Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 6 / 2 / c

Separate the transported initial datum from the gain term. Defineand the Boltzmann Volterra operatorThe Duhamel principle makes the linear Boltzmann equation equivalent, whenever the integrals and solution are legitimate, toThe gain samples velocities $v_{\ast }$ at the backward spatial position of the outgoing velocity $v$; replacing that spatial shift by one depending on $v_{\ast }$ would give a different equation.