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^{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}}\,

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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