Variable order fractional systems.

Highlights • It presents a revision of previously proposed variable order fractional derivatives. • It introduces a new approach based on the inverse Laplace transform. • Variable order fractional linear systems are defined. • A variable order Mittag-Leffler function is introduced. • Variable order two-sided fractional derivatives. Abstract Fractional Calculus had a remarkable evolution during recent decades, and paved the way towards the definition of variable order derivatives. In the literature we find several different alternative definitions of such operators. This paper presents an overview of the fundamentals of this topic and addresses the questions of finding out which of them are reasonable according to simple criteria used for constant order fractional derivatives. This approach leads to the definitions of variable order fractional derivative based on the Grünwald–Letnikov and the Liouville formulations defined on R , as well as to the definition of a Mittag-Leffler function for variable orders, and to the application of these definitions to dynamical systems. [ABSTRACT FROM AUTHOR]