Do different forms of Green's functions give same result?

[Atman]

I'm solving Helmholtz equation in a cylindrical coordinate. With boundary conditions be Neumann type, I can write several satisfactory forms of the Green's function, right? (for example, I can make the discontinuity of the derivative of the Green's function in the radial part, or in the z direction part.) Then I use them to get a solution for a specific problem with a source inside the cylinder, will they give the same solution?

Apparently, they are written in different forms. When I study the properties of these solutions, I find that in one form, it will give divergence at some point, while the other one won't at the same point. That makes me very confused! Should they be the same, even though I'm using different forms of the Green's function? I thought they should at least be equivalent. If that's not the case, I don't understand why people can choose whatever form of the Green's function they like when they tackle the problem.

FYI, the boundary condition is the derivative vanishes at the top, the bottom and also the inner surface of the cylinder.

Really appreciate your help if you can clarify it:)

 

[gtoguy97]

The solution to a Helmholtz equation with Neumann boundary conditions should be the same regardless of which form of Green's function is used. This is because the boundary conditions are the same, and the forms of the Green's functions will all satisfy the same boundary conditions. However, the Green's functions may differ in terms of their behavior inside the domain. For example, one might have a divergence at a certain point while the other does not. In this case, one should be careful when choosing which Green's function to use, since it can affect the behavior of the solution inside the domain.