Posted by peeterjoot on February 27, 2009
Lectures 12, 13, and perhaps 11 of Prof Brad Osgood’s Fourier transform lectures cover Schwartz’s distributions theory in a high level fashion, and this sounds particularily useful.
In trying to Fourier transform Maxwell’s equation and reduce the Green’s function to arrive at the retarded potential solutions, one has to do a whole lot of dubious stuff, pulling delta and unit step functions out of the air from integral representations.
I found the lecture notes on this material, and have now digested some of this distribution content.
As a test application I used these ideas to solve the 1D wave equation. After listening to the distribution lectures, I wasn’t convinced that this method would be practical, but the proof is in the application. I was surprised that it is actually simpler, with no “so many words” (as Osgood puts it) requirement to pull delta functions out of magic hats from PV sinc evaluations of the exponential integral. He’s done an excellent job at making this material accessible.
I plan to write up my wave equation distribution treatment for myself shortly, and try some variations (Poisson, higher dimensional wave equations, non-homogenous cases, heat equation, …). Will these ideas also extend naturally to higher grade objects such as the bivector electrodynamic field?