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

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In summary: Start by calculating the Green's function for the 2-d problem using the Neumann boundary conditions. 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.
  • #1
Green's Funk
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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
 
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  • #2
[tex]
\nabla^2 \phi(x,y) = - \frac{k} {I_0} \frac {dI(x,y)} {dz}
[/tex]

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

What is z? the z axis? or the xy plane?
 
Last edited:
  • #3
How about something like [tex]G = \frac{-1}{4\pi|r-r'|}[/tex]? Just a guess.
 
  • #4
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
 
  • #5
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:
[tex]
\nabla^2 \ \frac{1} {|r-r'|} = - 4 \pi \delta(r-r')
[/tex]

By the way, anyone knows how to combute it by the Fourier Transform Method? I am having a hard time with it...
 
  • #6
Magister said:
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:
[tex]
\nabla^2 \ \frac{1} {|r-r'|} = - 4 \pi \delta(r-r')
[/tex]

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.
 

1. What is a Green's function?

A Green's function is a mathematical tool used to solve differential equations, specifically the Poisson's equation in this context. It represents the response of a system to a point source or impulse at a specific location.

2. How is a Green's function used to solve Poisson's equation?

The Green's function is convolved with the source term in Poisson's equation, resulting in a solution that satisfies the boundary conditions. This approach is known as the Green's function method or the method of images.

3. What are the advantages of using Green's function to solve Poisson's equation?

Green's function allows for the solution to be expressed in terms of the source term, making it easier to handle non-uniform or complex boundary conditions. It also provides a more intuitive understanding of the physical behavior of the system.

4. Are there any limitations to using Green's function to solve Poisson's equation?

Green's function is not always applicable to all types of boundary conditions and source terms. It also requires the knowledge of the source term and the boundary conditions, which may not always be readily available.

5. How is Green's function related to the concept of impulse response in signal processing?

Green's function and impulse response are similar concepts, both describing the response of a system to a point source. In signal processing, the impulse response represents the output of a system when an impulse signal is applied as an input, while in Green's function, the impulse-like source is used to solve the Poisson's equation.

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