Expansion of fractional derivatives in terms of an integer derivative series: physical and numerical applications

We use the displacement operator to derive an infinite series of integer derivatives for the Grünwald-Letnikov fractional derivative and demonstrate that the infinite series of integer order derivatives is the same for Grünwald-Letnikov, Riemann-Liouville, and Caputo fractional derivatives. With the first few terms of the infinite series, we find that for functions with a finite radius of convergence of their Taylor series, the corresponding integer derivative expansion has by an infinite radius of convergence. Specifically, we demonstrate a robust convergence of integer derivative expansion for the hyperbolic secant and tangent functions, characterized by a finite radius of convergence of the Taylor series R=pi/2. Moreover, for a plane wave with an infinite radius of convergence, we show the truncation error decreases as the number of terms in the expansion increases. We find that our numerical results closely approximates the exact solutions given by the Mittag-Leffler and Fox-Wright functions. Thus, we demonstrate that the truncated expansion is a powerful method for solving linear fractional differential equations.