Title

Solutions of finitely smooth nonlinear singular differential equations and problems of diagonalization and triangularization

Keywords

Approximate solutions; Diagonalization; Finitely smooth nonlinear equations; Irregular singularities; Triangularization

Abstract

It is known that existence of a formal power series solution ŷ(x) to a system of nonlinear ordinary differential equations (ODEs) with analytic or infinitely smooth coefficients at an irregular singular point implies the existence of an actual solution y(x), which possesses the asymptotic expansion ŷ(x). In the present paper we extend this result for systems with finitely smooth coefficients. In this case one cannot speak about a formal power series solution ŷ(x); it has therefore to be replaced by the requirement of existence of an "approximate" solution y0(x). The existence of a corresponding actual solution is a subject of certain conditions that link the smoothness of the system, the "accuracy" of the approximation y0(x), and the "degeneracy" of the system, linearized with respect to y0(x). As applications, problems of reduction of linear time dependent systems of ODEs into diagonal and triangular forms, as well as some other problems, are considered. In particular, the well-known theorem on integration of linear systems with irregular singularities is extended from analytical to finitely smooth systems. In one of the simplest cases, our result is simultaneously a consequence of the classical Levinson theorem.

Publication Date

1-1-1998

Publication Title

SIAM Journal on Mathematical Analysis

Volume

29

Issue

3

Number of Pages

757-778

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1137/S0036141096307710

Socpus ID

0032391141 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/0032391141

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