As by now you should fully understand from working with the Poisson equation, one very general way to solve inhomogeneous partial differential equations (PDEs) is to build a Green's function11.1 and write the solution as an integral equation.
Let's very quickly review the general concept (for a further discussion
don't forget WIYF
,MWIYF). Suppose is a general (second order)
linear partial differential operator on e.g.
and one wishes to
solve the inhomogeneous equation:
If one can find a solution
to the associated
differential equation for a point source function11.2:
This solution can easily be verified:
It seems, therefore, that we should thoroughly understand the ways of building Green's functions in general for various important PDEs. I'm uncertain of how much of this to do within these notes, however. This isn't really ``Electrodynamics'', it is mathematical physics, one of the fundamental toolsets you need to do Electrodynamics, quantum mechanics, classical mechanics, and more. So check out Arfken, Wyld, WIYF , MWIYFand we'll content ourselves with a very quick review of the principle ones we need: