I would like some insight in how to solve the following equations analytically:
$$ \frac{\partial T}{\partial t} = \frac{\alpha}{r} \frac{\partial^2 (r T)}{\partial r^2} \\ \frac{\partial T}{\partial t} = \frac{\alpha}{r^2} \frac{\partial^2 (r^2 T)}{\partial r^2} $$
Subject to the following Dirichlet boundary conditions:
$$T(r=0,t) = T_0; \\ T(r=\infty,t) = 0 \hspace{1mm} \textrm{or} \hspace{1mm} T(r=\infty,t) = T_{\infty}$$
I already know that the solution for the cartesian problem, namely
$$ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial r^2} $$
Is something of the likes of $1-erf(r/(\sqrt{2\alpha t}))$ for $T_{\infty}=0$, so I would guess it is not that different for other coordinate systems? I'm trying to solve it myself but I figured I could find some help here. I'm only interested for the solution for $r>0, t>0$. I'm sorry I can't provide more input since I'm mostly concerned with numerical solutions of PDE's, but for my implementation it would be much quicker to have an analytical solution. Any help would be appreciated, thank you for your time.
A simple way to solve these equations is by variable separation. I will show this just for the first case being similar for the other. You have to choose your solution in the form $$ T(r,t)=R(r)\Theta(t). $$ By inserting this into the equation one gets $$ \frac{1}{\Theta(t)}\frac{\partial\Theta(t)}{\partial t}=\frac{\alpha}{rR(r)}\frac{\partial^2(rR(r))}{\partial r^2}. $$ By introducing a constant $\lambda$, one gets the following two equations $$ \frac{\partial\Theta(t)}{\partial t}=-\lambda\Theta(t) $$ and $$ \alpha\frac{\partial^2(rR(r))}{\partial r^2}=-\lambda rR(r). $$ Then, a particular solution has the form $$ T_\lambda(r,t)=\left(A\frac{e^{i\lambda\alpha r}}{r}+B\frac{e^{-i\lambda\alpha r}}{r}\right)e^{-\lambda t}. $$ The parameter $\lambda$ is fixed by the boundary conditions. In your case, the condition $T(r=0,t)=T_0$ yields only the possibility $A=-B$ but seems inconsistent otherwise.