Divergence property of formal solutions for singular first order linear partial differential equations
type: Item
instance of: scholarly article
title: Divergence property of formal solutions for singular first order linear partial differential equations
zbMATH Open document ID: 0956.35024
author: Masaki Hibino
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2001-03-07T00:00:00Z
review text: Summary: This paper is concerned with the study of the convergence and the divergence of formal power series solutions of the following first-order singular linear partial differential equation with holomorphic coefficients at the origin: \[ P(x, D)u(x)= \sum^d_{i= 1}a_i(x) D_iu(x)+ b(x) u(x)= f(x), \] where \(f(x)\) is holomorphic at the origin. Here the equation is said to be singular if \(a_i(0)= 0\) \((j= 1,\dots, d)\). In this case, it is known that under the so-called Poincaré condition, if \(\{a_i(x)\}^d_{i=1}\) generates a simple ideal, every formal solution is convergent. However, if we remove these conditions, we shall see that the formal solution, if it exists, may be divergent. More precisely, we will characterize the rate of divergence of formal solutions via Gevrey order of formal solutions determined by a Newton polyhedron, a generalization of Newton polygon which is familiar in the study of ordinary differential equations with an irregular singular point.
Mathematics Subject Classification ID: 35C10
Mathematics Subject Classification ID: 35A10
zbMATH DE Number: 1496030
zbMATH Keywords: rate of divergence
zbMATH Keywords: Gevrey orderr of formal solutions
zbMATH Keywords: Poincaré condition
zbMATH Keywords: Newton polyhedron
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195143361
OpenAlex ID: W1980241662
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DOI: 10.2977/PRIMS/1195143361
description: scientific article; zbMATH DE number 1496030