### Author Topic: Q7  (Read 1991 times)

#### Shentao YANG

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« on: November 24, 2016, 08:58:10 PM »
Consider Laplace equation $\Delta u=0$ in the cylinder$\{r\le a,\ 0<z<b,\ 0\le \theta \le 2\pi\}$. Separate variables $u=R(r)Z(z)\Theta(\theta)$.
1. Write down ODE which should satisfy $\Theta$ and solve it (using periodicity).
2. Write down ODE which should satisfy $Z$ and solve it using $Z(0)=Z(b)=0$.
3. Write down ODE which should satisfy $R$.

Ans:
\eqalign{ & \Delta u = {u_{rr}} + {1 \over r}{u_r} + {1 \over {{r^2}}}{u_{\theta \theta }} + {u_{zz}} = 0 \cr & \Rightarrow {{{r^2}R'' + rR'} \over R} + {{\Theta ''} \over \Theta } + {{{r^2}Z''} \over Z} = 0 \cr}
(1) Let:
$$\left. \matrix{ {{\Theta ''} \over \Theta } = - {m^2} < 0 \hfill \cr \Theta (0) = \Theta (2\pi ),\Theta '(0) = \Theta '(2\pi ) \hfill \cr} \right\}\matrix{ {{\Theta _{1,m}} = \cos (m\theta )} \cr {{\Theta _{2,m}} = \sin (m\theta )} \cr }$$
(2) Let:
$$\left. \matrix{ {{{r^2}Z''} \over Z} = - l \Rightarrow Z'' + {l \over {{r^2}}}Z = 0 \hfill \cr Z(0) = Z(b) = 0 \hfill \cr} \right\}\matrix{ {{l \over {{r^2}}} = {{{\pi ^2}{n^2}} \over {{b^2}}}} & { \Rightarrow l = {{{r^2}{\pi ^2}{n^2}} \over {{b^2}}}} \cr {z = \sin ({{\pi nz} \over b})} & {n = 1,2,...} \cr }$$
(3) from (1) and (2) we have:
$$\matrix{ {{r^2}R'' + rR' - \left( {{m^2} + {{{r^2}{\pi ^2}{n^2}} \over {{b^2}}}} \right)R = 0,} & {n = 1,2,...} \cr }$$