I'm looking for a closed form (if it exists) of the Fourier series $$u(x,t)=u_0 e^{-\lambda t} \sum_{n=1}^\infty \sin (n\rho) \,\frac{1-e^{(\lambda -\zeta n^2)t}}{\zeta n^2-\lambda}\, \sin\!\left(n \frac{\pi x}{L}\right)$$ and am stuck. The function $u(x,t)$ is defined for $x\in(0,L)$ and $t>0$. The parameters $\lambda$, $\rho$, $\zeta$, and $u_0$ are constant.
Background is: I am trying to solve the heat equation on a finite interval with a pointlike decaying heat source ($\sim e^{-\lambda t}$), and solution via Fourier transform. Maybe there's a better way?