Delsarte equation for Caputo operator of fractional calculus

A fractional order variant of the Delsarte equation is investigated involving the Caputo differential derivative. Solvability of the resulting fractional hyperbolic Cauchy problem is achieved in the sense of distributions. A regularity result shows that the solution may be a function of time. Rigorous Delsarte representations are established. The symmetry between the fractional operators acting on space and time, induced by the Delsarte equation, opens the door to new type of fractional partial differential equations.