Green's Functions with Applications

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.

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

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.