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Zh. Vychisl. Mat. Mat. Fiz., 2013, Volume 53, Number 5, Pages 702–726 (Mi zvmmf9852)  

This article is cited in 2 scientific papers (total in 2 papers)

A method for the numerical solution of the Painlevé equations

A. A. Abramova, L. F. Yukhnob

a Dorodnitsyn Computing Centre of the Russian Academy of Sciences, Moscow
b Institute for Mathematical Modelling, Russian Academy of Sciences, Moscow

Abstract: A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all 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. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.

Key words: Painlevé I–VI ordinary differential equation, pole of a solution, singularity of an equation, numerical method, method of the successive elimination of singularities.

DOI: https://doi.org/10.7868/S0044466913050025

Full text: PDF file (363 kB)
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English version:
Computational Mathematics and Mathematical Physics, 2013, 53:5, 540–563

Bibliographic databases:

UDC: 519.624.2
Received: 26.11.2012

Citation: A. A. Abramov, L. F. Yukhno, “A method for the numerical solution of the Painlevé equations”, Zh. Vychisl. Mat. Mat. Fiz., 53:5 (2013), 702–726; Comput. Math. Math. Phys., 53:5 (2013), 540–563

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Reeger J.A., Fornberg B., “Painlevé IV: a Numerical Study of the Fundamental Domain and Beyond”, Physica D, 280 (2014), 1–13  crossref  mathscinet  zmath  isi  scopus
    2. Peter A. Clarkson, “Open Problems for Painlevé Equations”, SIGMA, 15 (2019), 006, 20 pp.  mathnet  crossref
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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