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.

Original languageEnglish
Pages (from-to)231-243
Number of pages13
JournalCommunications In Nonlinear Science And Numerical Simulation
Publication statusPublished - 15 Jun 2019


  • Fractional derivative
  • Fractional integral
  • Variable order
  • Variable order linear system
  • Variable order Mittag-Leffler function

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