Green's function

In mathematics, 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.

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.

Motivation

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

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

While in particle physics, Green's functions are usually used as propagators in Feynman diagrams (and the phrase "Green's function" is actually often used for any correlation functions), one of GF main utilities in mathematics is to solve inhomogeneous boundary value problem.

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 solution u(x) which satisfies

and it is given by

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

1. g(x,s) is continuous in x and s.
2. For   , Lg(x,s) = 0.
3. For   , Dg(x,s) = 0.
4. Derivative "jump": g'(s _{+ 0},s) - g'(s _{- 0},s) = 1 / p(s).
5. Symmetry: g(x,s) = g(s,x).

Example

Given the problem

Lu = u'' + u = f(x)

 

Find Green's function.

First step: From demand-2 we see that

For x < s we see from demand-3 that the c₁(s) = 0, while for x > s we see from demand-3 that the c₂(s) = 0 (we leave to the reader to fill in the in-between steps).

Summerize the results:

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

Using demand-1 we get

a(s)sin(s) = b(s)cos(s).

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 automaticly satisfy demand-5 !

So, our Green's function to this problem is

Examples

• Let the manifold be R and L be d/dx. Then, the Heaviside 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)