An approximate solution method for ordinary fractional differential equations with the Riemann–Liouville fractional derivatives

•The Cauchy-type problem for ordinary fractional differential equations is considered.•A method for constructing of approximate solutions to this problem is proposed.•Method is based on the two scale technique.•Proposed method is illustrated by several examples.

A new method is proposed to construct the approximate solutions of ordinary fractional differential equations with the Riemann–Liouville fractional derivatives. The method is based on the two scale technique. A fractional part of the order of the fractional derivative is considered as a small parameter εε, and two different scales x   and xεxε are introduced. As a result, the fractional differential equation is reduced to a series of integer-order differential equations, all of that are linear, except may be first one. Two different approaches to initial conditions for this series of equations are discussed. Some examples illustrate the efficiency of the proposed method.