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^2R'' + 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-\cos2\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}\cos2\theta\left(\frac{r}{\rho}\right)^2\,

Partial Differential Equations

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