Stakgold / Holst Green's Functions and Boundary Value Problems

3. Auflage 2011
ISBN: 978-0-470-90652-1
Verlag: John Wiley & Sons
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 736 Seiten, E-Book

Reihe: Wiley Series in Pure and Applied Mathematics

Praise for the Second Edition
"This book is an excellent introduction to the wide field ofboundary value problems."--Journal of EngineeringMathematics
"No doubt this textbook will be useful for both students andresearch workers."--Mathematical Reviews
A new edition of the highly-acclaimed guide to boundary valueproblems, now featuring modern computational methods andapproximation theory
Green's Functions and Boundary Value Problems, Third Editioncontinues the tradition of the two prior editions by providingmathematical techniques for the use of differential and integralequations to tackle important problems in applied mathematics, thephysical sciences, and engineering. This new edition presentsmathematical concepts and quantitative tools that are essential foreffective use of modern computational methods that play a key rolein the practical solution of boundary value problems. With acareful blend of theory and applications, the authors successfullybridge the gap between real analysis, functional analysis,nonlinear analysis, nonlinear partial differential equations,integral equations, approximation theory, and numerical analysis toprovide a comprehensive foundation for understanding and analyzingcore mathematical and computational modeling problems.
Thoroughly updated and revised to reflect recent developments,the book includes an extensive new chapter on the modern tools ofcomputational mathematics for boundary value problems. The ThirdEdition features numerous new topics, including:
* Nonlinear analysis tools for Banach spaces
* Finite element and related discretizations
* Best and near-best approximation in Banach spaces
* Iterative methods for discretized equations
* Overview of Sobolev and Besov space linear
* Methods for nonlinear equations
* Applications to nonlinear elliptic equations
In addition, various topics have been substantially expanded,and new material on weak derivatives and Sobolev spaces, theHahn-Banach theorem, reflexive Banach spaces, the Banach Schauderand Banach-Steinhaus theorems, and the Lax-Milgram theorem has beenincorporated into the book. New and revised exercises foundthroughout allow readers to develop their own problem-solvingskills, and the updated bibliographies in each chapter provide anextensive resource for new and emerging research andapplications.
With its careful balance of mathematics and meaningfulapplications, Green's Functions and Boundary Value Problems, ThirdEdition is an excellent book for courses on applied analysis andboundary value problems in partial differential equations at thegraduate level. It is also a valuable reference for mathematicians,physicists, engineers, and scientists who use applied mathematicsin their everyday work.

Autoren/Hrsg.

Stakgold, Ivar

Holst, Michael J.

Inhaltsverzeichnis

Preface to Third Edition.
Preface to Second Edition.
Preface to First Edition.
0 Preliminaries.
0.1 Heat Conduction.
0.2 Diffusion.
0.3 Reaction-Diffusion Problems.
0.4 The Impulse-Momentum Law: The Motion of Rods andStrings.
0.5 Alternative Formulations of Physical Problems.
0.6 Notes on Convergence.
0.7 The Lebesgue Integral.
1 Green's Functions (Intuitive Ideas).
1.1 Introduction and General Comments.
1.2 The Finite Rod.
1.3 The Maximum Principle.
1.4 Examples of Green's Functions.
2 The Theory of Distributions.
2.1 Basic Ideas, Definitions, and Examples.
2.2 Convergence of Sequences and Series of Distributions.
2.3 Fourier Series.
2.4 Fourier Transforms and Integrals.
2.5 Differential Equations in Distributions.
2.6 Weak Derivatives and Sobolev Spaces.
3 One-Dimensional Boundary Value Problems.
3.1 Review.
3.2 Boundary Value Problems for Second-Order Equations.
3.3 Boundary Value Problems for Equations of Order p.
3.4 Alternative Theorems.
3.5 Modified Green's Functions.
4 Hilbert and Banach Spaces.
4.1 Functions and Transformations.
4.2 Linear Spaces.
4.3 Metric Spaces, Normed Linear Spaces, and Banach Spaces.
4.4 Contractions and the Banach Fixed-Point Theorem.
4.5 Hilbert Spaces and the Projection Theorem.
4.6 Separable Hilbert Spaces and Orthonormal Bases.
4.7 Linear Functionals and the Riesz RepresentationTheorem.
4.8 The Hahn-Banach Theorem and Reflexive BanachSpaces.
5 Operator Theory.
5.1 Basic Ideas and Examples.
5.2 Closed Operators.
5.3 Invertibility: The State of an Operator.
5.4 Adjoint Operators.
5.5 Solvability Conditions.
5.6 The Spectrum of an Operator.
5.7 Compact Operators.
5.8 Extremal Properties of Operators.
5.9 The Banach-Schauder and Banach-Steinhaus Theorems.
6 Integral Equations.
6.1 Introduction.
6.2 Fredholm Integral Equations.
6.3 The Spectrum of a Self-Adjoint Compact Operator.
6.4 The Inhomogeneous Equation.
6.5 Variational Principles and Related ApproximationMethods.
7 Spectral Theory of Second-Order DifferentialOperators.
7.1 Introduction; The Regular Problem.
7.2 Weyl's Classification of Singular Problems.
7.3 Spectral Problems with a Continuous Spectrum.
8 Partial Differential Equations.
8.1 Classification of Partial Differential Equations.
8.2 Well-Posed Problems for Hyperbolic and ParabolicEquations.
8.3 Elliptic Equations.
8.4 Variational Principles for Inhomogeneous Problems.
8.5 The Lax-Milgram Theorem.
9 Nonlinear Problems.
9.1 Introduction and Basic Fixed-Point Techniques.
9.2 Branching Theory.
9.3 Perturbation Theory for Linear Problems.
9.4 Techniques for Nonlinear Problems.
9.5 The Stability of the Steady State.
10 Approximation Theory and Methods.
10.1 Nonlinear Analysis Tools for Banach Spaces.
10.2 Best and Near-Best Approximation in Banach Spaces.
10.3 Overview of Sobolev and Besov Spaces.
10.4 Applications to Nonlinear Elliptic Equations.
10.5 Finite Element and Related Discretization Methods.
10.6 Iterative Methods for Discretized Linear Equations.
10.7 Methods for Nonlinear Equations.
Index.

Über Autor(innen)

IVAR STAKGOLD, PhD, is Professor Emeritus and former Chairof the Department of Mathematical Sciences at the University ofDelaware. He is former president of the Society for Industrial andApplied Mathematics (SIAM), where he was also named a SIAM Fellowin the inaugural class of 2009. Dr. Stakgold's research interestsinclude nonlinear partial differential equations,reaction-diffusion, and bifurcation theory.
MICHAEL HOLST, PhD, is Professor in the Departments ofMathematics and Physics at the University of California, San Diego,where he is also CoDirector of both the Center for ComputationalMathematics and the Doctoral Program in Computational Science,Mathematics, and Engineering. Dr. Holst has published numerousarticles in the areas of applied analysis, computationalmathematics, partial differential equations, and mathematicalphysics.