Green's function Details, Meaning Green's function Article and Explanation Guide

Green's function Guide, Meaning , Facts, Information and Description

In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. Technically, a Green's function of a linear operator L acting on distributions over a manifold M, at a point x₀, is any solution of (Lf)(x) = δ(x − x₀), where δ is the Dirac delta function. If the kernel of L is nontrivial, then the Green's function is not unique. However, in practice, some combination of symmetry, boundary conditions and/or other externally imposed criteria would give us a unique Green's function. Also, please note Green's functions are distributions in general, not functions, meaning they can have discontinuities.

Not every operator L admits a Green's function. A Green's function can also be thought of as a one-sided inverse of L.

The Green's function was named for British mathematician George Green, who first developed the concept in the 1830s.

Table of contents

1 Motivation
2 Green's Function for solving inhomogeneous Boundary value problem

2.1 Working Frame
2.2 Theorem

3 Example
4 Futher examples

Motivation

Convolving with a Green's function gives solutions to inhomogeneous differentio-integral equations, most commonly a Sturm-Liouville problem . If g is the Green's function of an operator L, then the solution for f of the equation Lf = h is given by

This can be thought as an expansion of f according to Dirac delta function basis (projecting f over δ(x − s)) and a superposition of the solution on each projection.

Green's Function for solving inhomogeneous Boundary value problem

The primary use of Green's functions in mathematics is to solve inhomogeneous boundary value problems. In particle physics, Green's functions are also usually used as propagators in Feynman diagrams (and the phrase "Green's function" is often used for any correlation function).

Working Frame

Let L be a linear differential operator in the form of

and let D be the boundary conditions operator

Let f(x) be a continuous function in [0,l]. We shall also suppose that the problem

is regular, i.e. only the trivial soluton exists for the homogenous problem.

Theorem

Then there is one and only one solution u(x) which satisfies

and it is given by

where g(x,s) is Green's function and satisfies the following demands:

g(x,s) is continuous in x and s.
For , .
For , .
Derivative "jump": .
Symmetry: g(x,s) = g(s,x).

Example

Given the problem

Find Green's function.

First step: From demand-2 we see that

For x < s we see from demand-3 that the , while for x > s we see from demand-3 that the (we leave it to the reader to fill in the in-between steps).

Summarize the results:

Second step: Now we shall determine a(s) and b(s).

Using demand-1 we get

Using demand-4 we get

Using Cramer's rule or by intelligent guess solve for a(s) and b(s) and obtain that .

Check that this automatically satisfies demand-5.

So our Green's function for this problem is:

Futher examples

Let the manifold be R and L be d/dx. Then, the Heaviside step function H(x − x₀) is a Green's function of L at x₀.
Let the manifold be the quarter-plane { (x, y) : x, y ≥ 0 } and L be the Laplacian. Also, assume a Dirichlet boundary condition is imposed at x=0 and a Neumann boundary condition is imposed at y=0. Then the Green's function is G(x, y ; x₀, y₀)

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