Green's Functions and Boundary Value Problems

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  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2011-02-08
  • Publisher: Wiley

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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
Heat Conduction
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
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
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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