2-D Poisson's equation - Green's function

  • Thread starter plasmoid
  • Start date
  • #1
15
0
In the x-y plane, we have the equation


[tex]\nabla^{2} \Psi = - 4\pi \delta(x- x_{0}) \delta (y- y_{0}) [/tex]

with [tex] \Psi = 0 [/tex] at the rectangular boundaries, of size L.

A paper I'm looking at says that for

[tex]R^{2}[/tex] = [tex](x-x_{0})^{2}[/tex] + [tex](y-y_{0})^{2}[/tex] << [tex]L^{2}[/tex] ,

that is, for points very close to the source, the solution must behave as if the boundaries were at infinity, and

[tex] \Psi \approx -2 ln R. [/tex]

I see that -2 ln R will satisfy the equation, but why should it be the only solution valid at R<<L? And how does it satisfy the boundary condition?

- 2 ln R goes to infinity as R goes to zero, does that have anything to do with it?

Morse and Feshbach have the same thing on Pg 798 of vol 1 of their Methods ... , but I cant see an explanation there either.
 
Last edited:

Answers and Replies

Related Threads on 2-D Poisson's equation - Green's function

Replies
1
Views
2K
  • Last Post
Replies
4
Views
7K
Replies
1
Views
1K
Replies
1
Views
4K
Replies
6
Views
129
Replies
4
Views
1K
Replies
0
Views
4K
Top