Question on using solution from Helmholtz equation in Poisson equation

[yungman]

Helmholtz equation stated that
[tex]\nabla^2 u(r,\theta,\phi) =-ku(r,\theta,\phi) = f(r,\theta,\phi) [/tex]
This is being used for Poisson equation with zero boundary:
[tex]\nabla^2 u(r,\theta,\phi) = f(r,\theta,\phi) [/tex]
and
[tex]u(a,\theta,\phi)=0[/tex]

I just don't see how this can work as [itex]k=m^2[/itex] is a number only.

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

If [itex]u(r,\theta,\phi)[/itex] is a constant number only, that cannot be right?

Please explain. Thanks

 

[gtoguy97]

.The Helmholtz equation can be used for the Poisson equation with zero boundary by setting k = 0. This means that \nabla^2 u(r,\theta,\phi) = f(r,\theta,\phi), and u(a,\theta,\phi)=0. This will result in a solution of the form u(r,\theta,\phi) = C_1 + C_2r + C_3r^2 + ..., where C_1, C_2, etc. are constants determined by the boundary conditions.