The diophantine nature for the convergence of formal solutions
type: Item
instance of: scholarly article
title: The diophantine nature for the convergence of formal solutions
zbMATH Open document ID: 0592.35006
author: Masafumi Yoshino
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1986-01-01T00:00:00Z
review text: The author studies the convergence of all formal solutions \(u(x)=x^{\omega}\sum_{\eta \in {\mathbb{N}}^ d}u_{\eta}x^{\eta}/\eta !\) of equations, essentially, of the form \[ Pu\in \sum_{| \alpha | =| \beta | \leq m}x^{\alpha}a_{\alpha \beta}(x)(\partial /\partial x)^{\beta}u(x)=f\quad (x)x^{\omega} \] where \(\omega \in {\mathbb{C}}^ d\), \(x\in {\mathbb{C}}^ d\), \(d\geq 2\), \(\alpha =(\alpha_ 1,...,\alpha_ d)\in {\mathbb{N}}^ d\), \({\mathbb{N}}=\{0,1,2,...\}\) and where \(a_{\alpha \beta}(x)\) is analytic at \(x=0\). He gives a necessary and sufficient condition for the convergence of all formal solutions, which is described by use of two diophantine functions.
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35A99
Mathematics Subject Classification ID: 35G05
zbMATH DE Number: 3951059
zbMATH Keywords: analytic partial differential equations
zbMATH Keywords: formal solutions
zbMATH Keywords: diophantine functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228414
OpenAlex ID: W2130756713
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DOI: 10.2748/TMJ/1178228414
description: scientific article; zbMATH DE number 3951059