Green's functions solution to Poisson's eqn (or something similar)

[Green's Funk]

Yikes!

I am having a problem finding a Green's function to solve the Intensity Transport Equation (ITE).

The ITE is nabla^2 * phase = -k/I0 * dI/dz

Where phase is a function of (x,y) as is I the intensity. k and I0 are constants and dI/dz is the partial derivative of I with respect to z.

I am measuring the whole of the RHS of the equation in one fell swoop, so I just need a Green's function to solve nabla^2 * phase = Constant, so I can retrieve the phase from an intensity gradient I have calculated.

Can anyone help? The Green's function needs to satisfy Neumann boundary conditions with G( (x,y) , (x',y') ) being defined by it's Laplacian.
ie. Nabla ^2 G = dirac (r - r').

Anyone know any good Green's functions or how to implement them?

I'm turning a little Green here!

Many thanks,

Bob

 

[ExtravagantDreams]

$$\nabla^2 \phi(x,y) = - \frac{k} {I_0} \frac {dI(x,y)} {dz}$$

hmm yeah, sorry, I really have no idea what you are talking about but thought I would clean it up a bit.

What is z? the z axis? or the xy plane?

 

[Lonewolf]

How about something like $$G = \frac{-1}{4\pi|r-r'|}$$? Just a guess.

 

[Dr Transport]

Start by calculating the Green's function for the 2-d problem using the Neumann boundary conditions. ) I would write it here, but I just moved halfway across the country and stil do not know where all of my notes are hidden away in my basement.) After finishing that, use it to integrate the non-homogeneous side of your equation. The form of the greens' function depends on the coordinate system and the basis functions used, so blindly writing a function is not the answer.

dt

 

[Magister]

The green funtion for poisson equation is in fact the one given by Lonewolf.
You can get it by comparison with the known equation of electrodynamics:
$$\nabla^2 \ \frac{1} {|r-r'|} = - 4 \pi \delta(r-r')$$

By the way, anyone knows how to combute it by the Fourier Transform Method? I am having a hard time with it...

 

[dextercioby]

The green funtion for poisson equation is in fact the one given by Lonewolf.
You can get it by comparison with the known equation of electrodynamics:
$$\nabla^2 \ \frac{1} {|r-r'|} = - 4 \pi \delta(r-r')$$

By the way, anyone knows how to combute it by the Fourier Transform Method? I am having a hard time with it...

But that's true only in 3 dimensions. In 2, the Green functions is totally different.