We started the discussion from the problem of the temperature on a circular disk, solved in polar coordinates, Since the initial conditions do not depend on , we expect the solution to be radially symmetric as well, , which satisfies the equation
With we found the equations
The equation for is clearly self-adjoint, it can be written as
(10.62) |
So how does the equation for relate to Bessel’s equation? Let us make the change of variables . We find
(10.63) |
and we can remove a common factor to obtain ( )
(10.64) |
which is Bessel’s equation of order , i.e.,
(10.65) |
The boundary condition shows that
(10.66) |
where are the points where . We thus conclude
(10.67) |
the first five solutions (for ) are shown in Fig. 10.5 .
From Sturm-Liouville theory we conclude that
(10.68) |
Together with the solution for the equation,
(10.69) |
we find a Fourier-Bessel series type solution
(10.70) |
with .
In order to understand how to determine the coefficients from the initial condition we need to study Fourier-Bessel series in a little more detail.