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

Abstract

We solve the Cauchy problem defined by the fractional partial differential equation $[\partial _{tt}-\kappa \mathbb {D}]u=0$, with $\mathbb {D}$ the pseudo-differential Riesz operator of first order, and the initial conditions $u(x,0)=\mu (\sqrt {\pi }x_0)^{-1}e^{-(x/x_0)^2}$, 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 φ(x) = μδ(x) is obtained as corollary.

To prof. Veronique Hussin on his 60th birthday.