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 \leq \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_{n}r^{n}+D_{n}r^{{-n}})\sin n\theta \,

u(10,\theta )={\frac  {1}{2}}A_{0}+B_{0}\log 10+\sum _{{n=1}}^{\infty }(A_{n}10^{n}+B_{n}10^{{-n}})\cos n\theta +\sum _{{n=1}}^{\infty }(C_{n}10^{n}+D_{n}10^{{-n}})\sin n\theta =15\cos(\theta )\,

u(20,\theta )={\frac  {1}{2}}A_{0}+B_{0}\log 20+\sum _{{n=1}}^{\infty }(A_{n}20^{n}+B_{n}20^{{-n}})\cos n\theta +\sum _{{n=1}}^{\infty }(C_{n}20^{n}+D_{n}20^{{-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  {r}{2}}+{\frac  {200}{r}})\cos \theta +(2r-{\frac  {200}{r}})\sin \theta \,

Partial Differential Equations

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