Green's Functions and Boundary Value Problems

by Stakgold, Ivar; Holst, Michael J.

• ISBN13: 9780470609705
• ISBN10: 0470609702
• Edition: 3rd
• Format: Hardcover
• Copyright: 2011-02-08
• Publisher: Wiley

Summary

This Third Edition includes basic modern tools of computational mathematics for boundary value problems and also provides the foundational mathematical material necssary to understand and use the tools. Central to the text is a down-to-earth approach that shows readers how to use differential and integral equations when tackling significant problems in the physical sciences, engineering, and applied mathematics, and the book maintains a careful balance between sound mathematics and meaningful applications. A new co-author, Michael J. Holst, has been added to this new edition, and together he and Ivar Stakgold incorporate recent developments that have altered the field of applied mathematics, particularly in the areas of approximation methods and theory including best linear approximation in linear spaces; interpolation of functions in Sobolev Spaces; spectral, finite volume, and finite difference methods; techniques of nonlinear approximation; and Petrov-Galerkin and Galerkin methods for linear equations.Additional topics have been added including weak derivatives and Sobolev Spaces, linear functionals, energy methods and A Priori estimates, fixed-point techniques, and critical and super-critical exponent problems.The authors have revised the complete book to ensure that the notation throughout remained consistent and clear as well as adding new and updated references. Discussions on modeling, Fourier analysis, fixed-point theorems, inverse problems, asymptotics, and nonlinear methods have also been updated.

Author Biography

Ivar Stakgold, PHD, is Professor Emeritus and former Chair of the Department of Mathematical Sciences at the University of Delaware. He is former president of the Society for Industrial and Applied Mathematics (SIAM), where he was also named a SIAM Fellow in the inaugural class of 2009. Dr. Stakgold's research interests include nonlinear partial differential equations, reaction-diffusion, and bifurcation theory. Michael Holst, PHD, is Professor in the Departments of Mathematics and Physics at the University of California, San Diego, where he is also CoDirector of both the Center for Computational Mathematics and the Doctoral Program in Computational Science, Mathematics, and Engineering. Dr. Holst has published numerous articles in the areas of applied analysis, computational mathematics, partial differential equations, and mathematical physics.

Table of Contents

Preface to Third Edition

Preface to Second Edition

Preface to First Edition

Preliminaries

Heat Conduction

Diffusion

Reaction-Diffusion Problems

The Impulse-Momentum Law: The Motion of Rods and Strings

Alternative Formulations of Physical Problems

Notes on Convergence

The Lebesgue Integral

Green's Functions (Intuitive Ideas)

Introduction and General Comments

The Finite Rod

Maximum Principle

Examples of Green's Functions

The Theory of Distributions

Basic Ideas, Definitions, Examples

Convergence of Sequences and Series of Distributions

Fourier Series

Fourier Transforms and Integrals

Differential Equations in Distributions

Weak Derivatives and Sobolev Spaces

One-Dimensional Boundary Value Problems

Review

Boundary Value Problems for Second-Order Equations

Boundary Value Problems for Equations of Order

Alternative Theorems

Modified Green?s Functions

Hilbert and Banach Spaces

Functions and Transformations

Linear Spaces

Metric Spaces, Normed Linear Spaces, Banach Spaces

Contractions and the Banach Fixed-Point Theorem

Hilbert Spaces, the Projection Theorem

Separable Hilbert Spaces and Orthonormal Bases

Linear Functionals, the Riesz Representation Theorem

The Hahn-Banach Theorem, Reflexive Banach Spaces

Operator Theory

Basic Ideas and Examples

Closed Operators

Invertibility--the State of an Operator

Adjoint Operators

Solvability Conditions

The Spectrum of an Operator

Compact Operators

Extremal Properties of Operators

The Banach-Schauder and Banach-Steinhaus Theorems

Integral Equations 353

Introduction

Fredholm Integral Equations

The Spectrum of a Self-Adjoint Compact Operator

The Inhomogeneous Equation

Variational Principles And Related Approximation Methods

Spectral Theory of Second-Order Differential Operators

Introduction; The Regular Problem

Weyl's Classification of Singular Problems

Spectral Problems with a Continuous Spectrum

Partial Differential Equations

Classification Of Partial Differential Equations

Typical Well-Posed Problems for Hyperbolic and Parabolic Equations

Elliptic Equations

Variational Principles for Inhomogeneous Problems

The Lax-Milgram Theorem

Nonlinear Problems

Introduction and Basic Fixed-Point Techniques

Branching Theory

Perturbation Theory for Linear Problems

Techniques For Nonlinear Problems

The Stability of the Steady State

Approximation Theory and Methods

Nonlinear Analysis Tools for Banach Spaces

Best and Near-Best Approximation in Banach Spaces

Overview of Sobolev and Besov Spaces

Applications to Elliptic Partial Differential Equations

Finite Element and Related Discretization Methods

Iterative Methods for Discretized Linear Equations

Methods for Nonlinear Equations

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