Fractional-Hyperbolic Systems
  We describe a class of evolution systems of linear partial differential equations with the Caputo-Dzhrbashyan fractional derivative of order $\alpha \in (0,1)$ in the time variable $t$ and the first order derivatives in spatial variables $x=(x_1,...,x_n)$, which can be considered as a fractional analogue of the class of hyperbolic systems. For such systems, we construct a fundamental solution of the Cauchy problem having exponential decay outside the fractional light cone $\{(t,x):\ |t^{-\alpha}x|\le 1\}$. 
Anatoly N. Kochubei
math.AP math-ph math.MP