Approximation of ordinary fractional differential equations by differential equations with a small parameter

An approach to approximation of ordinary fractional differential equations by integer-order differential equations is proposed. It is assumed that the order of fractional differentiation is close to integer. Perturbation expansions for the Riemann–Liouville and Caputo fractional derivatives are derived in terms of a suitable small parameter extracted from the order of fractional differentiation. The first-order term of these expansions is represented by series depending on integer-order derivatives of all integer orders. The expansions obtained permit one to approximate ordinary fractional differential equations, involving such types of fractional derivatives, by integer-order differential equations with a small parameter. It is proved that, for fractional differential equations belonging to a certain class, corresponding approximate equations contain only a finite number of integer-order derivatives. Approximate solutions to such equations can be obtained using well-known perturbation techniques. The proposed approach is illustrated by several examples.