This article is cited in 5 papers

Numerical solution of the Painlevé V equation

A. A. Abramov^a, L. F. Yukhno<sup>b

a</sup> Dorodnitsyn Computing Centre of the Russian Academy of Sciences, Moscow
^b M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Moscow

Abstract: A numerical method for solving the Cauchy problem for the fifth Painlevé equation is proposed. The difficulty of the problem is that the unknown function can have movable singular points of the pole type; moreover, the equation has singularities at the points where the solution vanishes or takes the value 1. The positions of all of these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Numerical results illustrating the potentials of this method are presented.

Key words: Painlevé V ordinary differential equation, pole of a solution, singularity of an equation, numerical method.

UDC: 519.624.2

Received: 11.05.2012

DOI: 10.7868/S004446691301002X

 English version: Computational Mathematics and Mathematical Physics, 2013, 53:1, 44–56

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