Exact solutions of linear Riemann–Liouville fractional differential equations with impulses

Abstract

Linear Riemann–Liouville fractional differential equations with impulses are studied in the case of scalar equations and the case of systems. Both cases are considered: the case when the lower limit of the fractional derivative is fixed on the whole interval of consideration and the case when the lower limit of the fractional derivative is changed at any point of impulse. Two types of initial conditions and impulsive conditions are applied to set up initial value problems for fractional differential equations with impulses. Explicit formulas for the solutions are obtained. The Mittag-Leffler function and the matrix generalization of the fractional exponential function are applied.