Melnikova / Filinkov Abstract Cauchy Problems

Three Approaches
Erscheinungsjahr 2010
ISBN: 978-1-4200-3549-0
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 264 Seiten

Reihe: Monographs and Surveys in Pure and Applied Mathematics

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.

Zielgruppe

Graduate students and researchers in functional analysis; Researchers in applied mathematics, science, and finance interested in functional analysis.

Autoren/Hrsg.

Melnikova, Irina V.

Filinkov, Alexei

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

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