PDE21

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


u_{r}(2,\theta )=0\,
u_{r}(1,\theta )=\sin ^{2}\theta \,

1<r<2,\,\,\,\,\,-\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(1,\theta )={\frac  {1}{2}}A_{0}+\sum _{{n=1}}^{\infty }\left[(A_{n}+B_{n})\cos n\theta +(C_{n}+D_{n})\sin n\theta \right]=\sin ^{2}\theta \,

Using the fact that \sin ^{2}\theta ={\frac  {1}{2}}(1-\cos 2\theta )\,,

We can compare the coefficients and get the identities:

A_{0}=1,\,\,A_{2}+B_{2}={\frac  {-1}{2}}\,

A_{n}+B_{n}=0,\,\,n=1,3,4,...\,

Imposing the boundary conditions,

u_{r}(r,\theta )={\frac  {1}{2}}\left({\frac  {B_{0}}{r}}\right)+\sum _{{n=1}}^{\infty }(nA_{n}r^{{n-1}}-nB_{n}r^{{-n-1}})\cos n\theta +(nC_{n}r^{{n-1}}-nD_{n}r^{{-n-1}})\sin n\theta \,

u_{r}(2,\theta )={\frac  {B_{0}}{4}}+\sum _{{n=1}}^{\infty }(nA_{n}2^{{n-1}}-nB_{n}2^{{-n-1}})\cos n\theta +(nC_{n}2^{{n-1}}-nD_{n}2^{{-n-1}})\sin n\theta =0\,

This gives the equations for the coefficients:

nA_{n}2^{{n-1}}-nB_{n}2^{{-n-1}}=0,\,\,n=1,2,...\,

nC_{n}2^{{n-1}}-nD_{n}2^{{-n-1}}=0,\,\,n=1,2,...\,

This system can be solved and gives

A_{0}=1,\,\,B_{0}=0,\,\,A_{2}={\frac  {-1}{34}},\,\,B_{2}={\frac  {-8}{17}}\,

The final solution is

u(r,\theta )={\frac  {1}{2}}-{\frac  {1}{17}}\left({\frac  {r^{2}}{2}}+{\frac  {8}{r^{2}}}\right)\cos 2\theta \,

Partial Differential Equations

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