One way to solve a linear partial differential equation is by using a Green's function. A Green's function is a kind of an inverse operator for the differential operator in question. So imagine you've got an equation that looks like this:

D f(x,y,z,t) = g(x,y,z,t)

where D is a linear differential operator, f is the function you want to solve and g is an excitation, for example a source field for sound waves. With a Green's function you can work out f as:

f = D-1g,

where D-1 is the Green's function operator. If you have studied some electromagnetic theory, you may have already solved the following equation by a Green's function

e0(d2/dx2 + d2/dy2 + d2/dz2) V(x,y,z) = q(x,y,z)

The Green's function is

G(r,r') = 1/(4 Pi e0 |r - r'|)

where r = (x,y,z), r' = (x',y',z'), || is Euclidean distance and in terms of G the solution is:

V(r) = \Int dr' G(r,r') q(r')

where \Int denotes integration. In this example we didn't specify any boundary conditions, so this is the solution without any external fields.

A Green's function always has twice as many variables as the space we are in, one set of variables (r') for the 'source' and one set of variables (r) for where its contribution on the solution is evaluated. G(r,r') = G(r',r). Another interesting property is that

D G = Deltan(r)

where D is the differential operator that G is the Green's function of and Deltan is the Dirac delta function. This can be seen as an analogy of the matrix equation A A-1 = I. It is easy to check for the example function G(r,r').

Green's functions are used a lot in quantum field theory, where they are called propagators. The function was discovered by George Green.

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