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

In summary, the problem requires finding the solution to Laplace's equation for a circular sector with given boundary conditions. This can be done using an eigenfunction expansion, with two different spectral representations due to the two varying variables. The first step is to separate the variables and solve for the eigenfunctions, with the radial eigenfunction satisfying a Bessel equation. However, this presents a problem as there is no way to satisfy the boundary conditions without setting one of the constants to zero. The solution may lie in considering negative values for the eigenvalue, resulting in oscillatory functions, but this may also cause issues as r approaches zero. Further investigation into the Bessel equation may provide a solution.
  • #1
tjackson3
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Homework Statement



Find the solution of Laplace's equation for [itex]\phi(r,\theta)[/itex] in the circular sector [itex]0 < r < 1; 0 < \theta < \alpha[/itex] with the boundary conditions [itex]\phi(r,0) = f(r), \phi(r,\alpha) = 0, \phi(1,\theta) = 0.[/itex] (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 [Broken]

Homework Equations



In polar coordinates, Laplace's equation is

[tex]\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}[/tex]

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 ([itex]\phi(r,\theta) = u(r)v(\theta)[/itex], and collect the r terms, you find that the r eigenfunction has to satisfy

[tex]r^2 \frac{d^2u}{dr^2} + r\frac{du}{dr} - \lambda u = 0[/tex]

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

[tex]u(r) = c_1 r^{\mu} + c_2 r^{-\mu} = c_1 e^{\mu\ln r} + c_2 e^{-\mu\ln r}[/tex]

Since [itex]u(0)[/itex] has to be bounded, we know that [itex]c_2 = 0[/itex]. But now the problem is that there's no way to make [itex]u(1) = 0[/itex] without setting [itex]c_1 = 0[/itex], 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: [itex]u(r) = c_1\sin(\mu\ln r) + c_2\cos(\mu\ln r)[/itex], which could solve the problem at r = 1, but obviously has trouble as r goes to zero.
 
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  • #2
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
 

1. What is Laplace's equation on a wedge?

Laplace's equation on a wedge is a mathematical equation that describes the distribution of electric potential, temperature, or any other scalar field in a wedge-shaped region. It is a partial differential equation that relates the second derivatives of the scalar field to its value at each point in the wedge.

2. What is the significance of Laplace's equation on a wedge?

Laplace's equation on a wedge has many applications in physics and engineering, including the study of electric fields, heat transfer, and fluid flow in wedge-shaped systems.

3. How is Laplace's equation on a wedge solved?

Laplace's equation on a wedge can be solved using various methods, such as separation of variables, conformal mapping, and Green's function techniques. The specific method used depends on the boundary conditions and geometry of the wedge.

4. What are the boundary conditions for Laplace's equation on a wedge?

The boundary conditions for Laplace's equation on a wedge typically include the known values of the scalar field at the edges of the wedge, as well as any symmetry or periodicity conditions that may apply.

5. What are some real-life examples of Laplace's equation on a wedge?

Laplace's equation on a wedge can be applied to various physical systems, such as the study of electric potential in a wedge-shaped capacitor, the temperature distribution in a wedge-shaped heat sink, or the flow of fluid in a wedge-shaped channel.

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