Fractal and Fractional

Research

13 pages, 1930 KiB

Open AccessArticle

Semi-Regular Continued Fractions with Fast-Growing Partial Quotients

by Shirali Kadyrov, Aiken Kazin and Farukh Mashurov

Fractal Fract. 2024, 8(8), 436; https://doi.org/10.3390/fractalfract8080436 - 24 Jul 2024

Viewed by 815

In number theory, continued fractions are essential tools because they provide distinct representations of real numbers and provide information about their characteristics. Regular continued fractions have been examined in great detail, but less research has been carried out on their semi-regular counterparts, which [...] Read more.

In number theory, continued fractions are essential tools because they provide distinct representations of real numbers and provide information about their characteristics. Regular continued fractions have been examined in great detail, but less research has been carried out on their semi-regular counterparts, which are produced from the sequences of alternating plus and minus ones. In this study, we investigate the structure and features of semi-regular continuous fractions through the lens of dimension theory. We prove a primary result about the Hausdorff dimension of number sets whose partial quotients increase more quickly than a given pace. Furthermore, we conduct numerical analyses to illustrate the differences between regular and semi-regular continued fractions, shedding light on potential future directions in this field. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

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24 pages, 376 KiB

Open AccessArticle

Further Hermite–Hadamard-Type Inequalities for Fractional Integrals with Exponential Kernels

by Hong Li, Badreddine Meftah, Wedad Saleh, Hongyan Xu, Adem Kiliçman and Abdelghani Lakhdari

Fractal Fract. 2024, 8(6), 345; https://doi.org/10.3390/fractalfract8060345 - 7 Jun 2024

Cited by 2 | Viewed by 1146

Abstract

This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example [...] Read more.

This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example with graphical representations and provides some practical applications, highlighting their relevance to special means. This study presents novel results, offering new insights into classical integrals as the fractional order

β

approaches 1, in addition to the fractional integrals we examined. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

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13 pages, 331 KiB

Open AccessArticle

Fractional Maclaurin-Type Inequalities for Multiplicatively Convex Functions

by Meriem Merad, Badreddine Meftah, Abdelkader Moumen and Mohamed Bouye

Fractal Fract. 2023, 7(12), 879; https://doi.org/10.3390/fractalfract7120879 - 12 Dec 2023

Cited by 3 | Viewed by 1337

Abstract

This paper’s major goal is to prove some symmetrical Maclaurin-type integral inequalities inside the framework of multiplicative calculus. In order to accomplish this and after giving some basic tools, we have established a new integral identity. Based on this identity, some symmetrical Maclaurin-type [...] Read more.

This paper’s major goal is to prove some symmetrical Maclaurin-type integral inequalities inside the framework of multiplicative calculus. In order to accomplish this and after giving some basic tools, we have established a new integral identity. Based on this identity, some symmetrical Maclaurin-type inequalities have been constructed for functions whose multiplicative derivatives are bounded as well as convex. At the end, some applications to special means are provided. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

18 pages, 423 KiB

Open AccessArticle

Subordination Principle for Generalized Fractional Zener Models

by Emilia Bazhlekova and Ivan Bazhlekov

Fractal Fract. 2023, 7(4), 298; https://doi.org/10.3390/fractalfract7040298 - 29 Mar 2023

Cited by 1 | Viewed by 1271

Abstract

The fractional Zener constitutive law is frequently used as a model of solid-like viscoelastic behavior. In this work, a class of linear viscoelastic models of Zener type, which generalize the fractional Zener model, is studied by the use of Bernstein functions technique. We [...] Read more.

The fractional Zener constitutive law is frequently used as a model of solid-like viscoelastic behavior. In this work, a class of linear viscoelastic models of Zener type, which generalize the fractional Zener model, is studied by the use of Bernstein functions technique. We prove that the corresponding relaxation moduli are completely monotone functions under appropriate thermodynamic restrictions on the parameters. Based on this property, we study the propagation function and establish the subordination principle for the corresponding Zener-type wave equation, which provides an integral representation of the solution in terms of the propagation function and the solution of a related classical wave equation. The analytical findings are supported by numerical examples. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

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30 pages, 1899 KiB

Open AccessArticle

Fractional Gradient Methods via ψ-Hilfer Derivative

by Nelson Vieira, M. Manuela Rodrigues and Milton Ferreira

Fractal Fract. 2023, 7(3), 275; https://doi.org/10.3390/fractalfract7030275 - 21 Mar 2023

Cited by 1 | Viewed by 1523

Abstract

Motivated by the increase in practical applications of fractional calculus, we study the classical gradient method under the perspective of the

ψ

-Hilfer derivative. This allows us to cover several definitions of fractional derivatives that are found in the literature in our study. [...] Read more.

Motivated by the increase in practical applications of fractional calculus, we study the classical gradient method under the perspective of the

ψ

-Hilfer derivative. This allows us to cover several definitions of fractional derivatives that are found in the literature in our study. The convergence of the

ψ

-Hilfer continuous fractional gradient method was studied both for strongly and non-strongly convex cases. Using a series representation of the target function, we developed an algorithm for the

ψ

-Hilfer fractional order gradient method. The numerical method obtained by truncating higher-order terms was tested and analyzed using benchmark functions. Considering variable order differentiation and step size optimization, the

ψ

-Hilfer fractional gradient method showed better results in terms of speed and accuracy. Our results generalize previous works in the literature. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

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8 pages, 285 KiB

Open AccessArticle

G-Fractional Diffusion on Bounded Domains in

R^{d}

by Luca Angelani and Roberto Garra

Fractal Fract. 2023, 7(3), 235; https://doi.org/10.3390/fractalfract7030235 - 7 Mar 2023

Cited by 1 | Viewed by 999

Abstract

In this paper, we study g-fractional diffusion on bounded domains in

R^{d}

with absorbing boundary conditions. A new general and explicit representation of the solution is obtained. We study the first-passage time distribution, showing the dependence on the particular choice of [...] Read more.

In this paper, we study g-fractional diffusion on bounded domains in

R^{d}

with absorbing boundary conditions. A new general and explicit representation of the solution is obtained. We study the first-passage time distribution, showing the dependence on the particular choice of the function g. Then, we specialize the analysis to the interesting case of a rectangular domain. Finally, we briefly discuss the connection of this general theory with the physical application to the so-called fractional Dodson diffusion model, recently discussed in the literature. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

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12 pages, 323 KiB

Open AccessArticle

A Rigorous Analysis of Integro-Differential Operators with Non-Singular Kernels

by Arran Fernandez and Mohammed Al-Refai

Fractal Fract. 2023, 7(3), 213; https://doi.org/10.3390/fractalfract7030213 - 24 Feb 2023

Cited by 8 | Viewed by 1143

Abstract

Integro-differential operators with non-singular kernels have been much discussed among fractional calculus researchers. We present a mathematical study to clearly establish the rigorous foundations of this topic. By considering function spaces and mapping results, we show that operators with non-singular kernels can be [...] Read more.

Integro-differential operators with non-singular kernels have been much discussed among fractional calculus researchers. We present a mathematical study to clearly establish the rigorous foundations of this topic. By considering function spaces and mapping results, we show that operators with non-singular kernels can be defined on larger function spaces than operators with singular kernels, as differentiability conditions can be removed. We also discover an analogue of the Sonine invertibility condition, giving two-sided inversion relations between operators with non-singular kernels that are not possible for operators with singular kernels. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

20 pages, 386 KiB

Open AccessArticle

On the 1st-Level General Fractional Derivatives of Arbitrary Order

by Yuri Luchko

Fractal Fract. 2023, 7(2), 183; https://doi.org/10.3390/fractalfract7020183 - 12 Feb 2023

Cited by 7 | Viewed by 1371

Abstract

In this paper, the 1st-level general fractional derivatives of arbitrary order are defined and investigated for the first time. We start with a generalization of the Sonin condition for the kernels of the general fractional integrals and derivatives and then specify a set [...] Read more.

In this paper, the 1st-level general fractional derivatives of arbitrary order are defined and investigated for the first time. We start with a generalization of the Sonin condition for the kernels of the general fractional integrals and derivatives and then specify a set of the kernels that satisfy this condition and possess an integrable singularity of the power law type at the origin. The 1st-level general fractional derivatives of arbitrary order are integro-differential operators of convolution type with the kernels from this set. They contain both the general fractional derivatives of arbitrary order of the Riemann–Liouville type and the regularized general fractional derivatives of arbitrary order considered in the literature so far. For the 1st-level general fractional derivatives of arbitrary order, some important properties, including the 1st and the 2nd fundamental theorems of fractional calculus, are formulated and proved. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

42 pages, 629 KiB

Open AccessArticle

Stability and Controllability Study for Mixed Integral Fractional Delay Dynamic Systems Endowed with Impulsive Effects on Time Scales

by Hasanen A. Hammad and Manuel De la Sen

Fractal Fract. 2023, 7(1), 92; https://doi.org/10.3390/fractalfract7010092 - 13 Jan 2023

Cited by 20 | Viewed by 1886

Abstract

In this article, we investigate a novel class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. Also, fixed-point techniques are applied to study the existence and uniqueness of a solution to the considered systems. Furthermore, sufficient conditions for [...] Read more.

In this article, we investigate a novel class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. Also, fixed-point techniques are applied to study the existence and uniqueness of a solution to the considered systems. Furthermore, sufficient conditions for Ulam–Hyers stability and controllability of the considered systems are established. It turns out that controllability is a very relevant property in dynamic systems and also in differential equations since, if controllability holds, then the solution of a system of differential equations also holds. Finally, an illustrative example of the obtained results is provided. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

11 pages, 293 KiB

Open AccessArticle

Boundedness of Fractional Integrals on Grand Weighted Herz–Morrey Spaces with Variable Exponent

by Babar Sultan, Fatima M. Azmi, Mehvish Sultan, Tariq Mahmood, Nabil Mlaiki and Nizar Souayah

Fractal Fract. 2022, 6(11), 660; https://doi.org/10.3390/fractalfract6110660 - 9 Nov 2022

Cited by 13 | Viewed by 1691

Abstract

In this paper, we introduce grand weighted Herz–Morrey spaces with a variable exponent and prove the boundedness of fractional integrals on these spaces. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

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