PDE22.5

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


u(10,\theta) = 15\cos\theta\,
u(20,\theta) = 30\sin\theta\,

10<r<20,\,\,\,\,\, -\pi<\theta\le\pi\,




Seperate variables: u(r,\theta) = R(r)\Theta(\theta)\,

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 of the form

u(r,\theta) = \frac{1}{2} A_0 + B_0 \log r + \sum_{n=1}^\infty (A_n r^n + B_n r^{-n})\cos n\theta + \sum_{n=1}^\infty (C_nr^n -D_nr^{-n})\sin n\theta\,

u(10,\theta) = \frac{1}{2}A_0 + B_0\log10+\sum_{n=1}^\infty(A_n 10^n + B_n 10^{-n})\cos n\theta + \sum_{n=1}^\infty (C_n10^n -D_n10^{-n})\sin n\theta=15\cos(\theta)\,

u(20,\theta) = \frac{1}{2}A_0 + B_0\log20+\sum_{n=1}^\infty(A_n 20^n + B_n 20^{-n})\cos n\theta + \sum_{n=1}^\infty (C_n20^n -D_n20^{-n})\sin n\theta=30\sin(\theta)\,

\frac{1}{2}A_0 + B_0 \log 10 = \frac{1}{\pi}\int_{-\pi}^\pi 15 \cos\theta\,d\theta = 0\,

\frac{1}{2}A_0 + B_0 \log 20 = \frac{1}{\pi}\int_{-\pi}^\pi 30 \sin\theta\,d\theta = 0\,

So A_0 = B_0 = 0\,

The boundary conditions only involve the n=1\, terms. The rest are zero.

A_1 10 + B_1 10^{-1} = \frac{1}{\pi}\int_{-\pi}^\pi 15 \cos \theta \cos \theta \,d\theta=15\,

A_1 20 + B_1 20^{-1} = \frac{1}{\pi}\int_{-\pi}^\pi 30 \sin \theta \cos \theta \,d\theta=0\,

So A_1 = \frac{-1}{2}, B_1 = 200\,

C_1 10 + D_1 10^{-1} = \frac{1}{\pi}\int_{-\pi}^\pi 15 \cos \theta \sin \theta \,d\theta=0\,

C_1 20 + D_1 20^{-1} = \frac{1}{\pi}\int_{-\pi}^\pi 30 \sin^2 \theta \,d\theta=30\,

So C_1 = 2, D_1 = -200\,

The solution is

  • u(r,\theta) = (-\frac{1}{2} + \frac{200}{r})\cos\theta + (2r - \frac{200}{r})\sin\theta\,

Partial Differential Equations

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