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

Abstract

We use the displacement operator to derive an infinite series of integer order derivatives for the Gr\"{u}nwald-Letnikov fractional derivative and show its correspondence to the Riemann-Liouville and Caputo fractional derivatives. We demonstrate that all three definitions of a fractional derivative lead to the same infinite series of integer order derivatives. We find that functions normally represented by Taylor series with a finite radius of convergence have a corresponding integer derivative expansion with an infinite radius of convergence. Specifically, we demonstrate robust convergence of the integer derivative series for the hyperbolic secant (tangent) function, characterized by a finite radius of convergence of the Taylor series $R=\pi/2$, which describes bright (dark) soliton propagation in non-linear media. We also show that for a plane wave, which has a Taylor series with an infinite radius of convergence, as the number of terms in the integer derivative expansion increases, the truncation error decreases. Finally, we illustrate the utility of the truncated integer derivative series by solving two linear fractional differential equations, where the fractional derivative is replaced by an integer derivative series up to the second order derivative. We find that our numerical results closely approximate 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, such as the fractional Schr\"{o}dinger equation.