I'm trying to solve the PDE given by $$\begin{array}\ u_t = \nabla^2 u, & 0 \lt r \lt 1, & 0 \lt \theta \lt \alpha \lt 2\pi\end{array} \\ u(r,\theta, 0) = f(\theta) \\ u(1,\theta,t) = u(r,0,t) = u(r,\alpha, t)=0$$
Here's what I've got so far:
Separating variables gets: $$T''+\lambda T = 0\\ r^2R'' + rR' + (\lambda r^2 - \gamma)R = 0 \\ \Theta'' + \gamma\Theta = 0$$ where $R(1)=0,\Theta(0) = \Theta(\alpha) = 0$.
So solving the $\Theta$ problem I get $$\Theta_n(\theta) = \sin\left(\frac {n\pi}{\alpha}\theta\right)$$
Then for the $R$ problem, make the change of variables $x=\sqrt{\lambda}r$ and $y=R$ to get $$x^2y'' +xy' + (x^2 - \frac {n^2\pi^2}{\alpha^2})y = 0$$
From here, I'd like to say that the solution is something like $R(r) = BJ_n(\sqrt{\lambda}r)$. But wouldn't the $\frac {n^2\pi^2}{\alpha^2}$ have to be an integer for this to be Bessel's equation. In general, this definitely won't be because $n$ is already an integer and $\pi/\alpha$ is not even necessarily rational, much less an integer itself. So what's my next step?
Also, as a side note, is there some simiplification I can make because $f(\theta)$ doesn't depend on $r$?