PDE22

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


u(\rho ,\theta )=\cos ^{2}\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 ),n=1,2,...\,

Evaluate this function at the boundary conditions.

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

Matching coefficients in like terms,

A_{0}=1,\,\,A_{2}={\frac  {-1}{2}},\,\,B_{2}=0,\,\,A_{n}=B_{n}=0,\,\,n=1,3,4,...\,

The final solution is

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

Partial Differential Equations

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