Abstract Cauchy Problems: Three Approaches, 1st Edition (Hardback) book cover

Abstract Cauchy Problems

Three Approaches, 1st Edition

By Irina V. Melnikova, Alexei Filinkov

Chapman and Hall/CRC

264 pages

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Hardback: 9781584882503
pub: 2001-03-27
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Description

Although the theory of well-posed Cauchy problems is reasonably understood, ill-posed problems-involved in a numerous mathematical models in physics, engineering, and finance- can be approached in a variety of ways. Historically, there have been three major strategies for dealing with such problems: semigroup, abstract distribution, and regularization methods. Semigroup and distribution methods restore well-posedness, in a modern weak sense. Regularization methods provide approximate solutions to ill-posed problems. Although these approaches were extensively developed over the last decades by many researchers, nowhere could one find a comprehensive treatment of all three approaches.

Abstract Cauchy Problems: Three Approaches provides an innovative, self-contained account of these methods and, furthermore, demonstrates and studies some of the profound connections between them. The authors discuss the application of different methods not only to the Cauchy problem that is not well-posed in the classical sense, but also to important generalizations: the Cauchy problem for inclusion and the Cauchy problem for second order equations.

Accessible to nonspecialists and beginning graduate students, this volume brings together many different ideas to serve as a reference on modern methods for abstract linear evolution equations.

Table of Contents

Preface

Introduction

ILLUSTRATION AND MOTIVATION

Heat Equation

The Reversed Cauchy Problem for the Heat Equation

Wave Equation

SEMIGROUP METHODS

C0-Semigroups

Integrated Semigroups

k-Convoluted Semigroups

C-Regularized Semigroups

Degenerate Semigroups

The Cauchy Problem for Inclusions

Second Order Equations

ABSTRACT DISTRIBUTION METHODS

The Cauchy Problem

The Degenerate Cauchy Problem

Ultradistributions and New Distributions

REGULARIZATION METHODS

The Ill-Posed Cauchy Problem

Regularization and C-Regularized Semigroups

Bibliographic Remark

Bibliography

Glossary of Notation

Index

About the Series

Monographs and Surveys in Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT007000
MATHEMATICS / Differential Equations
MAT037000
MATHEMATICS / Functional Analysis
SCI040000
SCIENCE / Mathematical Physics