Green's Function Article Index for
Green's 		Website Links For
Greens 		 

Information About

Green's Function
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



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.

Green functions are also a useful tool in condensed matter theory, where they allow the resolution of the Diffusion Equation - and in Quantum Mechanics , where the Green function of the Hamiltonian is a key concept, with important links to the concept of density of states. The Green functions used in those two domains are highly similar, due to the analogy in the mathematical structure of the Diffusion Equation and Schrödinger Equation .

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


MOTIVATION


Convolving with a Green's function gives solutions to inhomogeneous differential-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

: f(x) = \int{ h(s) g(x,s) \, ds}.

This can be thought of as an expansion of ''h'' according to a Dirac Delta Function basis (projecting ''h'' over δ(''x'' − ''s'')) and a superposition of the solution on each Projection . Such an integral is known as a Fredholm Integral Equation , the study of which constitutes Fredholm Theory .


GREEN'S FUNCTION FOR SOLVING INHOMOGENEOUS BOUNDARY VALUE PROBLEMS


The primary use of Green's functions in mathematics is to solve inhomogeneous Boundary Value Problem s. In Particle Physics , Green's functions are also usually used as Propagator s in Feynman Diagram s (and the phrase "Green's function" is often used for any Correlation Function ).


Working frame


Let L be the Sturm-Liouville operator, a linear differential operator of the form
: L = {d \over dx}\left p(x) {d \over dx} ight + q(x)
and let ''D'' be the Boundary Condition s operator
: Du = \left\{\begin{matrix} \alpha _1 u'(0) + \beta _1 u(0) \ \alpha _2 u'(l) + \beta _2 u(l) \end{matrix} ight.

Let f(x) be a Continuous Function in {Link without Title} . We shall also suppose that the problem
: \begin{matrix}Lu = f \ Du = 0 \end{matrix}
is regular, i.e. only the Trivial solution exists for the Homogeneous problem.


Theorem


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

: \begin{matrix}Lu = f \ Du = 0 \end{matrix}

and it is given by

: u(x) = \int_{0}^{l}{ f(s) g(x,s) \, ds}

where ''g(x,s)'' is Green's function and satisfies the following demands:
# ''g(x,s)'' is continuous in ''x'' and ''s''.
# For x
e s , L g ( x, s ) = 0 .
# For s
e 0, l , D g ( x, s ) = 0 .
# Derivative "jump": g ' ( s_{ + 0}, s ) - g ' (s_{ - 0}, s ) = 1 / p(s) .
# Symmetry: ''g''(''x'', ''s'') = ''g''(''s'', ''x'').


FINDING GREEN'S FUNCTIONS


Eigenvalue expansions


If a Differential Operator ''L'' admits a set of Eigenvectors \Psi_n(x) (i.e. a set of functions \Psi_n(x) and scalars \lambda_n such that L \Psi_n = \lambda_n \Psi_n) ) that are complete, then we can construct a Green's function from these eigenvectors and Eigenvalues .

By complete, we mean that the set of functions : \Psi_n(x) satisfies the following Completeness Relation :

: \delta(x - x') = \sum_{n=0}^\infty \Psi_n(x) \Psi_n(x').

We can prove the following:

: G(x, x') = \sum_{n=0}^\infty rac{\Psi_n(x) \Psi_n(x')}{\lambda_n}.

Now consider acting on this on each side with the operator L. We'll end up with the completeness relation, which was assumed true.

The general study of the Green's function written in the above form, and its relationship to the Function Space s formed by the eigenvectors, is known as Fredholm Theory .


GREEN'S FUNCTION FOR THE LAPLACIAN


Green's functions for linear differential operators involving the Laplacian may be readily put to use using the second of Green's Identities .

To derive Green's theorem, begin with the divergence theorem (otherwise known as gauss' law):

: \int_V
abla \cdot \hat A\ dV = \int_S \hat A \cdot d\hat\sigma

Let A = \phi
abla\psi - \psi
abla\phi and substitute into Gauss' law. Compute
abla\cdot\hat A and apply the chain rule for the
abla operator:


abla\cdot\hat A =
abla\cdot(\phi
abla\psi - \psi
abla\phi) = (
abla\phi)\cdot(
abla\psi) + \phi
abla^2\psi - (
abla\phi)\cdot(
abla\psi) - \psi
abla^2\phi = \phi
abla^2\psi - \psi
abla^2\phi

Plugging this into the divergence theorem, we arrive at Green's theorem:

: \int_V \phi
abla^2\psi - \psi
abla^2\phi\ dV = \int_S \phi
abla\psi - \psi
abla\phi \cdot d\hat\sigma

Suppose that our linear differential operator ''L'' is the Laplacian ,
abla^2, and that we have a Green's function ''G'' for the Laplacian. The defining property of the green's function still holds:

:L G(x,x') =
abla^2 G(x,x') = \delta(x-x')

Let \psi = G in Green's theorem. We get:

: \int_V \phi(x') \delta(x - x') - G(x,x')
abla^2\phi(x')\ d^3x' = \int_S \phi(x')
abla' G(x,x') - G(x,x')
abla'\phi(x') \cdot d\hat\sigma'

Using this expression, we can solve the Laplace Equation (
abla^2\phi(x)=0) or Poisson Equation ((
abla^2\phi(x)=-4\pi ho(x)) subject to either Neumann or Dirichlet boundary conditions. In other words, we can solve for \phi(x) everywhere inside a volume where either (1) the value of \phi(x) is specified on the bounding surface of the volume (Dirichlet boundary conditions), or (2) the normal derivative of \phi(x) is specified on the bounding surface.

Suppose we're interested in solving for \phi(x) inside the region. Then the integral \int_V\phi(x')\delta(x-x')\ d^3x' reduces to simply \phi(x) due to the defining property of the Dirac Delta Function and we have:

:\phi(x) = \int_V G(x,x') ho(x')\ d^3x' + \int_S \phi(x')
abla' G(x,x') - G(x,x')
abla'\phi(x') \cdot d\hat\sigma'

This form expresses the well-known property of Harmonic Function s, that if the value or normal derivative is known on a bounding surface, then the value of the function inside the volume is known everywhere.

In Electrostatics , we interpret \phi(x) as the Electric Potential , ho(x) as Electric Charge Density , and the normal derivative
abla\phi(x')\cdot d\hat\sigma' as the normal component of the electric field.

If we're interested in solving a Dirichlet boundary value problem, we choose our Green's function such that G(x,x') vanishes when either x or x' is on the bounding surface; conversely, if we're interested in solving a Neumann boundary value problem, we choose our Green's function such that its normal derivative vanishes on the bounding surface. Thus we are left with only one of the two terms in the surface integral.

With no boundary conditions, the Green's function for the Laplacian is: