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Green’s Functions with Applications
DEAN G. DUFFY
CHAPMAN & HALL/CRC
Introduction
Definitions of the Most Commonly Used Functions
1 Some Background Material
1.1 Historical Development
1.2 The Dirac Delta Function
1.3 Green’s Formulas
1.4 What Is a Green’s Function?
2 Green’s Functions for Ordinary Differential Equations
2.1 Initial-Value Problems
2.2 The Superposition Integral
2.3 Regular Boundary-Value Problems
2.4 Eigenfunction Expansion for Regular
Boundary-Value Problems
2.5 Singular Boundary-Value Problems
2.6 Maxwell’s Reciprocity
3 Green’s Functions for the Wave Equation
3.1 One-Dimensional Wave Equation
in an Unlimited Domain
3.2 One-Dimensional Wave Equation
on the Interval 0 < x < L
3.3 Axisymmetric Vibrations of a Circular Membrane
3.4 Two-Dimensional Wave Equation
in an Unlimited Domain
3.5 Three-Dimensional Wave Equation
in an Unlimited Domain
3.6 Asymmetric Vibrations of a Circular Membrane
3.7 Thermal Waves
3.8 Discrete Wavenumber Representation
3.9 Leaky Modes
3.10 Water Waves
4 Green’s Function for the Heat Equation
4.1 Heat Equation over Infinite
or Semi-Infinite Domains
4.2 Heat Equation Within a Finite Cartesian Domain
4.3 Heat Equation Within a Cylinder
4.4 Heat Equation Within a Sphere
4.5 Product Solution
4.6 Absolute and Convective Instability
5 Green’s Function for the Helmholtz Equation
5.1 Free-Space Green’s Functions for Helmholtz’s
and Poisson’s Equations
5.2 Two-Dimensional Poisson’s Equation
over Rectangular and Circular Domains
5.3 Two-Dimensional Helmholtz Equation
over Rectangular and Circular Domains
5.4 Poisson’s and Helmholtz’s Equations
on a Rectangular Strip
5.5 Three-Dimensional Problems in a Half-Space
5.6 Three-Dimensional Poisson’s Equation
in a Cylindrical Domain
5.7 Poisson’s Equation for a Spherical Domain
5.8 Improving the Convergence Rate of Green’s Functions
Appendix A: The Fourier Transform
A.1 Definition and Properties of Fourier Transforms
A.2 Inversion of Fourier Transforms
A.3 Solution of Ordinary Differential Equations
A.3 Solution of Partial Differential Equations
Appendix B: The Laplace Transform
B.1 Definition and Elementary Properties
B.2 The Shifting Theorems
B.3 Convolution
B.4 Solution of Linear Ordinary Differential Equations
with Constant Coefficients
B.5 Inversion by Contour Integration
B.6 Solution of Partial Differential Equations
Appendix C: Bessel Functions
C.1 Bessel Functions and Their Properties
C.2 Hankel Functions
C.3 Fourier-Bessel Series
Appendix D: Relationship between Solutions
of Helmholtz’s and Laplace’s Equations
in Cylindrical and Spherical Coordinates
Answers to Some of the Problems
Green’s Functions with Applications
DEAN G. DUFFY
CHAPMAN & HALL/CRC
Introduction
Definitions of the Most Commonly Used Functions
1 Some Background Material
1.1 Historical Development
1.2 The Dirac Delta Function
1.3 Green’s Formulas
1.4 What Is a Green’s Function?
2 Green’s Functions for Ordinary Differential Equations
2.1 Initial-Value Problems
2.2 The Superposition Integral
2.3 Regular Boundary-Value Problems
2.4 Eigenfunction Expansion for Regular
Boundary-Value Problems
2.5 Singular Boundary-Value Problems
2.6 Maxwell’s Reciprocity
3 Green’s Functions for the Wave Equation
3.1 One-Dimensional Wave Equation
in an Unlimited Domain
3.2 One-Dimensional Wave Equation
on the Interval 0 < x < L
3.3 Axisymmetric Vibrations of a Circular Membrane
3.4 Two-Dimensional Wave Equation
in an Unlimited Domain
3.5 Three-Dimensional Wave Equation
in an Unlimited Domain
3.6 Asymmetric Vibrations of a Circular Membrane
3.7 Thermal Waves
3.8 Discrete Wavenumber Representation
3.9 Leaky Modes
3.10 Water Waves
4 Green’s Function for the Heat Equation
4.1 Heat Equation over Infinite
or Semi-Infinite Domains
4.2 Heat Equation Within a Finite Cartesian Domain
4.3 Heat Equation Within a Cylinder
4.4 Heat Equation Within a Sphere
4.5 Product Solution
4.6 Absolute and Convective Instability
5 Green’s Function for the Helmholtz Equation
5.1 Free-Space Green’s Functions for Helmholtz’s
and Poisson’s Equations
5.2 Two-Dimensional Poisson’s Equation
over Rectangular and Circular Domains
5.3 Two-Dimensional Helmholtz Equation
over Rectangular and Circular Domains
5.4 Poisson’s and Helmholtz’s Equations
on a Rectangular Strip
5.5 Three-Dimensional Problems in a Half-Space
5.6 Three-Dimensional Poisson’s Equation
in a Cylindrical Domain
5.7 Poisson’s Equation for a Spherical Domain
5.8 Improving the Convergence Rate of Green’s Functions
Appendix A: The Fourier Transform
A.1 Definition and Properties of Fourier Transforms
A.2 Inversion of Fourier Transforms
A.3 Solution of Ordinary Differential Equations
A.3 Solution of Partial Differential Equations
Appendix B: The Laplace Transform
B.1 Definition and Elementary Properties
B.2 The Shifting Theorems
B.3 Convolution
B.4 Solution of Linear Ordinary Differential Equations
with Constant Coefficients
B.5 Inversion by Contour Integration
B.6 Solution of Partial Differential Equations
Appendix C: Bessel Functions
C.1 Bessel Functions and Their Properties
C.2 Hankel Functions
C.3 Fourier-Bessel Series
Appendix D: Relationship between Solutions
of Helmholtz’s and Laplace’s Equations
in Cylindrical and Spherical Coordinates
Answers to Some of the Problems