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Issue with Green's function for Poisson's equation

  1. Jan 12, 2010 #1
    Say we have a 3D function, [tex]p(x,y,z)[/tex] and we define it in terms of another function [tex]f(x,y,z)[/tex] via,

    [tex]\nabla ^2 p = f.[/tex]

    I know that if we are working in [tex]R^3[/tex] space (with no boundaries) we can say that,

    [tex]p= \frac{-1}{4\pi}\iiint \limits_R \frac{f(x',y',z')}{\sqrt{(x-x')^2 +(y-y')^2+(z-z')^2}} dx' dy' dz' [/tex].

    But here is my issue: Say that instead of [tex]R^3[/tex] , we are working in a twice-infinite channel-shaped domain called [tex]\Omega[/tex], defined via,

    [tex] \Omega =\left[ (x,y,z): \quad -\infty < x,y < \infty , \quad -\frac{1}{2} \leq z \leq -\frac{1}{2} \right] .[/tex]

    Call the boundary of this domain [tex]\partial \Omega[/tex]. Now say that we have no boundary conditions specified for [tex]p[/tex] on [tex]\partial \Omega[/tex], but that [tex]f[/tex] is defined on [tex] \Omega[/tex], and is not specified outside of [tex]\Omega[/tex].

    I can see no reason, however, that we couldn't say that [tex] f=0[/tex] outside of [tex] \Omega[/tex], and that way define [tex] f[/tex] on all of [tex]R^3[/tex]. (But I could be missing something.) I should further specify that [tex] f=0[/tex], on the boundary [tex]\partial \Omega[/tex], so we could extend the domain of [tex]f[/tex] in thids way without sacrificing smoothness. (We can assume that [tex]f[/tex] is smooth and finite everywhere on [tex]\Omega[/tex].)

    So, since [tex]p[/tex] is not defined on [tex]\partial \Omega[/tex], can I simply treat it as a function defined on [tex]R^3[/tex] with no boundaries, and hence solve for [tex]p[/tex] using the triple integral above? Or is this illegal, since [tex]f[/tex] is defined only on [tex] \Omega[/tex]?
     
    Last edited: Jan 12, 2010
  2. jcsd
  3. Jan 13, 2010 #2
    i think the problem (must) needs the boundary condition, if not it's ill-posed. The Green function of
    [tex] \nabla ^2 p = f \; , \; \text{ in } \mathbb{R} ^3 [/tex]​
    is

    [tex]G(x,y,z; \xi , \eta , \theta)= \frac{-1}{4\pi} \frac{1}{ \sqrt{(x-\xi )^2 +(y-\eta )^2+(z-\theta )^2}} [/tex]
    and the Green function of the problem
    [tex] \begin{align} \nabla ^2 p &= f \; , \; \text{ in } \Omega \; \text{(defined by you)} \\
    p &= g \; , \; \text{ on } \partial \Omega
    \end{align} [/tex]
    is

    [tex] G_0 (x,y,z;\xi , \eta , \theta )= \sum_{n= -\infty}^{\infty} [ G(x,y,z; \xi , \eta , k + \theta + 1/2) - G(x,y,z; \xi , \eta , k - \theta - 1/2 ) ] [/tex]
    where [tex] f \in L^2( \mathbb{R} ^3)[/tex] or [tex] f \in L^2 ( \Omega ) [/tex].
     
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