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Numerical solution of the Painlevé V equation

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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.

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References

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Authors and Affiliations

  1. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

    A. A. Abramov

  2. Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4a, Moscow, 125047, Russia

    L. F. Yukhno

Authors
  1. A. A. Abramov
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  2. L. F. Yukhno
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Correspondence to A. A. Abramov.

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Original Russian Text © A.A. Abramov, L.F. Yukhno, 2013, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2013, Vol. 53, No. 1, pp. 58–71.

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Abramov, A.A., Yukhno, L.F. Numerical solution of the Painlevé V equation. Comput. Math. and Math. Phys. 53, 44–56 (2013). https://doi.org/10.1134/S0965542513010028

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  • DOI: https://doi.org/10.1134/S0965542513010028

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