I often see solutions to radiating, center fed dipoles with "approximations" of the current distribution being sinusoidal. When describing this, people often refer to the "Boundary Conditions" as no current on the ends of the dipole, and peak current at the center, where the source is. But this seems hand wavy, I've never actually seen a mathematical description of this current distribution. Are these actual boundary conditions, or just intuitive ones?
Moreover, if these are actual boundary conditions, I would expect the solution to include "modes" similar to a cavity resonator. These might reflect the solutions of the lambda/4, lambda/2, lambda, 2/lambda ... etc. dipole antenna. Instead most people just assume a current distribution and find the radiation from it. Is there a rigorous way to find the current distribution on the conductor? I imagine it's some solution of the diffusion equation?