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Fractal Fract
Special Issues
New Trends on Generalized Fractional Calculus
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New Trends on Generalized Fractional Calculus

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: 15 May 2024 | Viewed by 9250

Special Issue Editors

Dr. Milton Ferreira

E-Mail Website
Guest Editor
1. School of Technology and Management, Polytechnic Institute of Leiria, Leiria, Portugal
2. CIDMA-Center for Research and Development in Mathematics and Applications, University of Aveiro, Aveiro, Portugal
Interests: Fractional calculus, Fractional partial differential equations; Clifford analysis; Group representation theory; Gyrogroups; Harmonic and Wavelet analysis
Prof. Dr. Maria Manuela Fernandes Rodrigues

E-Mail Website1 Website2
Guest Editor
Department of Mathematics & CIDMA, University of Aveiro, 3810-193 Aveiro, Portugal
Interests: fractional calculus; mathematical modelling; integral equations; integral transforms; special functions; partial differential equations
Dr. Nelson Felipe Loureiro Vieira

E-Mail Website
Guest Editor
Department of Mathematics & CIDMA, University of Aveiro, 3810-193 Aveiro, Portugal
Interests: fractional calculus; fractional Clifford analysis; linear and non-linear fractional ODEs and fractional PDEs; fractional boundary value problems; neural networks; deep learning

Special Issue Information

Dear Colleagues,

Recently, the study of general fractional differential operators has attracted the interest of the fractional calculus research community. In the literature, we can find several proposals for new definitions that categorise fractional calculus into general classes to unify different integro-differential operators. Examples of such classes are the ψ-fractional calculus with respect to a given function ψ, the weighted ψ-fractional calculus, and fractional calculus with general analytic kernels and Sonine kernels. This variety of classes is justified by the need for operators with different structures to successfully model a large number of processes and phenomena that exist in the real world.

For the study of fractional differential equations involving these general fractional derivatives, there is a need to develop new mathematical tools in the areas of integral transforms, special functions and their properties, generalised functions, and numerical methods, just to mention a few. This Special Issue intends to contribute to the development and deepening of these topics within the scope of generalised fractional calculus.

We invite researchers to submit their original work, as well as review articles that discuss recent developments, applications, and connections with other fields of science.

Topics include (but are not limited to):

  • Mathematical theory of generalised fractional calculus.
  • Integral transform methods.
  • Special functions and their properties.
  • Inequalities, maximum principles, and stability results.
  • Initial and boundary value problems.
  • Numerical analysis and algorithms.
  • Fixed-point theory and applications in fractional calculus.
  • Fractional differential equations arising in physical models. In particular, anomalous diffusion processes involving generalised fractional derivatives.
  • Fractional stochastic differential equations.
  • Fractional networks.

Dr. Milton Ferreira
Prof. Dr. Maria Manuela Fernandes Rodrigues
Dr. Nelson Felipe Loureiro Vieira
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • generalised fractional calculus
  • fractional calculus with general analytic kernels
  • ψ-fractional derivatives
  • weighted fractional derivatives
  • integral transforms
  • special functions
  • fractional inequalities
  • fractional ODE and PDE
  • initial and Boundary value problems

Published Papers (8 papers)

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Research

13 pages, 331 KiB  
Article
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 1 | Viewed by 931
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  
Article
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 866
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  
Article
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 1104
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  
Article
G-Fractional Diffusion on Bounded Domains in Rd
by Luca Angelani and Roberto Garra
Fractal Fract. 2023, 7(3), 235; https://doi.org/10.3390/fractalfract7030235 - 07 Mar 2023
Cited by 1 | Viewed by 733
Abstract
In this paper, we study g-fractional diffusion on bounded domains in Rd 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 Rd 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  
Article
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 3 | Viewed by 825
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  
Article
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 2 | Viewed by 925
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  
Article
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 12 | Viewed by 1350
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  
Article
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 - 09 Nov 2022
Cited by 11 | Viewed by 1258
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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