Investigation of the Cauchy problem for one fractional order equation with the Riemann–Liouville operator

The article is dedicated to solving the Cauchy problem for a differential equation with a Riemann–Liouville fractional derivative. The initial condition is formulated in a natural way and it is proven that the resulting solution is regular. Firstly, a fundamental solution is constructed and its properties are analyzed. Then, based on these properties, the solution to the homogeneous equation in the Cauchy problem is studied. Furthermore, unlike other problems of this type, the solution to the Cauchy problem presented for a nonhomogeneous equation is explicitly obtained in this work using the Duhamel's principle and the three-parameter Mittag–Leffler function. By applying additional conditions to these problems, it is also demonstrated that this solution is classical.