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Thread: Question of using solution of Helmholtz equation to solve Poisson equation

  1. #1
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    Question of using solution of Helmholtz equation to solve Poisson equation

    Helmholtz equation stated that

    $\displaystyle \nabla^2 u(r,\theta,\phi) =-ku(r,\theta,\phi) = f(r,\theta,\phi) $

    This is being used for Poisson equation with zero boundary:

    $\displaystyle \nabla^2 u(r,\theta,\phi) = f(r,\theta,\phi) $

    and

    $\displaystyle u(a,\theta,\phi)=0$

    I just don't see how this can work as $\displaystyle k=m^2$ is a number only.

    If $\displaystyle \nabla^2 u(r,\theta,\phi)=1$ which means $\displaystyle ku(r,\theta,\phi)$ is only constant numbers depending on $\displaystyle m$!!!

    If $\displaystyle u(r,\theta,\phi)$ is a constant number only, that cannot be right?

    Please explain. Thanks
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Question of using solution of Helmholtz equation to solve Poisson equation

    Maybe your question would be answered easily, if you wrote

    "...Poisson equation with zero boundary

    $\displaystyle \nabla^2 w=f, w(a,\theta,\phi)=0$

    etc"
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