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math121b:04-06 [2020/04/06 07:00] pzhou |
math121b:04-06 [2020/04/06 10:24] (current) pzhou [Steady State temperature distribution inside a unit ball] |
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=== Summary === | === Summary === | ||
The general eigenfunction is | The general eigenfunction is | ||
- | $$u(r, | + | $$u(r, |
+ | P_l^m(\cos \theta) \begin{cases} \cos(m \phi) \cr \sin(m \phi) \end{cases}, \quad \lambda = -k^2 | ||
+ | $$ | ||
+ | ===== Steady State temperature distribution inside a unit ball ===== | ||
+ | We solve $ \Delta u = 0$ for $r \leq 1$ with $u(r=1, \theta, \phi) = f(\theta, | ||
+ | $$ f(\theta, \phi) = \sum_{l=0}^\infty \sum_{m=0}^l [a_{l m} \cos(m\phi) + b_{l m} \sin(m \phi) ] P_l^m(\cos \theta) | ||
+ | with $b_{l 0}=0$. | ||
+ | Then the solulution for $u$ is obtained by setting $\lambda = \lambda_r=0$ | ||
+ | $$ u(r, \theta, \phi) = \sum_{l=0}^\infty \sum_{m=0}^l r^l [a_{l m} \cos(m\phi) + b_{l m} \sin(m \phi) ] P_l^m(\cos \theta). | ||
- | + | To obtain the coefficients $a_{lm}, b_{lm}$ from $f(\theta, \phi)$, we uses orthogonality of these functions $P_l^m(\cos \theta) \cos(m \phi), P_l^m(\cos \theta) \sin(m \phi)$ on the two sphere with volume form $\sin \theta d\theta d\phi$. For example, we claim that, if $(l, m) \neq (l', m')$, then | |
- | ===== Steady State | + | $$ \int_{\phi |
+ | Indeed, integrating $d\phi$, we see that if $m \neq m'$ the result is zero; if $m=m'$ but $l \neq l'$, then we use the orthogonality of associated Legendre functions (see section 12.10 of Boas), to show that the integral is zero. | ||