Exact results for a fractional derivative of elementary functions
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, Lorentzian, and shifted polynomial 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 hypergeometric 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 hypergeometric function with a power-law argument. We determine that the Caputo fractional derivative of elementary functions is given by the generalized hypergeometric 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 Coputo, 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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Jun 15, 2018
- Source ID
- 10.21468/scipostphys.4.6.029
Entities
People
- Anastasia Gladkina
- Gavriil Shchedrin
- Lincoln D Carr
- Nathanael Smith
Organizations
- Air Force Office of Scientific Research
- Colorado School of Mines
- National Science Foundation