What is the solution to Laplace's equation on a wedge?

[tjackson3]

Homework Statement

Find the solution of Laplace's equation for $\phi(r,\theta)$ in the circular sector $0 < r < 1; 0 < \theta < \alpha$ with the boundary conditions $\phi(r,0) = f(r), \phi(r,\alpha) = 0, \phi(1,\theta) = 0.$ (also, implicitly, the solution is bounded at r = 0). Use two different spectral representations. (note: this just means do the problem twice, expanding in a different variable each time). Below is a crude MS Paint drawing of the boundary conditions, just to summarize:

http://tjackson3.webs.com/laplace.png

Homework Equations

In polar coordinates, Laplace's equation is

$$\frac{\partial^2 \phi}{\partial r^2} + \frac{1}{r}\frac{\partial \phi}{\partial r} + \frac{1}{r}^2\frac{\partial^2\phi}{\partial \theta^2}$$

The Attempt at a Solution

As I said, the goal of this problem is to use an eigenfunction expansion. We have to solve the problem twice - once using the eigenfunction corresponding to r, once using the eigenfunction corresponding to θ. Despite the fact that the θ variable is the inhomogeneous one, I was able to complete the eigenfunction expansion in that one. The problem kicked in for the r variable.

If you do the usual separation of variables ($\phi(r,\theta) = u(r)v(\theta)$, and collect the r terms, you find that the r eigenfunction has to satisfy

$$r^2 \frac{d^2u}{dr^2} + r\frac{du}{dr} - \lambda u = 0$$

with the boundary conditions that $u(1) = 0, |u(0)| < \infty$. Unlike the normal case when you deal with Laplace's equation on a circle, we can't stipulate that $\lambda$ is an integer. There are two cases to consider, and they both seem unlikely. Prior experience leads me to assume $\lambda > 0$, so for simplicity, let $\lambda = \mu^2$. Then the solution to the above ODE is

$$u(r) = c_1 r^{\mu} + c_2 r^{-\mu} = c_1 e^{\mu\ln r} + c_2 e^{-\mu\ln r}$$

Since $u(0)$ has to be bounded, we know that $c_2 = 0$. But now the problem is that there's no way to make $u(1) = 0$ without setting $c_1 = 0$, which we obviously can't do. Does anyone have any experience with how to get around this problem?

Thanks!

edit: If you assume lambda is negative, then you would get oscillatory functions: $u(r) = c_1\sin(\mu\ln r) + c_2\cos(\mu\ln r)$, which could solve the problem at r = 1, but obviously has trouble as r goes to zero.

 

[sunjin09]

I have the feeling that your radial component eigenfunction is the Bessel function of some non-integer order, try to compare Bessel equation to your radial ODE and see if there is a match