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Newton polyhedra and power transformations. (English) Zbl 1017.34500
Summary: We give a simple presentation of an algorithm of selecting asymptotical first approximations of equations (algebraic and ordinary differential and partial differential). Here the first approximation of a solution of the initial equation is a solution of the corresponding first approximation of the equation. The algorithm is based on the geometry of power exponents including the Newton polyhedron. The geometry admits transformations induced by power transformations of coordinates. We give also a survey of applications of the algorithms in problems of Celestial Mechanics and Hydrodynamics.

MSC:
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A45 Theoretical approximation of solutions to ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
65L99 Numerical methods for ordinary differential equations
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