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Abstract

We propose algorithms that allow for nonlinear equations to obtain asymptotic expansions of solutions in the form of: (a) power series with constant coefficients, (b) power series with coefficients which are power series of logarithm and (c) series of powers of exponent of a power series with coefficients which are power series as well. These algorithms are applicable to nonlinear equations (A) algebraic, (B) ordinary differential and (C) partial differential, and to systems of such equations as well. We give the description of the method for one ordinary differential equation and we enumerate some applications of these algorithms.

Keywords

Expansions of solutions to ODE Power expansions Complicated expansions Exponential expansions 

MSC

Primary 33E17 Secondary 34E05 41E58 

Notes

Acknowledgements

The work was supported by RFBR, grant No. 18-01-00422A and by the Program of the Presidium of the Russian Academy of Sciences 01 “Fundamental Mathematics and its Applications” under grant PRAS-18-01.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Keldysh Institute of Applied Mathematics of RASMoscowRussia

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