Fractional Neutral Equations Research Articles

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.

Periodic solutions of a fractional neutral equation with finite delay

In this paper, we prove the maximal regularity property of an abstract fractional differential equation with finite delay on periodic Besov and Triebel–Lizorkin spaces and use these results to guarantee the existence and uniqueness of periodic solution of a neutral fractional differential equation with finite delay. The main tool used to achieve our goal is an operator-valued version of Miklhin’s Fourier multiplier theorem and fixed-point argument.