Green's Functions with Applications
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Summary
Since its introduction in 1828, using Green's functions has become a fundamental mathematical technique for solving boundary value problems. Most treatments, however, focus on its theory and classical applications in physics rather than the practical means of finding Green's functions for applications in engineering and the sciences.
Green's Functions with Applications systematically presents the various methods of deriving these useful functions. It leads readers through the process of developing Green's functions for ordinary and partial differential equations. In addition to exploring the classical problems involving the wave, heat, and Helmholtz equations, the book includes special sections on leaky modes, water waves, and absolute/convective instability.
The author gives special attention to the numerical evaluation of Green's functions. By illustrating many of the functions in the text and problem sets, he helps readers develop an intuition about the behavior of Green's function in certain problems. He also considers the questions of the computational efficiency and possible methods for accelerating the process.
With its wealth of examples and problems drawn from the literature, this book provides a treasure-trove of methods to construct and compute Green's functions. It is the most exhaustive source book of Green's functions yet available and the only one designed specifically for engineering and scientific applications.
Table of Contents
INTRODUCTION
DEFINITIONS OF THE MOST COMMONLY USED FUNCTIONS
SOME BACKGROUND MATERIAL
Historical Development
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
GREEN'S FUNCTIONS FOR THE WAVE EQUATION
One-Dimensional Wave Equation in an Unlimited Domain
One-Dimensional Wave Equation on the Interval 0
Editorial Reviews
"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
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