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13 pages, 273 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Some Fractional Integral and Derivative Formulas Revisited

by Juan Luis González-Santander and Francesco Mainardi

Mathematics 2024, 12(17), 2786; https://doi.org/10.3390/math12172786 - 9 Sep 2024

Cited by 2 | Viewed by 1248

In the most common literature about fractional calculus, we find that

{}_{a}D_{t}^{α} f (t) = {}_{a}I_{t}^{- α} f (t)

is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the [...] Read more.

In the most common literature about fractional calculus, we find that

{}_{a}D_{t}^{α} f (t) = {}_{a}I_{t}^{- α} f (t)

is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of

{}_{a}I_{t}^{α} f (t)

and

{}_{a}D_{t}^{α} f (t)

. In this sense, we prove that

{}_{0}D_{t} f (t) = {}_{0}I_{t}^{- α} f (t)

is true for

f (t) = t^{ν - 1} \log t

, and

f (t) = e^{λ t}

, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for

{}_{- \infty}D_{t}^{α} {|t|}^{- δ}

and

{}_{- \infty}I_{t}^{α} {|t|}^{- δ}

found in the literature are incorrect; thus, we derive the correct ones, proving in turn that

{}_{- \infty}D_{t}^{α} {|t|}^{- δ} = {}_{- \infty}I_{t}^{- α} {|t|}^{- δ}

holds true. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

16 pages, 1551 KiB

Open AccessArticle

On the Variable Order Fractional Calculus Characterization for the Hidden Variable Fractal Interpolation Function

by Valarmathi Raja and Arulprakash Gowrisankar

Fractal Fract. 2023, 7(1), 34; https://doi.org/10.3390/fractalfract7010034 - 28 Dec 2022

Cited by 6 | Viewed by 1802

Abstract

In this study, the variable order fractional calculus of the hidden variable fractal interpolation function is explored. It extends the constant order fractional calculus to the case of variable order. The Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus are investigated for hidden [...] Read more.

In this study, the variable order fractional calculus of the hidden variable fractal interpolation function is explored. It extends the constant order fractional calculus to the case of variable order. The Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus are investigated for hidden variable fractal interpolation function. Moreover, the conditions for the variable fractional order

μ

on a specified range are also derived. It is observed that, under certain conditions, the Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus of the hidden variable fractal interpolation function are again the hidden variable fractal interpolation functions interpolating the new data set. Full article

(This article belongs to the Special Issue Stability Analysis and Control of Fractional-Order Markovian Jump Systems)

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14 pages, 604 KiB

Open AccessArticle

A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses

by Jun-Sheng Duan, Yu-Jie Lan and Ming Li

Fractal Fract. 2022, 6(12), 692; https://doi.org/10.3390/fractalfract6120692 - 23 Nov 2022

Cited by 2 | Viewed by 1547

Abstract

The fractional oscillator equation with the sinusoidal excitation

m x^{″} (t) + b D_{t}^{α} x (t) + k x (t) = F sin (ω t)

, [...] Read more.

The fractional oscillator equation with the sinusoidal excitation

m x^{″} (t) + b D_{t}^{α} x (t) + k x (t) = F sin (ω t)

,

m, b, k, ω > 0

and

0 < α < 2

is comparatively considered for the Weyl fractional derivative and the Caputo fractional derivative. In the sense of Weyl, the fractional oscillator equation is solved to be a steady periodic oscillation

x_{W} (t)

. In the sense of Caputo, the fractional oscillator equation is solved and subjected to initial conditions. For the fractional case

α \in (0, 1) \cup (1, 2)

, the response to excitation,

S (t)

, is a superposition of three parts: the steady periodic oscillation

x_{W} (t)

, an exponentially decaying oscillation and a monotone recovery term in negative power law. For the two responses to initial values,

S_{0} (t)

and

S_{1} (t)

, either of them is a superposition of an exponentially decaying oscillation and a monotone recovery term in negative power law. The monotone recovery terms come from the Hankel integrals which make the fractional case different from the integer-order case. The asymptotic behaviors of the solutions removing the steady periodic response are given for the four cases of the initial values. The Weyl fractional derivative is suitable for a describing steady-state problem, and can directly lead to a steady periodic solution. The Caputo fractional derivative is applied to an initial value problem and the steady component of the solution is just the solution in the corresponding Weyl sense. Full article

(This article belongs to the Special Issue Application of Fractional-Calculus in Physical Systems)

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11 pages, 714 KiB

Open AccessArticle

Spatial Fractional Darcy’s Law on the Diffusion Equation with a Fractional Time Derivative in Single-Porosity Naturally Fractured Reservoirs

by Fernando Alcántara-López, Carlos Fuentes, Rodolfo G. Camacho-Velázquez, Fernando Brambila-Paz and Carlos Chávez

Energies 2022, 15(13), 4837; https://doi.org/10.3390/en15134837 - 1 Jul 2022

Cited by 4 | Viewed by 2005

Abstract

Due to the complexity imposed by all the attributes of the fracture network of many naturally fractured reservoirs, it has been observed that fluid flow does not necessarily represent a normal diffusion, i.e., Darcy’s law. Thus, to capture the sub-diffusion process, various tools [...] Read more.

Due to the complexity imposed by all the attributes of the fracture network of many naturally fractured reservoirs, it has been observed that fluid flow does not necessarily represent a normal diffusion, i.e., Darcy’s law. Thus, to capture the sub-diffusion process, various tools have been implemented, from fractal geometry to characterize the structure of the porous medium to fractional calculus to include the memory effect in the fluid flow. Considering infinite naturally fractured reservoirs (Type I system of Nelson), a spatial fractional Darcy’s law is proposed, where the spatial derivative is replaced by the Weyl fractional derivative, and the resulting flow model also considers Caputo’s fractional derivative in time. The proposed model maintains its dimensional balance and is solved numerically. The results of analyzing the effect of the spatial fractional Darcy’s law on the pressure drop and its Bourdet derivative are shown, proving that two definitions of fractional derivatives are compatible. Finally, the results of the proposed model are compared with models that consider fractal geometry showing a good agreement. It is shown that modified Darcy’s law, which considers the dependency of the fluid flow path, includes the intrinsic geometry of the porous medium, thus recovering the heterogeneity at the phenomenological level. Full article

(This article belongs to the Special Issue Characterization of Conventional and Unconventional Hydrocarbon Reservoirs)

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16 pages, 604 KiB

Open AccessEditor’s ChoiceArticle

Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia

by Jun-Sheng Duan and Di-Chen Hu

Fractal Fract. 2021, 5(3), 67; https://doi.org/10.3390/fractalfract5030067 - 12 Jul 2021

Cited by 6 | Viewed by 2652

Abstract

We considered forced harmonic vibration systems with the Liouville–Weyl fractional derivative where the order is between 1 and 2 and with a distributed-order derivative where the Liouville–Weyl fractional derivatives are integrated on the interval [1, 2] with respect to the order. Both types [...] Read more.

We considered forced harmonic vibration systems with the Liouville–Weyl fractional derivative where the order is between 1 and 2 and with a distributed-order derivative where the Liouville–Weyl fractional derivatives are integrated on the interval [1, 2] with respect to the order. Both types of derivatives enhance the viscosity and inertia of the system and contribute to damping and mass, respectively. Hence, such types of derivatives characterize the viscoinertia and represent an “inerter-pot” element. For such vibration systems, we derived the equivalent damping and equivalent mass and gave the equivalent integer-order vibration systems. Particularly, for the distributed-order vibration model where the weight function was taken as an exponential function that involved a parameter, we gave detailed analyses for the weight function, the damping contribution, and the mass contribution. Frequency–amplitude curves and frequency-phase curves were plotted for various coefficients and parameters for the comparison of the two types of vibration models. In the distributed-order vibration system, the weight function of the order enables us to simultaneously involve different orders, whilst the fractional-order model has a single order. Thus, the distributed-order vibration model is more general and flexible than the fractional vibration system. Full article

(This article belongs to the Special Issue Fractional Vibrations: Theory and Applications)

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12 pages, 276 KiB

Open AccessArticle

Some New Results Involving the Generalized Bose–Einstein and Fermi–Dirac Functions

by Rekha Srivastava, Humera Naaz, Sabeena Kazi and Asifa Tassaddiq

Axioms 2019, 8(2), 63; https://doi.org/10.3390/axioms8020063 - 21 May 2019

Cited by 3 | Viewed by 3522

Abstract

In this paper, we obtain a new series representation for the generalized Bose–Einstein and Fermi–Dirac functions by using fractional Weyl transform. To achieve this purpose, we obtain an analytic continuation for these functions by generalizing the domain of Riemann zeta functions from [...] Read more.

In this paper, we obtain a new series representation for the generalized Bose–Einstein and Fermi–Dirac functions by using fractional Weyl transform. To achieve this purpose, we obtain an analytic continuation for these functions by generalizing the domain of Riemann zeta functions from

(0 < ℜ (s) < 1)

to

(0 < ℜ (s) < μ) .

This leads to fresh insights for a new generalization of the Riemann zeta function. The results are validated by obtaining the classical series representation of the polylogarithm and Hurwitz–Lerch zeta functions as special cases. Fractional derivatives and the relationship of the generalized Bose–Einstein and Fermi–Dirac functions with Apostol–Euler–Nörlund polynomials are established to prove new identities. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

25 pages, 304 KiB

Open AccessArticle

Weyl and Marchaud Derivatives: A Forgotten History

by Fausto Ferrari

Mathematics 2018, 6(1), 6; https://doi.org/10.3390/math6010006 - 3 Jan 2018

Cited by 72 | Viewed by 8431

Abstract

In this paper, we recall the contribution given by Hermann Weyl and André Marchaud to the notion of fractional derivative. In addition, we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different [...] Read more.

In this paper, we recall the contribution given by Hermann Weyl and André Marchaud to the notion of fractional derivative. In addition, we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different fields of the mathematics. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

24 pages, 326 KiB

Open AccessArticle

Multiplicity of Homoclinic Solutions for Fractional Hamiltonian Systems with Subquadratic Potential

by Neamat Nyamoradi, Ahmed Alsaedi, Bashir Ahmad and Yong Zhou

Entropy 2017, 19(2), 50; https://doi.org/10.3390/e19020050 - 24 Jan 2017

Cited by 14 | Viewed by 4349

Abstract

In this paper, we study the existence of homoclinic solutions for the fractional Hamiltonian systems with left and right Liouville–Weyl derivatives. We establish some new results concerning the existence and multiplicity of homoclinic solutions for the given system by using Clark’s theorem from [...] Read more.

In this paper, we study the existence of homoclinic solutions for the fractional Hamiltonian systems with left and right Liouville–Weyl derivatives. We establish some new results concerning the existence and multiplicity of homoclinic solutions for the given system by using Clark’s theorem from critical point theory and fountain theorem. Full article

(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)

26 pages, 314 KiB

Open AccessArticle

The Fractional Orthogonal Derivative

by Enno Diekema

Mathematics 2015, 3(2), 273-298; https://doi.org/10.3390/math3020273 - 22 Apr 2015

Cited by 6 | Viewed by 4658

Abstract

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder [...] Read more.

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann–Liouville fractional derivative is considered by replacing the n-th derivative by its approximation in the formula for the fractional derivative. In the case of, for instance, Jacobi polynomials, an explicit formula for the kernel of this approximate fractional derivative can be given. Next, we consider the fractional derivative as a filter and compute the frequency response in the continuous case for the Jacobi polynomials and in the discrete case for the Hahn polynomials. The frequency response in this case is a confluent hypergeometric function. A different approach is discussed, which starts with this explicit frequency response and then obtains the approximate fractional derivative by taking the inverse Fourier transform. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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