Chapter 10.7, Problem 9E

Find a solution to the Neumann boundary value problem for a disk:

$\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}=0$, $0\le r<a$, $-\pi \le \theta \le \pi$,

$\frac{\partial u}{\partial r}\left(a,\theta \right)=f\left(\theta \right)$, $-\pi \le \theta \le \pi$.