Exact results for a fractional derivative of elementary functions

TL;DR

This paper derives exact analytical expressions for the Caputo fractional derivatives of various elementary functions using a generalized Euler's integral transform, with implications for modeling multi-scale physical systems.

Contribution

It introduces a unified method to evaluate the Caputo fractional derivative of elementary functions via a generalized hyper-geometric function, providing exact results and establishing key equivalences.

Findings

Exact formulas for Caputo fractional derivatives of elementary functions.

Demonstrates convergence of different fractional derivative definitions in the infinite limit.

Proves the equivalence between Liouville-Caputo and Fourier fractional derivatives.

Abstract

We present exact analytical results for the Caputo fractional derivative of a wide class of elementary functions, including trigonometric and inverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian, quartic Gaussian, and Lorentzian functions. These results are especially important for multi-scale physical systems, such as porous materials, disordered media, and turbulent fluids, in which transport is described by fractional partial differential equations. The exact results for the Caputo fractional derivative are obtained from a single generalized Euler's integral transform of the generalized hyper-geometric function with a power-law argument. We present a proof of the generalized Euler's integral transform and directly apply it to the exact evaluation of the Caputo fractional derivative of a broad spectrum of functions, provided that these functions can be expressed in terms of a generalized hyper-geometric function with a power-law argument. We determine that the Caputo fractional derivative of elementary functions is given by the generalized hyper-geometric function. Moreover, we show that in the most general case the final result cannot be reduced to elementary functions, in contrast to both the Liouville-Caputo and Fourier fractional derivatives. However, we establish that in the infinite limit of the argument of elementary functions, all three definitions of a fractional derivative - the Caputo, Liouville-Caputo, and Fourier- converge to the same result given by the elementary functions. Finally, we prove the equivalence between Liouville-Caputo and Fourier fractional derivatives.