Is the mathematical form of the acoustic diffusion equation present in other fields of physics? We are working in the field of High Performance Computing and we have developed a very efficient parallel implementation for solving the Acoustic Diffusion Equation as described below:
$$
\frac{\partial w(r,t)}{\partial t}-D\nabla^2w(r,t) + cmw(r, t) = P(t)\delta(r-r_s)
$$
We now wonder whether similar equations are important in other fields of physics; then we would just make somo slight modifications of our parallel implementation, and we could apply it to other relevant fields in physics. Any ideas?
 A: With $t\to-it$ and $P(t)=\delta(t-t_0)$ and some constants redefinition, this is an equation for the Green's function of the Schrödinger equation
$-i \hbar \partial_t \psi = \left(-\frac{\hbar^2}{2m} \Delta + V\right) \psi$
Where $V \sim cm$ is a constant. In most contexts, one needs either $V=0$ or $V=V(r)$. The solutions however become oscillatory which would probably imply significant changes to your algorithm.
With $cm=0$ and again $P(t)=\delta(t-t_0)$ this is the equation for the Green's function of the heat diffusion equation:
$\partial_t T - \alpha \Delta T = 0$
If we keep $P(t)$ a regular function, we have a solution of heat diffusion in an object which is in contact with a highly localized reservoir of temperature $\sim P(t)$ at $r=r_s$. This actually applies to any kind of a linear diffusion process.
(Note: the dimension of the problem and the boundary conditions are unclear, but the discussion applies to any case as long as the imposed boundary conditions are relevant in the physical problems.)
