Tutorial for Green's Functions, Materials Reliability Division, N.I.S.T

Tutorial for Green's Functions, Materials Reliability Division, N.I.S.T

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Green's Functions Tutorial Introduction to Green's Functions The basic idea of a Green's function is very familiar to students of mathematical physics. However, most engineers are not typically exposed to the concept of a Green's function until possibly at the upper-graduate level. Green's functions play an important role in the solution of linear ordinary and partial differential equations, and are a key component to the development of boundary integral equation methods. Here we present: A brief review of Green's functions An example calulation for Green's functions A review of some common techniques used for finding Green's functions A bibliography for reference texts on Green's functions Green's Functions: Basic Concepts Consider a linear differential equation written in the general form (1) where L(x) is a linear, self-adjoint differential operator, u(x) is the unknown function, and f(x) is a known non-homogeneous term. For a discussion of the concept of self-adjoint and non self- adjoint differential operators please refer, for example, to the text by Morse and Feshbach . Operationally, we can write a solution to equation (1) as (2) where L -1 is the inverse of the differential operator L . Since L is a differential operator, it is reasonable to expect its inverse to be an integral operator. We expect the usual properties of inverses to hold, (3)
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where I is the identity operator. More specifically, we define the inverse operator as (4) where the kernel G(x;x') is the Green's Function associated with the differential operator L . Note that G(x;x') is a two-point function which depends on x and x' . To complete the idea of the inverse operator L , we introduce the Dirac delta function as the identity operator I . Recall the properties of the Dirac delta function are (5) The Green's function G(x;x') then satisfies (6) The solution to equation (1) can then be written directly in terms of the Green's function as
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This note was uploaded on 05/28/2010 for the course EE EE564 taught by Professor Runyiyu during the Spring '10 term at Eastern Mediterranean University.

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Tutorial for Green's Functions, Materials Reliability Division, N.I.S.T

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