Lecture Notes

Today we consider PDE problem in cylindrical coordinate and spherical coordinate.

Recall that Laplacian in cylindrical coordinate $r, \theta, z$ is written as $$ \Delta u = \frac{1}{r} \d_r(r \d_r(u)) + \frac{1}{r^2} \d_\theta^2 u + \d_z^2 u. $$ We shall look for eigenfunctions of $\Delta$ of the following form $$ u(r,\theta, z) = R( r ) \Theta(\theta) Z(z), $$ Then we have $$ \frac{1}{u}\Delta u = \frac{1}{R} \frac{1}{r} \d_r(r \d_r(R)) + \frac{1}{\Theta} \frac{1}{r^2} \d_\theta^2 \Theta + \frac{1}{Z} \d_z^2 Z. $$ such that $$ \Theta^{-1} \d_\theta^2 \Theta = \lambda_\theta $$ $$ \frac{1}{R} \frac{1}{r} \d_r(r \d_r(R)) + r^{-2} \lambda_\theta = \lambda_r $$ $$ \frac{1}{Z} \d_z^2 Z = \lambda_Z. $$

Step 1: Solve the eigenvalue problem for $\Theta$. Since $\Theta(\theta) = \Theta(\theta+2\pi)$, the eigenvalues of $\theta$ are $\sin(n\theta), \cos(n\theta)$ with eigenvalues $\lambda_\theta = -n^2$ for integer $n \geq 0$. More precisely, for $\lambda_\theta=0$, the eigen-space is one-dimensional, and is generated by the constant function $1$; for $\lambda_\theta = -n^2$ for positive integer $n$, we have 2-dimensional eigenspaces $V_{\lambda_\theta}$, generated by $\sin(n\theta), \cos(n\theta)$.

Step 2: Solve the eigenvalue problem for $R( r)$. The eigenvalue problem is $$ \frac{1}{R} \frac{1}{r} \d_r(r \d_r(R)) + r^{-2} (-n^2) = \lambda_r $$ where we plugged in $\lambda_\theta$'s possible values. Re-arranging terms, we get $$ r \d_r(r \d_r(R)) + (-\lambda_r r^2 - n^2) R = 0 $$ Compare with Bessel equation (12.2) from Boas $$ x(xy')' + (x^2 - p^2) y = 0 \Rightarrow y(x) = a J_p(x) + b Y_p(x).$$ We see if $\lambda_r = -k^2$, $p=n$, then $J_n(kr)$ and $Y_n(kr)$ are solution to the above equation. However, $|Y_n(kr)| \to -\infty$ as $r \to 0$, which is not allowed, hence the only solution that satisfies the boundary condition $| R(0)| < \infty$ is $J_n(kr)$.

Step 3: Solve the eigenvalue for $Z$. At this moment, $\lambda_Z$ can be anything. To narrow down the choices, we need to see the actual problem.