# Finding Green's Function for Half Space Neumann Problem

• NYUmathgeek
In summary, the half-space Neumann problem is a mathematical problem that involves finding the solution to a partial differential equation (PDE) on a half-space domain, where the boundary of the domain has Neumann boundary conditions. The Green's function is important in solving this problem as it provides an exact solution and can be used for other types of boundary conditions. It is calculated using the method of images and has limitations such as only applying to linear PDEs and not being applicable for complex geometries.
NYUmathgeek
Hi all...need a little help with this one...

I need to find the Green's function for the half space Neumann problem in the domain z>0. i.e. Laplacian u=f in D, du/dn=h on the boundary of D.

Any ideas?

ya of corse
you can see http://www.ma3n.org/pages/jazar/
and problemes elliptiques

Last edited by a moderator:
Thanks!

Finding the Green's function for the half space Neumann problem can be a challenging task, but there are a few approaches you can take to solve it. One method is to use the method of images, where you create a mirror image of the problem in the lower half space and use the reflection principle to find the solution. Another approach is to use the Fourier transform to convert the problem into an integral equation, and then solve for the Green's function using this equation. Additionally, you can also use the method of separation of variables to find the Green's function in terms of a series solution. Whichever method you choose, it is important to carefully consider the boundary conditions and use appropriate techniques to solve for the Green's function. I hope this helps and good luck with your problem!

## 1. What is the half-space Neumann problem?

The half-space Neumann problem is a mathematical problem that involves finding the solution to a partial differential equation (PDE) on a half-space domain, where the boundary of the domain has Neumann boundary conditions. In simple terms, it is a problem of finding the function that satisfies both the PDE and the boundary conditions on a half-space.

## 2. Why is finding the Green's function for this problem important?

The Green's function is a fundamental concept in solving PDEs, as it allows us to express the solution to a PDE in terms of simpler functions. In the case of the half-space Neumann problem, finding the Green's function provides an exact solution to the problem, which can then be used to solve more complex problems or to verify numerical methods.

## 3. How is the Green's function for the half-space Neumann problem calculated?

The Green's function for the half-space Neumann problem is calculated using the method of images. This involves creating a mirror image of the original problem across the boundary, which simplifies the problem and allows for an analytical solution to be obtained. The Green's function is then expressed as a sum of two parts: one that satisfies the boundary conditions and one that satisfies the PDE.

## 4. Can the Green's function be used to find solutions for other boundary conditions?

Yes, the Green's function for the half-space Neumann problem can be used to find solutions for other types of boundary conditions, such as Dirichlet or Robin boundary conditions. This is because the Green's function captures the behavior of the underlying PDE and is independent of the specific boundary conditions.

## 5. Are there any limitations to using the Green's function for the half-space Neumann problem?

One limitation of using the Green's function for the half-space Neumann problem is that it only applies to linear PDEs. Additionally, the method of images may not always be applicable, especially for more complex geometries. In these cases, numerical methods may be a more suitable approach for finding solutions to the problem.

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