I am trying to solve the following equation in spherical coordinates:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\left( \nabla f \right) \cdot \vec{B} = g

[/tex]

where [itex]g[/itex] is a known scalar function, [itex]\vec{B}[/itex] is a known vector field, and [itex]f[/itex] is the unknown function.

I think the best way to approach this is to expand everything into a spherical harmonic basis:

[tex]

f(r,\theta,\phi) = \sum_{lm} f_{lm}(r) Y_{lm}(\theta,\phi)

[/tex]

[tex]

g(r,\theta,\phi) = \sum_{lm} g_{lm}(r) Y_{lm}(\theta,\phi)

[/tex]

[tex]

\vec{B}(r,\theta,\phi) = \sum_{lm} \left[ B_{lm}^r(r) \vec{Y}_{lm} + B_{lm}^{(1)} \vec{\Psi}_{lm} + B_{lm}^{(2)} \vec{\Phi}_{lm} \right]

[/tex]

where [itex]\vec{Y}_{lm}, \vec{\Psi}_{lm}, \vec{\Phi}_{lm}[/itex] are the vector spherical harmonics (VSH) defined here:

http://en.wikipedia.org/wiki/Vector_spherical_harmonics

Then, to evaluate the dot product between [itex]\nabla f[/itex] and [itex]\vec{B}[/itex], it is necessary to integrate over the unit sphere since the VSH orthogonality relations are defined in terms of integrals over [itex]d\Omega[/itex].

So, integrating the original equation over [itex]d\Omega[/itex] will yield the following ODE equation for the unknown [itex]f_{lm}(r)[/itex]:

[tex]

B_{lm}^r(r) \frac{d}{dr} f_{lm}(r) + \frac{l(l+1)}{r} B_{lm}^{(1)}(r) f_{lm}(r) = c_{lm} g_{lm}(r)

[/tex]

with

[tex]

c_{lm} = \int d\Omega Y_{lm} e^{-im\phi}

[/tex]

This ODE should be straightforward to solve numerically. However, my question is the ODE equation will determine [itex]f_{lm}[/itex] values which satisfy the equation:

[tex]

\int d\Omega \left( \nabla f \right) \cdot \vec{B} = \int d\Omega g

[/tex]

Is it true that these [itex]f_{lm}[/itex] will also satisfy the original equation?

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# Question about PDE solution

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