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Title: Exact results for a fractional derivative of elementary functions
We present exact analytical results for the Caputo fractional derivative ofa wide class of elementary functions, including trigonometric andinverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian,quartic Gaussian, Lorentzian, and shifted polynomial functions. Theseresults are especially important for multi-scale physical systems, suchas porous materials, disordered media, and turbulent fluids, in whichtransport is described by fractional partial differential equations. Theexact results for the Caputo fractional derivative are obtained from a singlegeneralized Euler’s integral transform of the generalized hypergeometricfunction with a power-law argument. We present a proof of thegeneralized Euler’s integral transform and directly apply it to theexact evaluation of the Caputo fractional derivative of a broad spectrum offunctions, provided that these functions can be expressed in terms of ageneralized hypergeometric function with a power-law argument. Wedetermine that the Caputo fractional derivative of elementary functions isgiven by the generalized hypergeometric function. Moreover, we show thatin the most general case the final result cannot be reduced toelementary functions, in contrast to both the Liouville-Caputo and Fourier fractionalderivatives. However, we establish that in the infinite limit of theargument of elementary functions, all three definitions of a fractionalderivative - the Coputo, Liouville-Caputo, and Fourier - converge to the same result given by theelementary functions. Finally, we prove the equivalence between Liouville-Caputo and Fourierfractional derivatives. more »« less
Ichiba, Tomoyuki; Pang, Guodong; Taqqu, Murad S.(
, Journal of Theoretical Probability)
null
(Ed.)
The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power-law shape function and non-stationary noises with a power-law variance function. In this paper, we study sample path properties of the generalized fractional Brownian motion, including Hölder continuity, path differentiability/non-differentiability, and functional and local law of the iterated logarithms.
A basic tenet of linear invariant systems is that they are sufficiently described by either the impulse response function or the frequency transfer function. This implies that we can always obtain one from the other. However, when the transfer function contains uncanceled poles, the impulse function cannot be obtained by the standard inverse Fourier transform method. Specifically, when the input consists of a uniform train of pulses and the output sequence has a finite duration, the transfer function contains multiple poles on the unit cycle. We show how the impulse function can be obtained from the frequency transfer function for such marginally stable systems. We discuss three interesting discrete Fourier transform pairs that are used in demonstrating the equivalence of the impulse response and transfer functions for such systems. The Fourier transform pairs can be used to yield various trigonometric sums involving sinπk/NsinπLk/N, where k is the integer summing variable and N is a multiple of integer L.
Dong, Hongjie; Kim, Doyoon(
, International Mathematics Research Notices)
Abstract We consider time fractional parabolic equations in divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$, which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, and on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.
Casper, W. Riley; Grünbaum, F. Alberto; Yakimov, Milen; Zurrián, Ignacio(
, Proceedings of the National Academy of Sciences)
Commuting integral and differential operators connect the topics of signal processing, random matrix theory, and integrable systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the Korteweg–de Vries (KdV) equation. We prove a general theorem that the integral operator associated to every wave function in the infinite-dimensional adelic Grassmannian G r a d of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of Kadomtsev–Petviashvili (KP) wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson’s sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A 9 0 ○ rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by [Airault, McKean, and Moser, Commun. Pure Appl. Math. 30, 95–148 (1977)] and [Krichever, Funkcional. Anal. i Priložen. 12, 76–78 (1978)], respectively, in the late 1970s. Many examples are presented.
We study shot noise processes with Poisson arrivals and nonstationary noises. The noises are conditionally independent given the arrival times, but the distribution of each noise does depend on its arrival time. We establish scaling limits for such shot noise processes in two situations: (a) the conditional variance functions of the noises have a power law and (b) the conditional noise distributions are piecewise. In both cases, the limit processes are self‐similar Gaussian with nonstationary increments. Motivated by these processes, we introduce new classes of self‐similar Gaussian processes with nonstationary increments, via the time‐domain integral representation, which are natural generalizations of fractional Brownian motions.
@article{osti_10074704,
place = {Country unknown/Code not available},
title = {Exact results for a fractional derivative of elementary functions},
url = {https://par.nsf.gov/biblio/10074704},
DOI = {10.21468/SciPostPhys.4.6.029},
abstractNote = {We present exact analytical results for the Caputo fractional derivative ofa wide class of elementary functions, including trigonometric andinverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian,quartic Gaussian, Lorentzian, and shifted polynomial functions. Theseresults are especially important for multi-scale physical systems, suchas porous materials, disordered media, and turbulent fluids, in whichtransport is described by fractional partial differential equations. Theexact results for the Caputo fractional derivative are obtained from a singlegeneralized Euler’s integral transform of the generalized hypergeometricfunction with a power-law argument. We present a proof of thegeneralized Euler’s integral transform and directly apply it to theexact evaluation of the Caputo fractional derivative of a broad spectrum offunctions, provided that these functions can be expressed in terms of ageneralized hypergeometric function with a power-law argument. Wedetermine that the Caputo fractional derivative of elementary functions isgiven by the generalized hypergeometric function. Moreover, we show thatin the most general case the final result cannot be reduced toelementary functions, in contrast to both the Liouville-Caputo and Fourier fractionalderivatives. However, we establish that in the infinite limit of theargument of elementary functions, all three definitions of a fractionalderivative - the Coputo, Liouville-Caputo, and Fourier - converge to the same result given by theelementary functions. Finally, we prove the equivalence between Liouville-Caputo and Fourierfractional derivatives.},
journal = {SciPost Physics},
volume = {4},
number = {6},
author = {Shchedrin, Gavriil and Smith, Nathanael and Gladkina, Anastasia and Carr, Lincoln},
}
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