Green's function for Poisson Equation

[bhatiaharsh]

Hi,

I am working on finding a solution to Poisson equation through Green's function in both 2D and 3D. For the equation: $$\nabla^2 D = f$$, in 3D the solution is:
$$D(\mathbf x) = \frac{1}{4\pi} \int_V \frac{f(\mathbf x')}{|\mathbf x - \mathbf x'|} d\mathbf{x}'$$, and in 2D the solution is:
$$D(\mathbf x) = \frac{1}{2\pi}\int_V \log(|\mathbf x - \mathbf x'|) f(\mathbf x') d\mathbf{x}'$$.

Now, my question is that where these solutions hold true only for infinite domains?

If I have a small rectangular domain, can I still use these equations to solve the Poisson's equation without any boundary conditions ?

Can someone help me with this, or point me to a reference which I should read ?

 

[Muphrid]

I believe you should be able to use these even on a finite domain provided that you have no boundary conditions on that domain, yes. What this means is your solution will not be unique--you could add a Laplace equation solution to it (which corresponds to the effects of sources lying outside your finite domain) without loss of generality.

 

[Marioeden]

These are what you call "Free Space Greens Functions" and are valid for infinite domains.
If you have a region without any boundary conditions then obviously these will still hold.

However, if you do have Boundary Conditions, you want to construct Dirichlet Greens Functions which are of the form G = H + G_f
where H is a harmonic function (i.e. solves laplace's equation) and G_f is the Free Space Green's Function.
Solving these problems is usually done using the method of images.

 

[bhatiaharsh]

Thanks for the pointers. If I understand right, and am not worried about a unique solution I should be able to use the integral solution of the equation. I tried a simple example in 1D and 3D, but the 3D example doesn't work out fine, and I am not sure what the problem is.

In either case, the source function $f$ does not decay (is constant). Could this be a problem ?

I am attaching my two examples:

 

[hunt_mat]

As someone said before, this is the free space Greens function and the Greens function is highly dependent upon the domain, you can perhaps use the method of images to obtain an answer if you want, but if it is a finite domain then I would investigate the use of a Fourier series for part of your solution.