Green’s Functions with Applications, Second Edition

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

Mr. Green’s Essay

Potential Equation

Heat Equation

Helmholtz’s Equation

Wave Equation

Ordinary Differential Equations

Fourier Transform

Laplace Transform

Bessel Functions

Legendre Polynomials

The Dirac Delta Function

Green’s Formulas

What Is a Green’s Function?

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

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

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

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

Hybrid Method

Galerkin Method

Evaluation of the Superposition Integral

Mixed Boundary Value Problems

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.

Reviews

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)