Heat transfer in a cylinder. Inhomogeneous PDE

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.

1. $u(0,r)=T_s=\text{const}$,

2. $u(t,R)=T_e=\text{const}$,

3. $0{\le}r{\le}R$,

4. $a=\text{const}$.

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.

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Can you include some of your working? – Chris Taylor Nov 21 '11 at 8:04
There should be parentheses around the $\partial^2 u$ and $\partial u$ terms on the right. Apart from that I don't understand your modeling of the heater. When the heater is a cylinder of radius $H<R$ then it causes a boundary condition $u(t, H)=T_H$. – Christian Blatter Nov 21 '11 at 11:42