PDE17

From Exampleproblems

$\Delta u = u_{rr} + \frac{1}{r}u_r + \frac{1}{r^2}u_{\theta\theta}=0\,$

$u(r,-\pi) = u(r,\pi)\,$
$u_\theta(r,-\pi) = u_\theta(r,\pi)\,$
$\lim_{r\rightarrow 0^+}u(r,\theta) < \infty\,$

$u(\rho,\theta) = f(\theta)\,$

$0<r<\rho,\,\,\,\,\, -\pi<\theta<\pi\,$

Let $u(r,\theta) = R(r)H(\theta)\,$

Plug into the original DE and seperate variables. Set both sides equal to a constant $\lambda\,$ . The two ODEs are

$r^2R'' + rR' - \lambda R = 0\,$

$H''+\lambda H = 0\,$

The solutions are

$R(r) = \begin{cases} c_1 + c_2 \log r & n=0 \\ c_3 r^n + c_4 r^{-n} & n=1,2,... \end{cases}\,$

$H(\theta) = \begin{cases} \frac{1}{2}a_0 & n=0 \\ a_n \cos(n \theta) + b_n \sin(n \theta) & n=1,2,... \end{cases}\,$

To eliminate infinite discontinuities as $r\rightarrow 0^+\,$ , set $c_2 = c_4 = 0\,$ . For mathematical ease, set $c_1 = 1, c_3 = \frac{1}{\rho^n}\,$