Theory of Differential Equations - 1st Edition - ISBN: 9781483229782, 9781483262321

Theory of Differential Equations

1st Edition

Authors: I. M. Gel'fand G. E. Shilov
eBook ISBN: 9781483262321
Imprint: Academic Press
Published Date: 1st January 1967
Page Count: 232
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Description

Generalized Functions, Volume 3: Theory of Differential Equations focuses on the application of generalized functions to problems of the theory of partial differential equations.

This book discusses the problems of determining uniqueness and correctness classes for solutions of the Cauchy problem for systems with constant coefficients and eigenfunction expansions for self-adjoint differential operators. The topics covered include the bounded operators in spaces of type W, Cauchy problem in a topological vector space, and theorem of the Phragmén-Lindelöf type. The correctness classes for the Cauchy problem, systems that are Petrovski?-correct, and generalized eigenfunctions of self-adjoint operators are also reviewed. This text likewise covers the differentiation of functionals of strongly and weakly bounded variation.

This volume is beneficial to students and researchers interested in the theory of differential equations.

Table of Contents


Translator's Note

Preface to the Russian Edition

Chapter I Spaces of Type W

1. Definitions

2. Bounded Operators in Spaces of Type W

3. Fourier Transforms

4. The Case of Several Variables

Chapter II Uniqueness Classes for the Cauchy Problem

1. Introduction

2. The Cauchy Problem in a Topological Vector Space

3. The Cauchy Problem for Systems of Partial Differential Equations. The Operator Method

4. The Cauchy Problem for Systems of Partial Differential Equations. The Method of Fourier Transforms

5. Examples

6. The Connection Between the Reduced Order of a System and its Characteristic Roots

7. A Theorem of the Phragmén-Lindelöf Type

Appendix 1. Convolution Equations

Appendix 2. Equations with Coefficients Which Depend on x

Appendix 3. Systems with Elliptic Operators

Chapter III Correctness Classes for the Cauchy Problem

1. Introduction

2. Parabolic Systems

3. Hyperbolic Systems

4. Systems Which are Petrovskiĭ-Correct

5. On the Solutions of Incorrect Systems

Chapter IV Generalized Eigenfunction Expansions

1. Introduction

2. Differentiation of Functionals of Strongly Bounded Variation

3. Differentiation of Functionals of Weakly Bounded Variation

4. Existence and Completeness Theorems for the System of Eigenfunctionals

5. Generalized Eigenfunctions of Self-Adjoint Operators

6. The Structure of the Generalized Eigenfunctions

7. Dynamical Systems

Notes and References

Bibliography

Index

Details

No. of pages:
232
Language:
English
Copyright:
© Academic Press 1967
Published:
Imprint:
Academic Press
eBook ISBN:
9781483262321

About the Author

I. M. Gel'fand

G. E. Shilov

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