A collection of correct fractional calculus for discontinuous functions

Abstract

In this paper, an important property of fractional order operators involving discontinuous functions is discussed, First, a pioneering work of impulsive fractional differential equations is recalled to illuminate the incorrectness of notation \({^C_{t_k}D}^{q}_t\). Second, a class of piecewise-defined equations with Caputo fractional derivative is contrastively investigated, and it is revealed that the additivity of integration on intervals for integer-order integral does not hold for fractional integrals, not to mention fractional derivatives. Third, by utilizing the Heaviside step function, an interesting property of fractional integral involving piecewise-defined functions is correspondingly presented. Finally, illustrative examples are given for validation of the derived results, which may lead to a new way to reconsider the dynamic behavior of fractional hybrid systems, as well as discontinuous control design for fractional systems.