Let us now turn to a different two-dimensional problem. A circular disk is prepared in such a way that its initial temperature is radially symmetric,
(10.1) |
Then it is placed between two perfect insulators and its circumference is connected to a freezer that keeps it at , as sketched in Fig. 10.2 .
Since the initial conditions do not depend on , we expect the solution to be radially symmetric as well, , which satisfies the equation
Once again we separate variables, , which leads to the equation
(10.3) |
This corresponds to the two equations
The radial equation (which has a regular singular point at ) is closely related to one of the most important equation of mathematical physics, Bessel’s equation. This equation can be reached from the substitution , so that with we get the equation
(10.5) |