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Singular Nonlinear Partial Differential Equations

  • Raymond Gérard
  • Hidetoshi Tahara

Part of the Aspects of Mathematics book series (ASMA, volume 28)

Table of contents

About this book

Introduction

The aim of this book is to put together all the results that are known about the existence of formal, holomorphic and singular solutions of singular non linear partial differential equations. We study the existence of formal power series solutions, holomorphic solutions, and singular solutions of singular non linear partial differential equations. In the first chapter, we introduce operators with regular singularities in the one variable case and we give a new simple proof of the classical Maillet's theorem for algebraic differential equations. In chapter 2, we extend this theory to operators in several variables. The chapter 3 is devoted to the study of formal and convergent power series solutions of a class of singular partial differential equations having a linear part, using the method of iteration and also Newton's method. As an appli­ cation of the former results, we look in chapter 4 at the local theory of differential equations of the form xy' = 1(x,y) and, in particular, we show how easy it is to find the classical results on such an equation when 1(0,0) = 0 and give also the study of such an equation when 1(0,0) #- 0 which was never given before and can be extended to equations of the form Ty = F(x, y) where T is an arbitrary vector field.

Keywords

Maillet's type theorems differential equation holomorphic solutions non linear singular partial differential equations partial differential equation regular singularities singular partial differential equations

Authors and affiliations

  • Raymond Gérard
    • 1
  • Hidetoshi Tahara
    • 2
  1. 1.Institut de Recherche Mathématique AlsacienUniversité Louis PasteurStrasbourgFrance
  2. 2.Dept. of MathematicsSophia University102 TokyoJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-322-80284-2
  • Copyright Information Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH, Wiesbaden 1996
  • Publisher Name Vieweg+Teubner Verlag
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-322-80286-6
  • Online ISBN 978-3-322-80284-2
  • Series Print ISSN 0179-2156
  • Buy this book on publisher's site