Comparison of Definition of Several Fractional Derivatives

Abstract

The idea of fractional derivatives was raised first by L'Hospital in 1695. The fractional calculus and its mathematical consequences attracted many mathematicians such as Fourier, Euler, Laplace. Various definitions of non-integer order integral or derivative was given by many mathematicians. In this paper, firstly, we discuss the positive integer higher order derivative of a function, and obtain general formula of high order derivative. Secondly, we introduce the definitions of Gr\"unwald-Letnikov, Riemann-Liouville and Caputo. Finally, we point out the relationship between these definitions. Caputo's integral definition and Gr\"unwald-Letnikov integral definition are consistent with the Riemann-Liouville integral definition. When f has 1 m order continuous derivative and