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