Setting and Solving of the Cauchy type problems for the Second Order Differential Equations with Riemann–Liouville Fractional Derivatives

The correctness of the Cauchy problems in local (classical) and nonlocal staging for two linear ordinary second order differential equations with Riemann–Liouville fractional derivatives is substantiated. The explicit solutions in terms of some special functions related Mittag–Leffler type function are found out. The continuos dependence from the fractional parameter $\beta$ for these solutions is indicated. For the second equation the changing statement of the Cauchy type problem coinciding with classical when $\beta=0$ is considered. These equations are proposed such as some model fractional oscillating equation.