I would like to solve the diffusion (or heat) equation $$ f_t(x,t) = D f_{xx}(x,t)$$ with the boundary conditions $$f(0,t)=f(H,t)=0$$ and $$ f(x,0)=\delta(x-x_0)$$ That is, the initial distribution starts somewhere in the interval $0 < x_0 < H$, and the heat is absorbed at the boundaries $x=0$ and $x=H$.
Standard Approach:
The usual approach is to construct an Eigenfunction expansion. One sets $f(x,t) \propto e^{-\lambda t} g(x)$ and finds $g_\lambda''(x) + (\lambda/D)g_\lambda'(x)=0$, finding ultimately that $$ f(x,t) = \sum_\lambda A_\lambda g_\lambda(x),$$ where the eigenmodes and coefficients are selected to meet the boundary and initial conditions.
Desired Approach: In my case, I would like the solve this problem by Laplace transforms. I believe this should be possible despite the bounded domain of $x$ due to the successful application of this technique to a similar (but much more difficult) problem by Masoliver and Porra (1996).
According to these authors, we can simply take the Laplace transform over space $\mathcal{L}\{f(x,t)\}(g) = \int_0^\infty dx e^{-g x} f(x,t) $,despite the upper boundary. This obtains the differential equation $$\tilde{P}_t(g,t) = - g^2 \tilde{P}, $$ giving solution in Laplace space $$\tilde{P}(g,t) = A(t) e^{-g^2 t}.$$ The initial condition $\tilde{P}(g,0)=1$ gives $A(0)=1$, but the transform over $g$ does not invert to a known solution, so I am not sure how to proceed. Any thoughts are appreciated on how to solve the diffusion equation with Laplace transforms.