Solution of Initial Value Problems in Classes of Generalized Analytic Functions

by Wolfgang Tutschke

Overview

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 concept 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.

Product Details

ISBN-13:	9783540502166
Publisher:	Springer Berlin Heidelberg
Publication date:	05/01/1989
Pages:	188
Product dimensions:	0.00(w) x 0.00(h) x 0.02(d)

Table of Contents

0. Introduction.- 1. Initial Value Problems in Banach Spaces.- 2. Scales of Banach Spaces.- 3. Solution of Initial Value Problems in Scales of Banach Spaces.- 4. The Classical Cauchy-Kovalevskaya Theorem.- 5. The Holmgren Theorem.- 6. Basic Properties of Generalized Analytic Functions.- 7. Initial Value Problems with Generalized Analytic Initial Functions.- 8. Contraction-Mapping Principles in Scales of Banach Spaces.- 9. Further Existence Theorems for Initial Value Problems in Scales of Banach Spaces.- 10. Further Uniqueness Theorems.- References.

Editorial Reviews

A sound introduction to the generalization of the Cauchy-Kovalevska theorem. Solutions of initial value problems are given for the cases where the initial functions are generalized analytic functions. Two methods are used: Banach space scales, and a modification of W. Walter's elementary proof of the classical theorem by Cauchy- Kovalevska. Printed from typescript. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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