I am trying to solve the heat equation on a finite interval with a localised decaying heat source, but I am stuck.
Specifically: Consider the equation $$ u_t(t,x)=\kappa u_{xx}(t,x) +q(t,x) \,,$$ where $x\in (0,L)$ and $t>0$. The boundary and initial conditions are simply $u(t,0)=u(t,L)=0$ and $u(0,x)=0$. The heat source is localised at $x_h$ somewhere in the interval and decays exponentially, $$ q(t,x)=q_0 \delta(x-x_h) e^{-\lambda t} .$$
I have tried to solve the problem via Fourier expanding $u$ and $q$ in $\sin \frac{n\pi x}{L}$ modes, which leads me to $$u(x,t)=u_0 e^{-\lambda t} \sum_{n=1}^\infty \sin (n\rho) \,\frac{1-e^{(\lambda -\zeta n^2)t}}{\zeta n^2-\lambda}\, \sin\left(n \frac{\pi x}{L}\right) ,$$ where $\zeta=\pi^2\kappa/L^2$, but I was not able to find a closed-form solution. (For my situation, numerically $\lambda/\zeta$ generally is in the range of $10^2$ to $10^3$.)
Is there a closed solution, or a better way to solve the problem?
Bonus: What I'm really interested in is the case where $\kappa$, which encodes the thermal properties of the material, changes, i.e. $$ \kappa=\left\{ \begin{matrix}\kappa_1 & 0<x<x_s\\ \kappa_2 & x_s\leq x<L \end{matrix} \right. .$$ Then, I probably cannot just equate the Fourier modes on both sides of the original equation. What's the best way to proceed?