On the relation between sources and initial conditions for the wave and diffusion equations

In seismic forward modeling and fluid-flow simulations, sources and initial conditions are two approaches to initiate the perturbation of the medium in order to compute synthetic seismograms and pore-pressure maps of a reservoir, respectively. Assuming delta functions in time and space, source and initial particle velocity are equivalent in the first case (wave equation), while in the second case (diffusion equation) source and diffusion field are equivalent. The differential equation based on fractional derivatives unifies both cases but those equivalences break down when the order of the derivative is not a natural number. A simulation example, based on the fractional wave equation, illustrates the implementation of the source with a band-limited time history on a numerical mesh. In this case, the implementation of the initial condition requires a numerical calculation since the medium is heterogeneous. Body forces and stress sources (e.g., earthquakes) can easily be related in uniform media. A 1D example shows such a relation.