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Power-elliptic expansions of solutions to an ordinary differential equation

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Abstract

A rather general ordinary differential equation is considered that can be represented as a polynomial in variables and derivatives. For this equation, the concept of power-elliptic expansions of its solutions is introduced and a method for computing them is described. It is shown that such expansions of solutions exist for the first and second Painlevé equations.

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References

  1. A. D. Bruno, “Asymptotic Behavior and Expansions of Solutions of an Ordinary Differential Equation,” Russ. Math. Surv. 59, 429–480 (2004).

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  2. A. D. Bruno, “Space Power Geometry for an ODE and Painlevé Equations,” International Conference on Painlevé Equations and Related Topics, St. Petersburg, June. 2011 (St. Petersburg, 2011), pp. 36–41; http://www.pdmi.ras.ru/EIMI/2011/PC/proceedings.pdf.

  3. A. D. Bruno, “Power-Exponential Expansions of Solutions to an Ordinary Differential Equation,” Dokl. Math. 85, 336–340 (2012).

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  4. A. D. Bruno, Preprint No. 60, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2011); http://www.keldysh.ru/papers/2011/source/prep2011-60.pdf.

  5. A. D. Bruno, “Space Power Geometry for One ODE and P 1-P 4, P 6,” Painlevé Equations and Related Topics, Ed. by A.D. Bruno and A.B. Batkhin (Walter de Gruyter, Berlin/Boston, 2012), pp. 41–51.

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  6. A. D. Bruno and A. V. Parusnikova, “Elliptic and Periodic Asymptotic Forms of Solution to P 5,” Ibid, pp. 53–65.

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  7. A. D. Bruno, “Regular Asymptotic Expansions of Solutions to One ODE and P 1-P 5,” Ibid, pp. 67–82.

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047, Russia

    A. D. Bruno

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  1. A. D. Bruno
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Correspondence to A. D. Bruno.

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Original Russian Text © A.D. Bruno, 2012, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2012, Vol. 52, No. 12, pp. 2206–2218.

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Bruno, A.D. Power-elliptic expansions of solutions to an ordinary differential equation. Comput. Math. and Math. Phys. 52, 1650–1661 (2012). https://doi.org/10.1134/S0965542512120056

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  • DOI: https://doi.org/10.1134/S0965542512120056

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