PDE19

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


u_{r}(\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 equal to a constant \lambda \,. The two ODEs are

r^{2}R''+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}}}\,

  • u(r,\theta )={\frac  {1}{2}}a_{0}+\sum _{{n=1}}^{\infty }\left(r/\rho \right)^{n}(a_{n}\cos n\theta +b_{n}\sin n\theta )\,

Evaluate this function at the boundary conditions.

{\frac  {\partial u(r,\theta )}{\partial r}}={\frac  {1}{\rho }}\sum _{{n=1}}^{\infty }\left(r/\rho \right)^{{n-1}}n(a_{n}\cos n\theta +b_{n}\sin n\theta )\,

{\frac  {\partial }{\partial r}}u(\rho ,\theta )={\frac  {1}{\rho }}\sum _{{n=1}}^{\infty }na_{n}\cos n\theta +nb_{n}\sin n\theta =f(\theta )\,

The Fourier coefficients are

a_{n}={\frac  {\rho }{n\pi }}\int _{{-\pi }}^{\pi }f(\theta )\cos n\theta \,d\theta \,

b_{n}={\frac  {\rho }{n\pi }}\int _{{-\pi }}^{\pi }f(\theta )\sin n\theta \,d\theta \,

a_{0}\, is undetermined.

Partial Differential Equations

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