The neutral-fractional telegraph equation

In this paper, the neutral-fractional telegraph equation is introduced and discussed. This equation is a natural fractional generalization of the conventional telegraph equation and contains two time-fractional Caputo derivatives of the orders α and α∕2, respectively, and the Riesz space-fractional derivative of the order α, 1 < α ≤ 2. In this paper, we derive some analytical representations of the fundamental solution to this equation and discuss its properties. A special focus is put to two prominent particular cases of the neutral-fractional telegraph equation, namely, to the α-fractional wave equation and to the α-fractional diffusion equation that contain only one time-fractional Caputo derivative of the order α or α∕2, respectively.