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We use the displacement operator to derive an infinite series of integerorder derivatives for the Gr\"{u}nwald-Letnikov fractional derivative and showits correspondence to the Riemann-Liouville and Caputo fractional derivatives.We demonstrate that all three definitions of a fractional derivative lead tothe same infinite series of integer order derivatives. We find that functionsnormally represented by Taylor series with a finite radius of convergence havea corresponding integer derivative expansion with an infinite radius ofconvergence. Specifically, we demonstrate robust convergence of the integerderivative series for the hyperbolic secant (tangent) function, characterizedby a finite radius of convergence of the Taylor series $R=\pi/2$, whichdescribes bright (dark) soliton propagation in non-linear media. We also showthat for a plane wave, which has a Taylor series with an infinite radius ofconvergence, as the number of terms in the integer derivative expansionincreases, the truncation error decreases. Finally, we illustrate the utilityof the truncated integer derivative series by solving two linear fractionaldifferential equations, where the fractional derivative is replaced by aninteger derivative series up to the second order derivative. We find that ournumerical results closely approximate the exact solutions given by theMittag-Leffler and Fox-Wright functions. Thus, we demonstrate that thetruncated expansion is a powerful method for solving linear fractionaldifferential equations, such as the fractional Schr\"{o}dinger equation.

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