, the modified Kortewegde Vries equation, the cylindrical Kortewegde Vries equation, the Boussinesq and Kadomtsev Petviashvili type equations, the nonlinear Schrdinger equation, the sine-Gordon equation, the equations of Einstein type, and so on. Connections are given briefly in [10,13]. Alternatively the Painlevé equations can be introduced as equations of isomonodromic deformations of auxiliary linear systems of differential equations [5].…”

“…Also, making use of the value of y and y ′ in the neighbored of x = 1, estimates of polar values can be obtained [7]. Recently, Dehghan and Shakeri solved problem (1.1) -(1.2) by Adomian decomposition method, homotopy perturbation method, and Legendre Tau method [10]. Also very recently, the authors of [12] solved this problem by homotopy analysis method.…”