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.