Wave–diffusion dualism of the neutral-fractional processes

In this paper, a neutral-fractional equation is introduced and analyzed. In contrast to the general time- and space-fractional diffusion equation, the neutral-fractional equation contains fractional derivatives of the same order α,1≤α≤2 both in space and in time. As it has been shown earlier, solutions of the neutral-fractional equation can be interpreted as damped waves with the constant propagation velocities that means that this equation inherits some characteristics of the wave equation. Otherwise, the first fundamental solution of the one-dimensional neutral-fractional equation is known to be a spatial probability density function evolving in time and is thus related to the diffusion processes. In this paper, we investigate the entropy and the entropy production rate of the neutral-fractional equation and show that both of them are strongly connected to those of the diffusion processes. Thus a wave–diffusion dualism of the processes described by the neutral-fractional equation is established.