PDE20

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

$u(a,\theta) = f(\theta)\,$
$u(b,\theta) = g(\theta)\,$

$a<r<b,\,\,\,\,\, -\pi<\theta<\pi\,$

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

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

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

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

The solutions are

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

$H(\theta) = \begin{cases} \frac{1}{2}A_0 & n=0 \\ A_n \cos(n \theta) + B_n \sin(n \theta) & n=1,2,... \end{cases}\,$

The solution is

$u(r,\theta) = \frac{1}{2}(A_0 + B_0\log r) + \sum_{n=1}^\infty\left[(A_n r^n + B_n r^{-n}) \cos n\theta + (C_n r^n + D_n r^{-n})\sin n\theta\right]\,$

$u(a,\theta) = \frac{1}{2}(A_0 + B_0\log a) + \sum_{n=1}^\infty\left[(A_n a^n + B_n a^{-n}) \cos n\theta + (C_n a^n + D_n a^{-n})\sin n\theta\right]=f(\theta)\,$

$u(b,\theta) = \frac{1}{2}(A_0 + B_0\log b) + \sum_{n=1}^\infty\left[(A_n b^n + B_n b^{-n}) \cos n\theta + (C_n b^n + D_n b^{-n})\sin n\theta\right]=g(\theta)\,$

$A_0 + B_0\log a = \frac{1}{\pi}\int_{-\pi}^\pi f(\theta)\,d\theta\,$

$A_0 + B_0\log b = \frac{1}{\pi}\int_{-\pi}^\pi g(\theta)\,d\theta\,$

$A_n a^n + B_n a^{-n} = \frac{1}{\pi}\int_{-\pi}^\pi f(\theta)\cos n\theta\,d\theta,\,\,n=1,2,...\,$

$C_n b^n + D_n b^{-n} = \frac{1}{\pi}\int_{-\pi}^\pi g(\theta)\sin n\theta\,d\theta,\,\,n=1,2,...\,$

The constants can be solved for.

Partial Differential Equations

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