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Zh. Vychisl. Mat. Mat. Fiz., 2018, Volume 58, Number 3, Pages 346–364 (Mi zvmmf10688)  

On complicated expansions of solutions to ODES

A. D. Bruno

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

Abstract: Polynomial ordinary differential equations are studied by asymptotic methods. The truncated equation associated with a vertex or a nonhorizontal edge of their polygon of the initial equation is assumed to have a solution containing the logarithm of the independent variable. It is shown that, under very weak constraints, this nonpower asymptotic form of solutions to the original equation can be extended to an asymptotic expansion of these solutions. This is an expansion in powers of the independent variable with coefficients being Laurent series in decreasing powers of the logarithm. Such expansions are sometimes called psi-series. Algorithms for such computations are described. Six examples are given. Four of them are concern with Painlevé equations. An unexpected property of these expansions is revealed.

Key words: ordinary differential equation, asymptotic expansion, solution with logarithms, Painlevé equation.

Funding Agency Grant Number
Russian Academy of Sciences - Federal Agency for Scientific Organizations PRAS-18-01


DOI: https://doi.org/10.7868/S0044466918030043

References: PDF file   HTML file

English version:
Computational Mathematics and Mathematical Physics, 2018, 58:3, 328–347

Bibliographic databases:

UDC: 519.62
Received: 29.12.2016
Revised: 25.07.2017

Citation: A. D. Bruno, “On complicated expansions of solutions to ODES”, Zh. Vychisl. Mat. Mat. Fiz., 58:3 (2018), 346–364; Comput. Math. Math. Phys., 58:3 (2018), 328–347

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  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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