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Question on Laplace eq. in a ball

  1. Nov 15, 2013 #1
    For ##\nabla^2 u(r,\theta, \phi)=0##, ##u(r,\theta, \phi)=r^{n}Y_{nm}(\theta,\phi)##.

    But I have issue with this, for spherical coordinates:

    [tex]\nabla^2u=\frac{\partial^2{u}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial{u}}{\partial {r}}+\frac {1}{r^{2}}\left(\frac{\partial^2{u}}{\partial {\theta}^2}+\cot\theta\frac{\partial{u}}{\partial {\theta}}+\csc\theta\frac{\partial^2{u}}{\partial {\phi}^2}\right)[/tex]

    Let ##u=R(r)Y(\theta,\phi)## where ##Y(\theta,\phi)## is the spherical harmonics.

    [tex]\Rightarrow\; r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-\mu R=0[/tex]
    and
    [tex]\frac{\partial^2{Y}}{\partial {\theta}^2}+\cot\theta\frac{\partial{Y}}{\partial {\theta}}+\csc^2\theta\frac{\partial^2{Y}}{\partial {\phi}^2}+\mu Y=0[/tex]

    For Euler equation: ##r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+r\frac{\partial{R}}{\partial {r}}-\mu R=0## where ##\mu=n^2##. and the solution is ##R=r^n##.

    Here, because of the condition, only ##\mu=n(n+1)## is used for bounded solution.
    [tex] r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-\mu R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-n(n+1) R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+r\frac{\partial{R}}{\partial {r}}-(n+1/2)^2R[/tex]

    Which gives

    [tex]R=r^{(n+1/2)}[/tex]

    What have I done wrong?
     
    Last edited: Nov 15, 2013
  2. jcsd
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