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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Question about PDE solution

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**