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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)

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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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 2654

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 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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15 pages, 311 KiB

Open AccessArticle

Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

by Pedro J. Miana and Natalia Romero

Mathematics 2021, 9(9), 984; https://doi.org/10.3390/math9090984 - 27 Apr 2021

Cited by 1 | Viewed by 1948

Abstract

Generalized Laguerre polynomials,

L_{n}^{(α)}

Generalized Laguerre polynomials,

L_{n}^{(α)}

, verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis, Semigroup Theory and Difference-Differential Equations)

14 pages, 302 KiB

Open AccessArticle

On the Solutions of a Class of Integral Equations Pertaining to Incomplete H-Function and Incomplete H-Function

by Manish Kumar Bansal, Devendra Kumar, Jagdev Singh and Kottakkaran Sooppy Nisar

Mathematics 2020, 8(5), 819; https://doi.org/10.3390/math8050819 - 19 May 2020

Cited by 20 | Viewed by 2471

Abstract

The main aim of this article is to study the Fredholm-type integral equation involving the incomplete H-function (IHF) and incomplete H-function in the kernel. Firstly, we solve an integral equation associated with the IHF with the aid of the theory of fractional calculus and Mellin transform. Next, we examine an integral equation pertaining to the incomplete H-function with the help of theory of fractional calculus and Mellin transform. Further, we indicate some known results by specializing the parameters of IHF and incomplete H-function. The results computed in this article are very general in nature and capable of giving many new and known results connected with integral equations and their solutions hitherto scattered in the literature. The derived results are very useful in solving various real world problems. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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 4660

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 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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