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Green's Functions and Boundary Value Problems, 3rd Edition

ISBN: 978-0-470-60970-5
888 pages
February 2011
Green
Praise for the Second Edition

"This book is an excellent introduction to the wide field of boundary value problems."—Journal of Engineering Mathematics

"No doubt this textbook will be useful for both students and research workers."—Mathematical Reviews

A new edition of the highly-acclaimed guide to boundary value problems, now featuring modern computational methods and approximation theory

Green's Functions and Boundary Value Problems, Third Edition continues the tradition of the two prior editions by providing mathematical techniques for the use of differential and integral equations to tackle important problems in applied mathematics, the physical sciences, and engineering. This new edition presents mathematical concepts and quantitative tools that are essential for effective use of modern computational methods that play a key role in the practical solution of boundary value problems. With a careful blend of theory and applications, the authors successfully bridge the gap between real analysis, functional analysis, nonlinear analysis, nonlinear partial differential equations, integral equations, approximation theory, and numerical analysis to provide a comprehensive foundation for understanding and analyzing core mathematical and computational modeling problems.

Thoroughly updated and revised to reflect recent developments, the book includes an extensive new chapter on the modern tools of computational mathematics for boundary value problems. The Third Edition 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, the Hahn-Banach theorem, reflexive Banach spaces, the Banach Schauder and Banach-Steinhaus theorems, and the Lax-Milgram theorem has been incorporated into the book. New and revised exercises found throughout allow readers to develop their own problem-solving skills, and the updated bibliographies in each chapter provide an extensive resource for new and emerging research and applications.

With its careful balance of mathematics and meaningful applications, Green's Functions and Boundary Value Problems, Third Edition is an excellent book for courses on applied analysis and boundary value problems in partial differential equations at the graduate level. It is also a valuable reference for mathematicians, physicists, engineers, and scientists who use applied mathematics in their everyday work.

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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 and Strings.

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 Representation Theorem.

4.8 The Hahn-Banach Theorem and Reflexive Banach Spaces.

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 Approximation Methods.

7 Spectral Theory of Second-Order Differential Operators.

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 Parabolic Equations.

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. 

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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.

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