I have a problem of heat distribution in a solid cylinder with the heater in the middle, which I take as $\exp(-r^2)$.
$$\frac{\partial u(t,r)}{\partial t}=a^2\frac{\partial^2 u(t,r)}{\partial r^2}+\frac{\partial u(t,r)}{r\partial r}+\exp(-r^2)$$
The initial and boundary conditions are the following.
$u(0,r)=T_s=\text{const}$,
$u(t,R)=T_e=\text{const}$,
$0{\le}r{\le}R$,
$a=\text{const}$.
I tried using Fourier series, but only the complex one seems to give the solution, but that gives complex values for temperature that is not what I expect.
Can anybody help me solve this? Thanks.
