# Solving spherical PDE with separation of variables

The problem is (where $$T=T(r,t)$$ in spherical coordinates, it doesn't depend on $$\theta, \phi$$)

$$PDE: T_t (r,t) -a \Delta T=0, \quad 00$$ $$BC: T(R,t)=\frac{T_0}{5}$$ $$IC: T(r,0)=T_0$$

To solve this, I first rewrite the problem as $$T(r,t)=u(r,t)+\frac{T_0}{5}$$ so that I have a homogeneous PDE with homogeneous BC instead of a homogeneous PDE with inhomogeneous BC. The new problem looks like this

$$PDE: u_t (r,t) - a\Delta u=0$$ $$BC: u(R,t)=0$$ $$IC: u(r,0)=\frac{4T_0}{5}$$

This is where I start running into issues. To solve this I try using variable of separation $$u(r,t)=f(r)g(t)$$ which helps me rewrite the PDE as

$$g'(t)f(r)-ag(t)\Delta f(r)=0 \implies \frac{g'(t)}{ag(t)}=\frac{\Delta f(r)}{f(r)}=\lambda$$

This is how I usually work with separation of variables, and I know that for spherical equations, $$\Delta f(r)$$ can be written in some different ways: $$\Delta f(r) = \frac{1}{r} \frac{\partial^2 }{\partial r^2}r f(r) = \frac{1}{r^2} \frac{\partial}{\partial r}r^2 \frac{\partial}{\partial r} f(r) = \frac{\partial^2}{\partial r^2} f(r) + \frac{2}{r} \frac{\partial}{\partial r} f(r)$$.

When I am working with regular cartesian coordinates I know that I should try all solutions $$\lambda=0, \lambda<0, \lambda>0$$ to see which ones are trivial and which one contains the solution. But in this case with spherical coordinates I have no idea how to solve for $$f(r)$$. Any help would be greatly appreciated.

The solution should look like this;

$$T(r,t) = \frac{T_0}{5}+\sum a_n \frac{1}{r} sin(k_nr) e^{-ak_n^2 t}$$

where $$a_n$$ is a Fourier coefficient and $$k_n = \frac{n \pi}{R}$$