Features
- Provides a systematic approach to the various methods available for deriving a Green’s function
- Reviews the historical development of the Green’s function, the Fourier and Laplace transforms, the classical special functions of Bessel functions and Legendre polynomials, and the Dirac delta function
- Presents Green’s functions for each class of differential equation (ordinary differential, wave, heat, and Helmholtz equations) according to the number of spatial dimensions and the geometry of the domain
- Includes worked examples, problem sets, and illustrations from acoustics, applied mechanics, antennas, and the stability of fluids and plasmas
- Contains a new numerical methods chapter as well as hundreds of references to the literature on the construction and use of Green's functions
Summary
Since publication of the first edition over a decade ago, Green’s Functions with Applications has provided applied scientists and engineers with a systematic approach to the various methods available for deriving a Green’s function. This fully revised Second Edition retains the same purpose, but has been meticulously updated to reflect the current state of the art.
The book opens with necessary background information: a new chapter on the historical development of the Green’s function, coverage of the Fourier and Laplace transforms, a discussion of the classical special functions of Bessel functions and Legendre polynomials, and a review of the Dirac delta function.
The text then presents Green’s functions for each class of differential equation (ordinary differential, wave, heat, and Helmholtz equations) according to the number of spatial dimensions and the geometry of the domain. Detailing step-by-step methods for finding and computing Green’s functions, each chapter contains a special section devoted to topics where Green’s functions particularly are useful. For example, in the case of the wave equation, Green’s functions are beneficial in describing diffraction and waves.
To aid readers in developing practical skills for finding Green’s functions, worked examples, problem sets, and illustrations from acoustics, applied mechanics, antennas, and the stability of fluids and plasmas are featured throughout the text. A new chapter on numerical methods closes the book.
Included solutions and hundreds of references to the literature on the construction and use of Green's functions make Green’s Functions with Applications, Second Edition a valuable sourcebook for practitioners as well as graduate students in the sciences and engineering.
Table of Contents
Acknowledgments
Author
Preface
List of Definitions
Historical Development
Mr. Green’s Essay
Potential Equation
Heat Equation
Helmholtz’s Equation
Wave Equation
Ordinary Differential Equations
Background Material
Fourier Transform
Laplace Transform
Bessel Functions
Legendre Polynomials
The Dirac Delta Function
Green’s Formulas
What Is a Green’s Function?
Green’s Functions for Ordinary Differential Equations
Initial-Value Problems
The Superposition Integral
Regular Boundary-Value Problems
Eigenfunction Expansion for Regular Boundary-Value Problems
Singular Boundary-Value Problems
Maxwell’s Reciprocity
Generalized Green’s Function
Integro-Differential Equations
Green’s Functions for the Wave Equation
One-Dimensional Wave Equation in an Unlimited Domain
One-Dimensional Wave Equation on the Interval 0 < x < L
Axisymmetric Vibrations of a Circular Membrane
Two-Dimensional Wave Equation in an Unlimited Domain
Three-Dimensional Wave Equation in an Unlimited Domain
Asymmetric Vibrations of a Circular Membrane
Thermal Waves
Diffraction of a Cylindrical Pulse by a Half-Plane
Leaky Modes
Water Waves
Green’s Functions for the Heat Equation
Heat Equation over Infinite or Semi-Infinite Domains
Heat Equation within a Finite Cartesian Domain
Heat Equation within a Cylinder
Heat Equation within a Sphere
Product Solution
Absolute and Convective Instability
Green’s Functions for the Helmholtz Equation
Free-Space Green’s Functions for Helmholtz’s and Poisson’s Equation
Method of Images
Two-Dimensional Poisson’s Equation over Rectangular and Circular Domains
Two-Dimensional Helmholtz Equation over Rectangular and Circular Domains
Poisson’s and Helmholtz’s Equations on a Rectangular Strip
Three-Dimensional Problems in a Half-Space
Three-Dimensional Poisson’s Equation in a Cylindrical Domain
Poisson’s Equation for a Spherical Domain
Improving the Convergence Rate of Green’s Functions
Mixed Boundary Value Problems
Numerical Methods
Discrete Wavenumber Representation
Laplace Transform Method
Finite Difference Method
Hybrid Method
Galerkin Method
Evaluation of the Superposition Integral
Mixed Boundary Value Problems
Appendix: Relationship between Solutions of Helmholtz’s and Laplace’s Equations in Cylindrical and Spherical Coordinates
Answers to Some of the Problems
Author Index
Subject Index
Author(s) Bio
Dean G. Duffy received his bachelor of science in geophysics from Case Institute of Technology, Cleveland, Ohio, USA, and his doctorate of science in meteorology from the Massachusetts Institute of Technology, Cambridge, USA. He served in the US Air Force for four years as a numerical weather prediction officer. After his military service, he began a twenty-five year association with the National Aeronautics and Space Administration’s Goddard Space Flight Center, Greenbelt, Maryland, USA. Widely published, Dr. Duffy has taught courses at the US Naval Academy, Annapolis, Maryland, and the US Military Academy, West Point, New York.
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