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Solution of Initial Value Problems in Classes of Generalized Analytic Functions

  • Book
  • © 1989

Overview

Authors:
  1. Wolfgang Tutschke

    1. Sektion Mathematik, Martin-Luther-Universität, Halle, GDR

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Table of contents (11 chapters)

  1. Front Matter

    Pages 1-6
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  2. Introduction

    • Wolfgang Tutschke
    Pages 7-9

  3. Initial Value Problems in Banach Spaces

    • Wolfgang Tutschke
    Pages 9-18

  4. Scales of Banach Spaces

    • Wolfgang Tutschke
    Pages 19-25

  5. Solution of Initial Value Problems in Scales of Banach Spaces

    • Wolfgang Tutschke
    Pages 26-50

  6. The Classical Cauchy-Kovalevskaya Theorem

    • Wolfgang Tutschke
    Pages 51-68

  7. The Holmgren Theorem

    • Wolfgang Tutschke
    Pages 69-84

  8. Basic Properties of Generalized Analytic Functions

    • Wolfgang Tutschke
    Pages 84-111

  9. Initial Value Problems with Generalized Analytic Initial Functions

    • Wolfgang Tutschke
    Pages 112-128

  10. Contraction-Mapping Principles in Scales of Banach Spaces

    • Wolfgang Tutschke
    Pages 128-158

  11. Further Existence Theorems for Initial Value Problems in Scales of Banach Spaces

    • Wolfgang Tutschke
    Pages 158-177

  12. Further Uniqueness Theorems

    • Wolfgang Tutschke
    Pages 177-180

  13. Back Matter

    Pages 181-188
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Keywords

About this book

The purpose of the present book is to solve initial value problems in classes of generalized analytic functions as well as to explain the functional-analytic background material in detail. From the point of view of the theory of partial differential equations the book is intend­ ed to generalize the classicalCauchy-Kovalevskayatheorem, whereas the functional-analytic background connected with the method of successive approximations and the contraction-mapping principle leads to the con­ cept of so-called scales of Banach spaces: 1. The method of successive approximations allows to solve the initial value problem du CTf = f(t,u), (0. 1) u(O) = u , (0. 2) 0 where u = u(t) ist real o. r vector-valued. It is well-known that this method is also applicable if the function u belongs to a Banach space. A completely new situation arises if the right-hand side f(t,u) of the differential equation (0. 1) depends on a certain derivative Du of the sought function, i. e. , the differential equation (0,1) is replaced by the more general differential equation du dt = f(t,u,Du), (0. 3) There are diff. erential equations of type (0. 3) with smooth right-hand sides not possessing any solution to say nothing about the solvability of the initial value problem (0,3), (0,2), Assume, for instance, that the unknown function denoted by w is complex-valued and depends not only on the real variable t that can be interpreted as time but also on spacelike variables x and y, Then the differential equation (0.

Authors and Affiliations

  • Sektion Mathematik, Martin-Luther-Universität, Halle, GDR

    Wolfgang Tutschke

Bibliographic Information

  • Book Title: Solution of Initial Value Problems in Classes of Generalized Analytic Functions

  • Authors: Wolfgang Tutschke

  • DOI: https://doi.org/10.1007/978-3-662-09943-8

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1989

  • Softcover ISBN: 978-3-540-50216-6Published: 30 March 1989

  • eBook ISBN: 978-3-662-09943-8Published: 09 March 2013

  • Edition Number: 1

  • Number of Pages: 188

  • Additional Information: Jointly published with BSB B.G. Teubner Verlagsgesellschaft, Leipzig

  • Topics: Analysis, Theoretical, Mathematical and Computational Physics

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