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Solution that doesn't diverge at origin

  1. Jun 4, 2014 #1
    Hi - wondering if you can help me find a solution of:

    [itex]\nabla^{2}u-\frac{u}{\lambda^{2}}=a\delta(r)[/itex]

    for spherical symmetry in 3D with the condition that [itex]\lim_{r\rightarrow \infty}u=0[/itex]. It can be rewritten in spherical coordinates as

    [itex]\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial u}{\partial r}\right)-\frac{u}{\lambda^{2}}=a\delta(r)[/itex].

    Any help would be much appreciated! :)
     
  2. jcsd
  3. Jun 4, 2014 #2

    pasmith

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    You are solving [tex]\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial u}{\partial r}\right)-\frac{u}{\lambda^{2}} = 0[/tex] in [itex]r > 0[/itex]. Setting [tex]
    u(r) = \frac{f(r)}r[/tex] yields [tex]
    f'' - \lambda^{-2} f = 0
    [/tex] and the only way to not have [itex]u[/itex] diverge at the origin is to take [itex]f(0) = 0[/itex], which yields [itex]f(r) = A\sinh(\lambda^{-1} r)[/itex] and thus [tex]
    u(r) = \frac{A\sinh(\lambda^{-1} r)}{r}.
    [/tex] L'hopital confirms that [tex]
    \lim_{r \to 0} u(r) = \lim_{r \to 0} \frac{A\lambda^{-1}\cosh(\lambda^{-1} r)}{1} = A\lambda^{-1}.
    [/tex] Unfortunately [itex]|u| \to \infty[/itex] as [itex]r \to \infty[/itex]. To get a solution which decays at infinity you must take [itex]f(r) = e^{-r/\lambda}[/itex], and the resulting [itex]u[/itex] diverges at the origin.

    (Usually this setup is an abstraction of "there is a small sphere at the origin". Within the sphere you use a solution which is bounded at the origin, and outside the sphere you use a solution which decays as [itex]|r| \to \infty[/itex].)
     
    Last edited: Jun 4, 2014
  4. Jul 13, 2014 #3
    Bessel or Hankel functions.
     
  5. Jul 17, 2014 #4

    jasonRF

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    I wouldn't expect a physical solution that does not diverge at the origin. That is the exact equation for the electric potential of a point charge in a hot plasma, where $\lambda$ would be the Debye length. Potentials of point charges always diverge at the location of the charge...
     
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