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Zh. Vychisl. Mat. Mat. Fiz., 2013, Volume 53, Number 1, Pages 58–71 (Mi zvmmf9793)  

This article is cited in 5 scientific papers (total in 5 papers)

Numerical solution of the Painlevé V equation

A. A. Abramova, L. F. Yukhnob

a 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.

DOI: https://doi.org/10.7868/S004446691301002X

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

Bibliographic databases:

UDC: 519.624.2
Received: 11.05.2012

Citation: A. A. Abramov, L. F. Yukhno, “Numerical solution of the Painlevé V equation”, Zh. Vychisl. Mat. Mat. Fiz., 53:1 (2013), 58–71; Comput. Math. Math. Phys., 53:1 (2013), 44–56

Citation in format AMSBIB
\by A.~A.~Abramov, L.~F.~Yukhno
\paper Numerical solution of the Painlev\'e~V equation
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2013
\vol 53
\issue 1
\pages 58--71
\jour Comput. Math. Math. Phys.
\yr 2013
\vol 53
\issue 1
\pages 44--56

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    This publication is cited in the following articles:
    1. A. A. Abramov, L. F. Yukhno, “Numerical solution of the Painlevé VI equation”, Comput. Math. Math. Phys., 53:2 (2013), 180–193  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. A. Abramov, L. F. Yukhno, “A method for the numerical solution of the Painlevé equations”, Comput. Math. Math. Phys., 53:5 (2013), 540–563  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. Abramov A.A. Yukhno L.F., “a Method For Calculating the Painlevé Transcendents”, Appl. Numer. Math., 93:SI (2015), 262–269  crossref  mathscinet  zmath  isi  elib  scopus
    4. Bermudez D., Fernandez C D.J., Negro J., “Solutions to the Painlevé V equation through supersymmetric quantum mechanics”, J. Phys. A-Math. Theor., 49:33 (2016), 335203  crossref  mathscinet  zmath  isi  elib  scopus
    5. 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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