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math121b:04-06 [2020/04/06 06:41] pzhou created |
math121b:04-06 [2020/04/06 10:24] (current) pzhou [Steady State temperature distribution inside a unit ball] |
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$$ u (r, \theta, \phi) = R(r) \Theta(\theta) \Phi(\phi). $$ | $$ u (r, \theta, \phi) = R(r) \Theta(\theta) \Phi(\phi). $$ | ||
Then $ \Delta u = \lambda u$ is equivalent to | Then $ \Delta u = \lambda u$ is equivalent to | ||
- | $$ \frac{1}{u} \Delta u = \frac{1}{R} \frac{1}{r^2} \left( \frac{\d}{\d r} r^2 \frac{\d R}{\d r} \right) + | + | $$ \lambda = \frac{1}{u} \Delta u = \frac{1}{R} \frac{1}{r^2} \left( \frac{\d}{\d r} r^2 \frac{\d R}{\d r} \right) + |
\frac{1}{\Theta} \frac{1}{r^2} \frac{1}{\sin \theta} \frac{\d }{\d \theta} ( \sin \theta \frac{\d \Theta }{\d \theta}) + \frac{1}{\Phi } \frac{1}{r^2 \sin^2 \theta} \frac{\d^2 \Phi}{\d \phi^2}. $$ | \frac{1}{\Theta} \frac{1}{r^2} \frac{1}{\sin \theta} \frac{\d }{\d \theta} ( \sin \theta \frac{\d \Theta }{\d \theta}) + \frac{1}{\Phi } \frac{1}{r^2 \sin^2 \theta} \frac{\d^2 \Phi}{\d \phi^2}. $$ | ||
which in turns breaks into equations | which in turns breaks into equations | ||
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$$ \frac{1}{\sin \theta} \frac{\d }{\d \theta} ( \sin \theta \frac{\d \Theta}{\d \theta}) + \frac{\lambda_\phi}{\sin^2 \theta} \Theta | $$ \frac{1}{\sin \theta} \frac{\d }{\d \theta} ( \sin \theta \frac{\d \Theta}{\d \theta}) + \frac{\lambda_\phi}{\sin^2 \theta} \Theta | ||
$$ \frac{1}{R} \frac{1}{r^2} \left( \frac{\d}{\d r} r^2 \frac{\d R}{\d r} \right) + \frac{\lambda_\theta}{r^2} = \lambda_r \quad \Rightarrow | $$ \frac{1}{R} \frac{1}{r^2} \left( \frac{\d}{\d r} r^2 \frac{\d R}{\d r} \right) + \frac{\lambda_\theta}{r^2} = \lambda_r \quad \Rightarrow | ||
+ | The total eigenvalue $\lambda = \lambda_r$. | ||
The equation about $\Phi$ has $\sin(m\phi), | The equation about $\Phi$ has $\sin(m\phi), | ||
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The equation about $\Theta$ has associated Legendre polynomial as solution, recall that $y(\theta) = P_l^m (\cos \theta)$ solves equation (Problem 12.10.2) | The equation about $\Theta$ has associated Legendre polynomial as solution, recall that $y(\theta) = P_l^m (\cos \theta)$ solves equation (Problem 12.10.2) | ||
$$ \frac{1}{\sin \theta} \frac{\d }{\d \theta} ( \sin \theta \frac{\d y}{\d \theta}) + \left( l(l+1) - \frac{m^2}{\sin^2 \theta} \right) y = 0 $$ | $$ \frac{1}{\sin \theta} \frac{\d }{\d \theta} ( \sin \theta \frac{\d y}{\d \theta}) + \left( l(l+1) - \frac{m^2}{\sin^2 \theta} \right) y = 0 $$ | ||
- | By comparison, we get eigenvalues $\lambda_\theta = l (l+1)$. | + | By comparison, we get eigenvalues $\lambda_\theta = -l (l+1)$. |
- | The equation about $R$ is related to the spherical Bessel function' | + | The equation about $R$ is related to the spherical Bessel function' |
$$ \left( \frac{\d}{\d r} r^2 \frac{\d y}{\d r} \right) + (r^2 - n(n+1) ) y = 0 $$ | $$ \left( \frac{\d}{\d r} r^2 \frac{\d y}{\d r} \right) + (r^2 - n(n+1) ) y = 0 $$ | ||
+ | Hence, for $\lambda_\theta = l(l+1)$, we can solve $R(r)$ to get | ||
+ | $$ R(r) = \begin{cases} | ||
+ | j_l( k r), y_l(kr) & \lambda_r = -k^2 \cr | ||
+ | j_l( i k r), y_l(i kr) & \lambda_r = k^2 \cr | ||
+ | r^l, r^{-l-1} & \lambda_r=0 | ||
+ | \end{cases} | ||
+ | $$ | ||
+ | Usually, we require $R(0)$ to be finite, and $y_l(r) \sim r^{-l-1}$ as $r \to 0$, hence we only keep the $j_l(kr), j_l(ikr), r^l$ solution above. The case $\lambda_r$ can be obtained as the leading order behavior of $j_l(kr)$ after rescaling by a constant | ||
+ | $$ r^l = \lim_{k \to 0} k^{-l} j_l( k r). $$ | ||
+ | |||
+ | In some cases, suppose we solve the equation in the exterior of a ball and require the solution to decay as $r \to \infty$, then we will not ask $R(0)$ to be finite, instead we require $R(\infty) = 0$. Then the $y_l(kr), y_l(ikr), r^{-l-1}$ solutions will come into play. | ||
+ | |||
+ | === Summary === | ||
+ | The general eigenfunction is | ||
+ | $$u(r, | ||
+ | | ||
+ | $$ | ||
+ | |||
+ | ===== 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 | ||
+ | $$ \int_{\phi = 0}^{2\pi} \int_{\theta=0}^\pi P_l^m(\cos \theta) \cos(m \phi) P_{l' | ||
+ | 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. | ||
+ | |||
+ |