I want to solve the following PDE defined in 3D-space and time:
$(z \partial_t-F(t)\partial_z)f(t,\vec{x})+C[f]=S(t,z,r),$
where $r=\sqrt{x^2+y^2+z^2}$ and $C[f]$ is a linear integral operator. The details of this equation are not very important. What is important is that, by symmetry, the solution should not depend on the azimuthal angle $\phi=\arctan(y/x)$.
To solve this PDE, I wanted to use a spectral method and use the expansion $f(t,\vec{x})=\sum\limits_{m,n} a_{mn}(t)b_m(r)c_n(\theta)$, which already respects the symmetry of the equation. For the angular part, I think the most natural choice would be the spherical harmonics of order $0$: $c_n(\theta)=Y_n^0(\theta)=P_n(\cos\theta)$, but I'm not sure what basis functions to use for the radial part. I've looked in the literature, and I wasn't able to find a simple solution: most studies focus on problems defined in polar coordinates or on a spherical shell, which avoids the singularity of spherical coordinates at the origin.
I did find similar radial basis functions for problems in polar coordinates in this paper. Would it be possible to use the same basis in my situation? If not, is there another set of basis functions that would be appropriate for this problem?
Thanks!
