Do you have any ideas on how to analytically solve the diffusion equation in spherical coordinates (radial component only) for nonhomogeneous boundary conditions as follows?
$\frac{\partial u(r,t)}{\partial t}=\frac{D}{r^2}\frac{\partial }{\partial r} \left ( r^2 \frac{\partial u(r,t) }{\partial r}\right)$ $0<r<1$, $0\le t<\infty$
ICs $u(r,0)=u0$
BCs $\frac{\partial u(r,t)}{\partial r}|_{r=0}=0 $ and $\frac{\partial u(r,t)}{\partial r}|_{r=1}=f(t) $.
I found some similar equations, but here we have two Neumann boundary conditions.
Solution to diffusion equation in spherical coordinates
Solving the heat equation in spherical polars with nonhomogeneous boundary conditions
