An Integro-Differential Equation of the Fractional Form: Cauchy Problem and Solution

We solve the Cauchy problem defined by the fractional partial differential equation $[{\mathrm{\partial }}_{tt}-\kappa \mathbb{D}]u=0$, with $\mathbb{D}$ the pseudo-differential Riesz operator of first order, and the initial conditions $u(x,0)=\mu (\sqrt{\pi }{x}_0{)}^{-1}{e}^{-(x\mathrm{/}{x}_0{)}^2}$, $u_t(x,0)=0$. The solution of the Cauchy problem resulting from the substitution of the Gaussian pulse $u(x,0)$ by the Dirac delta distribution $\phi (x)=\mu \delta (x)$ is obtained as corollary.