Boundary Value Problems of Mathematical Physics I & II 0898714567

Table of contents :
Title page (I)
PREFACE TO THE CLASSICS EDITION
PREFACE
1. THE GREEN'S FUNCTION
1.1 The String Subject to Transverse Loading
1.2 The Dirac Delta Function
1.3 The Theory of Distributions
1.4 Preliminary Results on Linear Equations of the Second Order
1.5 Boundary Value Problems
1.6 Alternative Theorems and the Modified Green's Function
Suggested Readings for Chapter 1
2. INTRODUCTION TO LINEAR SPACES
2.1 Functions and Transformations
2.2 Linear Spaces
2.3 Metrlc Spaces, Normed Linear Spaces, and Inner Product Spaces
2.4 Properties of a Separable Hilbert Space
2.5 Functionals
2.6 Transformations
2.7 Linear Transformations on E_n^{(c)}
2.8 The Inverse of a Linear Transformation in Hilbert Space
2.9 The Spectrum of an Operator
2.10 Completely Continuous Operators
2.11 Extremal Properties of Bounded Operators
Suggested Readings for Chapter 2
3. LINEAR INTEGRAL EQUATIONS
3.1 Introduction
3.2 The Neumann Series (Method of Successive Approximations)
3.3 The Spectrum of a Self-adjoint Hilbert-Schmidt Operator
3.4 The Solution of the Inhomogeneous Equation wlth a Symmetric Hilbert-Schmidt Kernel
3.5 Extremal Principles
3.6 Approximations Based on Extremal Principles
3.7 Questions Relating to Continuity and Uniform Convergence - The Bilinear Series for the Kernel and the Iterated Kernels
3.8 Approximate Methods for the Solution of Integral Equations
3.9 Nonsymmetric Hilbert-Schmidt Operators
Suggested Readings for Chapter 3
4. SPECTRAL THEORY OF SECOND-ORDER DIFFERENTIAL OPERATORS
4.1 Introduction
4.2 The Regular Boundary Value Problem
4.3 Introductory Examples of Singular Problems
4.4 The General Singular Problem
Suggested Readings for Chapter 4
APPENDIX A
A.1 Statlc and Dynamic Problems for Strings and Membranes
A.2 Static and Dynamic Problems for Beams and Plates
A.3 The Equation of Heat Conduction
APPENDIX B
B.1 Bessel Functions
B.2 Wronskian Relationships
B.3 The Modified Bessel Function
B.4 The Behavior of Cylinder Functions at Zero and at Infinlty
INDEX
Title page (II)
PREFACE TO THE CLASSICS EDITION
PREFACE
5. DISTRIBUTIONS AND GENERALIZED SOLUTIONS
5.1 Introduction
5.2 Test Functions
5.3 Distributions
5.4 Convergence of Distributions
5.5 Additional Properties of Distributions
5.6 Fourier Transforms
5.7 Partial Differentiai Equations for Distributions
5.8 Fundamental Solutions
5.9 Classification of Partial Differentiai Equations
6. POTENTIAL THEORY
6.1 Introduction
6.2 Interior Dirichlet Problem for the Unit Circle
6.3 Some Properties of Harmonic Functions
6.4 Surface Layers
6.5 Integral Equations of Potential Theory
6.6 Green's Function for the Negative Laplacian
6.7 Methods for Determining the Green's Function
6.8 Some Physical Applications of Potentlal Theory
7. EQUATIONS OF EVOLUTION
7.1 Introduction
7.2 Causal Green's Function for Heat Conduction
7.3 Methods for Flnding the Causal Green's Function
7.4 Uniqueness and Continuous Dependence on the Data
7.5 Miscellaneous Topies Related to the Heat Equation
7.6 Preliminary Considerations for the Undamped Wave Equation
7.7 Causal Green's Function for the Wave Equation
7.8 Problems in One Space Dimension
7.9 Problems in More than One Dimension
7.10 Wave Equation with External Damping
7.11 Monochromatic Excitation and the Principle of Limiting Absorption
7.12 Green's Function for the Helmholtz Operator and Applications
7.13 Half-Plane Excited by a Line Source or a Plane Wave
7.14 Representation of Solutions of the Helmholtz Equation in Exterior Domains
7.15 Scattering Problem
7.16 Wlener-Hopf Method
8. VARIATIONAL AND RELATED METHODS
8.1 Introduction
8.2 Best Approximation in a Subspace
8.3 Maximum Theorem
8.4 Ritz-Rayleigh Method
8.5 Complementary Variational Principles
8.6 Capacity Problem
8.7 Natural Boundary Conditions
8.8 Indefinite and Nonsymmetric Operators
8.9 Other Methods for Upper Bounds to Functionals Associated with Positive Operators
8.10 Method of Least Squares
8.11 Extremal Principles for Eigenvalue Problems on Euclidean n Space
8.12 Eigenvalue Problems in Hilbert Space
8.13 Lower Bounds to Eigenvalues
APPENDIX A. SPHERICAL HARMONICS
APPENDIX B. ASYMPTOTIC EXPANSIONS
SUGGESTED ADDITIONAL READINGS
INDEX