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

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

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 $k_n = \frac{n \pi}{R}$

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