The Cauchy problem / H.O. Fattorini ; foreword by Felix Browder.

Format:

Book

Author/Creator:

Fattorini, H. O. (Hector O.), 1938- author.

Contributor:

Browder, Felix E., writer of foreword.

Series:

Encyclopedia of mathematics and its applications ; v. 18.

Encyclopedia of mathematics and its applications ; volume 18

Language:

English

Subjects (All):

Cauchy problem.

Initial value problems.

Physical Description:

1 online resource (xxii, 636 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 1983.

Language Note:

English

Summary:

This volume deals with the Cauchy or initial value problem for linear differential equations. It treats in detail some of the applications of linear space methods to partial differential equations, especially the equations of mathematical physics such as the Maxwell, Schrödinger and Dirac equations. Background material presented in the first chapter makes the book accessible to mathematicians and physicists who are not specialists in this area as well as to graduate students.

Contents:

Cover; Half-title; Title; Copyright; Contents; Editor's Statement; Foreword; Preface; Chapter 0 Elements of Functional Analysis; 1 Banach and Hilbert Spaces. Bounded Operators.Integration of Vector-Valued Functions; 2 Linear Functionals: The Dual Space. Vector-Valued Analytic Functions; 3 Unbounded Operators; the Resolvent. Closed Operators; 4 Adjoints; 5 Generalized Sequences. Weak Convergence; 6 Normal, Symmetric, Self-Adjoint, and Unitary Operators in Hilbert Space; 7 Some Special Spaces and Their Duals; 8 Convolution and Mollifiers. Sobolev Spaces

Chapter 1 The Cauchy Problem for Some Equations of Mathematical Physics: The Abstract Cauchy Problem1.1 The Heat Equation in a Square; 1.2 The Abstract Cauchy Problem; 1.3 The Diffusion Equation in a Square; 1.4 The Schrodinger Equation; 1.5 The Cauchy Problem in (- oo, oo); 1.6 The Maxwell Equations; 1.7 Miscellaneous Notes; Chapter 2 Properly Posed Cauchy Problems: General Theory; 2.1 The Cauchy Problem in / ^ 0; 2.2 The Cauchy Problem in - oo < t < oo. The Adjoint Equation; 2.3 Semigroup Theory; 2.4 The Inhomogeneous Equation; 2.5 Miscellaneous Comments

Chapter 3 Dissipative Operators and Applications3.1 Dissipative Operators; 3.2 Ordinary Differential Operators in the Whole Line; 3.3 Ordinary Differential Operators in a Closed Interval.Semi-Infinite Intervals; 3.4 Ordinary Differential Operators in a Compact Interval; 3.5 Symmetric Hyperbolic Systems in the Whole Space; 3.6 Isometric Propagators and Conservative Operators.Dissipative Operators in Hilbert Space; 3.7 Differential Equations in Banach Lattices. Positive Solutions. Dispersive Operators; 3.8 Miscellaneous Comments

Chapter 4 Abstract Parabolic Equations: Applications to Second Order Parabolic Equations4.1 Abstract Parabolic Equations; 4.2 Abstract Parabolic Equations; Analytic Propagators; 4.3 Applications to Ordinary Differential Operators; 4.4 Second Order Partial Differential Operators.Dissipativity; 4.5 Second Order Partial Differential Operators.Assignation of Boundary Conditions; 4.6 Second Order Partial Differential Operators.Construction of m-Dissipative Extensions in L2; 4.7 Regularity Theorems; 4.8 Construction of m-Dissipative Extensions in Lp(Ω) and C(Ω)

4.9 Analyticity of Solution Operators4.10 Positivity and Compactness of Solution Operators; 4.11 Miscellaneous Comments; Chapter 5 Perturbation and Approximation of Abstract Differential Equations; 5.1 A Perturbation Result; 5.2 The Neutron Transport Equation; 5.3 Perturbation Results for Operators in 6(1,0) and in &; 5.4 Perturbation Results for the Schrodinger and Dirac Equations; 5.5 Second Order Differential Operators; 5.6 Symmetric Hyperbolic Systems in Sobolev Spaces; 5.7 Approximation of Abstract Differential Equations

5.8 Approximation of Abstract Differential Equations by Finite Difference Equations

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references (p. 510-626) and index.

ISBN:

1-139-88414-X

0-511-94128-5

1-107-10234-0

1-107-09373-2

1-107-08752-X

0-511-66279-3