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A fixed point method to compute solvents of matrix polynomials

Fernando Marcos, Edgar Pereira (2010)

Mathematica Bohemica

Matrix polynomials play an important role in the theory of matrix differential equations. We develop a fixed point method to compute solutions of matrix polynomials equations, where the matricial elements of the matrix polynomial are considered separately as complex polynomials. Numerical examples illustrate the method presented.

A new proof of multisummability of formal solutions of non linear meromorphic differential equations

Jean-Pierre Ramis, Yasutaka Sibuya (1994)

Annales de l'institut Fourier

We give a new proof of multisummability of formal power series solutions of a non linear meromorphic differential equation. We use the recent Malgrange-Ramis definition of multisummability. The first proof of the main result is due to B. Braaksma. Our method of proof is very different: Braaksma used Écalle definition of multisummability and Laplace transform. Starting from a preliminary normal form of the differential equation x d y d x = G 0 ( x ) + λ ( x ) + A 0 y + x μ G ( x , y ) , the idea of our proof is to interpret a formal power series solution...

Additive groups connected with asymptotic stability of some differential equations

Árpád Elbert (1998)

Archivum Mathematicum

The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient λ 2 q ( s ) , s [ s 0 , ) is investigated, where λ and q ( s ) is a nondecreasing step function tending to as s . Let S denote the set of those λ ’s for which the corresponding differential equation has a solution not tending to 0. It is proved that S is an additive group. Four examples are given with S = { 0 } , S = , S = 𝔻 (i.e. the set of dyadic numbers), and S .

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