Solution of Initial Value Problems in Classes of Generalized Analytic Functions

  • Wolfgang Tutschke

Table of contents

  1. Front Matter
    Pages 1-6
  2. Wolfgang Tutschke
    Pages 7-9
  3. Wolfgang Tutschke
    Pages 9-18
  4. Wolfgang Tutschke
    Pages 19-25
  5. Wolfgang Tutschke
    Pages 51-68
  6. Wolfgang Tutschke
    Pages 69-84
  7. Wolfgang Tutschke
    Pages 177-180
  8. Back Matter
    Pages 181-188

About this book

Introduction

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.

Keywords

Banach Space analytic function banach spaces derivative differential equation partial differential equation

Authors and affiliations

  • Wolfgang Tutschke
    • 1
  1. 1.Sektion MathematikMartin-Luther-UniversitätHalleGDR

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-09943-8
  • Copyright Information Springer-Verlag Berlin Heidelberg 1989
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-540-50216-6
  • Online ISBN 978-3-662-09943-8
  • About this book