Green's function in elliptic box

In summary, the conversation discusses the search for a solution to Poisson's equation on an ellipse in 2 dimensions, where the Green's function vanishes on the ellipse. The solution for a circle is well-known, but the group is looking for a generalization. One potential method is to use elliptical coordinates and Mathieu functions, but there may be other approaches as well. Googling "elliptical greens function" can provide more information.
  • #1
Einj
470
59
Hello everyone! Does anyone know if there is a know expression for the Green's function for Poisson's equation that vanishes on an ellipse in 2 dimensions?
I'm essentially looking for a solution to:
$$
\nabla^2G(\vec x-\vec x_0)=\delta^2(\vec x-\vec x_0)
$$
in 2 dimensions where
$$G(\vec x-\vec x_0)=0$$
when [itex]\vec x[/itex] lies on an ellipse.
The solution for a circle is well know but I wanted to know if there any kind of generalization.

Thanks a lot!
 
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  • #2
If I remember correctly Laplace's equation on an ellipse is separable if you use elliptical coordinates, and the solution can be expressed in terms of Mathieu functions. From this one should be able to construct the Green's function.
 
  • #3
Well, I don't really want to solve Laplace equation since the Laplacian of the Green's function is not equal to zero but to the Dirac delta. Do you think the same technique still holds? Do you have any reference?
 
  • #4
If you think about it the Green's function equation is equal to Laplace's equation everywhere except for at one point. You can use the solution to Laplace's equation to construct the Green's function. It takes a little bit of algebra, but it is tractable. Of course there are other ways to solve for the Green's function. I bet most of them will probably use elliptical coordinates and Mathieu funtions.

I don't have references on hand, but Googling elliptical greens function produces some promising hits.
 
  • Like
Likes Einj
  • #5
Ok I will definitely do that! Thanks a lot.
 

1. What is the purpose of using Green's function in an elliptic box?

The purpose of using Green's function in an elliptic box is to solve for the general solution of a boundary value problem. It allows for the decomposition of a complex problem into a simpler, more manageable one by breaking it down into a series of simpler problems.

2. How is Green's function related to the boundary conditions of the elliptic box?

Green's function is directly related to the boundary conditions of the elliptic box. The boundary conditions determine the behavior of the solution at the boundaries and Green's function incorporates these conditions to provide a unique solution for the problem.

3. Can Green's function be used for any type of boundary value problem?

Yes, Green's function can be used for any type of boundary value problem as long as the problem is linear and the boundary conditions are well-defined. It is a powerful tool for solving a wide range of problems in mathematics and physics.

4. How is Green's function different from the fundamental solution of the elliptic box?

Green's function and the fundamental solution of an elliptic box are closely related but have some key differences. The fundamental solution is the solution to the homogeneous problem, while Green's function incorporates the effects of the boundary conditions. Additionally, Green's function is a unique solution, while the fundamental solution may have multiple solutions.

5. Can Green's function be used in higher dimensions?

Yes, Green's function can be used in higher dimensions. It is a versatile tool that can be applied to problems in any number of dimensions. However, as the number of dimensions increases, the complexity of the problem also increases and the use of Green's function may become more challenging.

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