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Math Help - Question on Laplace eq. on a ball.

  1. #1
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    Question on Laplace eq. on a ball.

    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:

    \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)

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

    \Rightarrow\; r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-\mu R=0

    and

    \frac{\partial^2{Y}}{\partial {\theta}^2}+\cot\theta\frac{\partial{Y}}{\partial {\theta}}+\csc^2\theta\frac{\partial^2{Y}}{\partia  l {\phi}^2}+\mu Y=0



    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.

     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

    Which gives

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

    What have I done wrong?
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Question on Laplace eq. on a ball.

    Quote Originally Posted by Alan0354 View Post
    Which gives

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


    No, it doesn't
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