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Singular Nonlinear Partial Differential Equations

  • Book
  • © 1996

Overview

Authors:
  1. Raymond Gérard

    1. Institut de Recherche Mathématique Alsacien, Université Louis Pasteur, Strasbourg, France

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  2. Hidetoshi Tahara

    1. Dept. of Mathematics, Sophia University, 102 Tokyo, Japan

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    You can also search for this author in PubMed   Google Scholar

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

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Table of contents (10 chapters)

  1. Front Matter

    Pages I-VIII
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  2. Operators with regular singularities: One variable case

    • Raymond Gérard, Hidetoshi Tahara
    Pages 1-41

  3. Operators with regular singularities: Several variables case

    • Raymond Gérard, Hidetoshi Tahara
    Pages 42-76

  4. Formal and convergent solutions of singular partial differential equations

    • Raymond Gérard, Hidetoshi Tahara
    Pages 77-110

  5. Local study of differential equations of the form xy′ = f(x,y) near x = 0

    • Raymond Gérard, Hidetoshi Tahara
    Pages 111-137

  6. Holomorphic and singular solutions of non linear singular first order partial differential equations

    • Raymond Gérard, Hidetoshi Tahara
    Pages 138-160

  7. Maillet’s type theorems for non linear singular partial differential equations

    • Raymond Gérard, Hidetoshi Tahara
    Pages 161-186

  8. Maillet’s type theorems for non linear singular partial differential equations without linear part

    • Raymond Gérard, Hidetoshi Tahara
    Pages 187-202

  9. Holomorphic and singular solutions of non linear singular partial differential equations

    • Raymond Gérard, Hidetoshi Tahara
    Pages 203-233

  10. On the existence of holomorphic solutions of the Cauchy problem for non linear partial differential equations

    • Raymond Gérard, Hidetoshi Tahara
    Pages 234-245

  11. Maillet’s type theorems for non linear singular integro—differential equations

    • Raymond Gérard, Hidetoshi Tahara
    Pages 246-263

  12. Back Matter

    Pages 264-272
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Keywords

About this book

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.

Authors and Affiliations

  • Institut de Recherche Mathématique Alsacien, Université Louis Pasteur, Strasbourg, France

    Raymond Gérard

  • Dept. of Mathematics, Sophia University, 102 Tokyo, Japan

    Hidetoshi Tahara

About the authors

Prof. Raymond Gerard ist am Institut de Recherche Mathématique Alsacien an der Université Louis Pasteur in Strasbourg beschäftigt. Prof. Hidetoshi Tahara lehrt an der Sophia Universität in Tokyo.

Bibliographic Information

  • Book Title: Singular Nonlinear Partial Differential Equations

  • Authors: Raymond Gérard, Hidetoshi Tahara

  • Series Title: Aspects of Mathematics

  • DOI: https://doi.org/10.1007/978-3-322-80284-2

  • Publisher: Vieweg+Teubner Verlag Wiesbaden

  • eBook Packages: Springer Book Archive

  • Copyright Information: Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 1996

  • Softcover ISBN: 978-3-322-80286-6Published: 16 December 2011

  • eBook ISBN: 978-3-322-80284-2Published: 06 December 2012

  • Series ISSN: 0179-2156

  • Edition Number: 1

  • Number of Pages: VIII, 272

  • Topics: Partial Differential Equations, Analysis

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