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$$ \frac{1}{u}\Delta u = \frac{1}{R} \frac{1}{r} \d_r(r \d_r(R)) + \frac{1}{\Theta} \frac{1}{r^2} \d_\theta^2 \Theta + \frac{1}{Z} \d_z^2 Z. $$ | $$ \frac{1}{u}\Delta u = \frac{1}{R} \frac{1}{r} \d_r(r \d_r(R)) + \frac{1}{\Theta} \frac{1}{r^2} \d_\theta^2 \Theta + \frac{1}{Z} \d_z^2 Z. $$ | ||
such that | such that | ||
- | $$ \d_\theta^2 \Theta(\theta) | + | $$ \Theta^{-1} |
$$ \frac{1}{R} \frac{1}{r} \d_r(r \d_r(R)) | $$ \frac{1}{R} \frac{1}{r} \d_r(r \d_r(R)) | ||
- | $$ \frac{1}{Z} \d_z^2 Z = \lambda_Z. $$ | + | $$ \frac{1}{Z} \d_z^2 Z = \lambda_z. $$ |
- | Step 1: Solve the eigenvalue problem for $\Theta$. | + | ** eigenvalue problem for $\Theta$.** \\ |
+ | $$\lambda_\theta = -n^2, \quad \Theta(\theta) | ||
+ | |||
+ | ** eigenvalue problem | ||
+ | Assume | ||
+ | $$ r \d_r(r \d_r(R)) + (-\lambda_r r^2 - n^2) R = 0. $$ | ||
- | Step 2: Solve the eigenvalue problem for $R( r)$. The eigenvalue problem is | ||
- | $$ | ||
- | where we plugged in $\lambda_\theta$' | ||
- | $$ r \d_r(r \d_r(R)) + (-\lambda_r r^2 - n^2) R = 0 $$ | ||
Compare with Bessel equation (12.2) from Boas | Compare with Bessel equation (12.2) from Boas | ||
- | $$ x(xy' | + | $$ x(xy' |
+ | And notice that we want $R(0)$ finite, and $|Y_p(0)| = \infty$, we see | ||
+ | $$ R( r) = \begin{cases} | ||
+ | | ||
+ | r^n & \lambda_r = 0 \cr | ||
+ | \end{cases} | ||
+ | $$ | ||
+ | Note that, the $\sqrt{-\lambda_r}$ has an ambiguity of $\pm 1$ sign, but $J_n(x)=(-1)^n J_n(-x)$, hence you can take either sign for the square root. | ||
+ | |||
+ | If $\lambda_r < 0$, say $\lambda_r = -k^2$ for $k>0$, then | ||
+ | $$ R( r) = J_n(\sqrt{-\lambda_r} r) = J_n(kr) $$ | ||
+ | $R( r) will oscillate as $r$ increase. | ||
+ | |||
+ | If $\lambda_r > 0$, say $\lambda_r = k^2$ for $k>0$, then we will take | ||
+ | $$ R( r) = I_n(kr) = i^n J_n(i k r) $$ | ||
+ | where $I_n(kr)$ is the ' | ||
+ | |||
+ | ** Eigenvalue problem for $Z( z)$ ** \\ | ||
+ | $$ Z(z) = \begin{cases} | ||
+ | | ||
+ | 1, z & \lambda_z = 0 | ||
+ | \end{cases} | ||
+ | $$ | ||
+ | |||
+ | Hence, for any $\lambda_\theta = -n^2, \lambda_r \in \R, \lambda_z \in \R$, we have an eigen-function of Laplacian | ||
+ | $$ u = R_{\lambda_r}( r) \Theta_n(\theta) Z_{\lambda_z}(z), | ||
+ | where the $R, \Theta, Z$ are given as above. | ||
+ | |||
+ | ==== Solve the Steady State Temperature problem ==== | ||
+ | Suppose we are given a cylinder of radius $1$ and height $1$ ($z \in [0,1]$), and suppose we specify the temperature on the boundary of the cylinder. We need to solve the equation | ||
+ | $$ \Delta u = 0 $$ | ||
+ | satisfying the boundary condition. | ||
+ | |||
+ | Note that $\lambda_z + \lambda_r = 0$, hence we will use $\lambda_r$ and $n$ to label the solutions. | ||
+ | |||
+ | We consider two special cases of boundary value specification. | ||
+ | |||
+ | ** Case 1: Bottom and Side boundary value = 0. $u(r, | ||
+ | Since $R( r)$ will have a zero at $r=1$, we should have $\lambda_r = -k^2 < 0$, and | ||
+ | $$ u_{k,n}(r, \theta, z) = J_n(kr) \cos(n\theta) \sinh(k z), \quad J_n(kr) \sin(n\theta) \sinh(k z)$$ | ||
+ | For each $n$, we need to choose those $k$ such that $J_n(k 1) = 0$, hence the choice of $k$ is discrete as well. | ||
+ | |||
+ | More precisely, let $k_{n,m} > 0$ be the $m$-th root of $J_n(x)$. Then the general solution is | ||
+ | $$ u(r, | ||
+ | Set $z=1$, we can use boundary condition to determine the coefficients $a_{n,m}, b_{n,m}$. Recall that | ||
+ | $$ \int_0^1 J_n(k_{n, | ||
+ | Hence | ||
+ | $$ a_{n,m} = \frac{\int_0^1 \int_{0}^{2\pi} u(r, | ||
+ | and $b_{n,m}$ is obtained similarly, replacing $\cos(n\theta)$ by $\sin(n \theta)$. | ||
+ | |||
+ | |||
+ | ** Case 2: Bottom and Top face boundary value = 0. $u(r=1, | ||
+ | This forces $\lambda_z < 0$. In particular, $Z(z) = \sin(m \pi z)$, hence $\lambda_z = -(m\pi)^2$ and $\lambda_r = (m \pi)^2$. | ||
+ | $$ u(r,\theta, z) = \sum_{n=1}^\infty \sum_{m=1}^\infty I_n( m \pi r) \sin(m \pi z) (a_{n,m} \cos(n\theta) + b_{n,m} \sin(n\theta)) $$. | ||
+ | |||
+ | We figure out the coefficient $a_{n,m}, b_{n,m}$ using the boundary value at $r=1$. | ||
+ | $$ u(1, \theta, z) = \sum_{n=1}^\infty \sum_{m=1}^\infty I_n( m \pi) \sin(m \pi z) (a_{n,m} \cos(n\theta) + b_{n,m} \sin(n\theta)) $$. | ||
+ | Hence by integrating $\theta, z$ we get | ||
+ | $$ a_{n,m} = \frac{ \int_0^1 \int_0^{2\pi} u(1, | ||
+ | and similarly for $b_{n,m}$. | ||
+ | |||
+ | ** General case ** | ||
+ | Decompose the boundary condition into three parts, top face, bottom face and side face. Break the original problem into 3 smaller problems in which only one part is non-zero. Then add up the solution from the three smaller problems. You end up with a solution that satisfies $\Delta u =0$ and has the desired boundary value. | ||