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
List of Definitions
Mr. Green’s Essay
Ordinary Differential Equations
The Dirac Delta Function
What Is a Green’s Function?
Green’s Functions for Ordinary Differential Equations
The Superposition Integral
Regular Boundary-Value Problems
Eigenfunction Expansion for Regular Boundary-Value Problems
Singular Boundary-Value Problems
Generalized Green’s Function
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
Diffraction of a Cylindrical Pulse by a Half-Plane
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
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
Discrete Wavenumber Representation
Laplace Transform Method
Finite Difference 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
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.
About the Previous Edition
"Roughly speaking, Green's functions constitute infinitesimal matrix coefficients that one can use to solve linear nonhomogeneous differential equations in an approach alternative to that which depends on eigenvalue analysis. These techniques receive a mention in many books on differential equations. Duffy goes much further toward exposing the detailed workings of important examples (wave equation, heat equation, Hemholtz equation on various domains). … Many plots help the reader picture the behavior of these functions. … a valuable sourcebook."
—CHOICE Magazine, March 2002
"The focus of this book is predominantly on low-temperature plasmas, but it contains a wonderful depth of technical material and background for understanding in general much of the laboratory generated plasmas and various applications using laboratory generated plasmas…. Because it is so well written and illustrated, readers will be quickly able to understand and benefit from this book.
–IEEE Electrical Insulation (Nov/Dec 2016)