Due to planned maintenance work on our platforms, there might be short service disruptions on Saturday, December 3rd, between 15:00 and 16:00 (CET).

Special Issue "2020 Selected Papers from Fractal Fract’s Editorial Board Members"

A special issue of Fractal and Fractional (ISSN 2504-3110).

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 21018

Special Issue Editor

Prof. Dr. Carlo Cattani
E-Mail Website1 Website2
Guest Editor
Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: numerical and computational methods; mathematical physics; nonlinear systems; Artificial Intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

I am pleased to announce a new Special Issue that is quite different from our typical ones, which mainly focus on either selected areas of research or special techniques. Being creative in many ways, with this Special Issue, Fractal Fract is compiling a collection of papers submitted exclusively by its Editorial Board Members (EBMs) covering different areas of fractals and fractional calculus in 2020. The main idea behind this Special Issue is to turn the tables and allow our readers to be the judges of our board members. With this Special Issue, we also want to celebrate our acceptance into ESCI (WoS), which we earned due to years of hard work, dedication, and commitment from our EBMs.

Our new Special Issue can be also viewed as a way of introducing Fractal Fract’s EBMs to top-notch researchers, so they will consider our journal a first-class platform for exchanging their scientific research.

Prof. Dr. Carlo Cattani
Guest Editor

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 1800 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.

Published Papers (11 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

Jump to: Other

Article
Analysis of Hilfer Fractional Integro-Differential Equations with Almost Sectorial Operators
Fractal Fract. 2021, 5(1), 22; https://doi.org/10.3390/fractalfract5010022 - 08 Mar 2021
Cited by 9 | Viewed by 1334
Abstract
In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing [...] Read more.
In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
Article
Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm
Fractal Fract. 2021, 5(1), 8; https://doi.org/10.3390/fractalfract5010008 - 13 Jan 2021
Cited by 4 | Viewed by 1463
Abstract
This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, [...] Read more.
This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
Show Figures

Figure 1

Article
The Impact of Anomalous Diffusion on Action Potentials in Myelinated Neurons
Fractal Fract. 2021, 5(1), 4; https://doi.org/10.3390/fractalfract5010004 - 05 Jan 2021
Cited by 1 | Viewed by 1579
Abstract
Action potentials in myelinated neurons happen only at specialized locations of the axons known as the nodes of Ranvier. The shapes, timings, and propagation speeds of these action potentials are controlled by biochemical interactions among neurons, glial cells, and the extracellular space. The [...] Read more.
Action potentials in myelinated neurons happen only at specialized locations of the axons known as the nodes of Ranvier. The shapes, timings, and propagation speeds of these action potentials are controlled by biochemical interactions among neurons, glial cells, and the extracellular space. The complexity of brain structure and processes suggests that anomalous diffusion could affect the propagation of action potentials. In this paper, a spatio-temporal fractional cable equation for action potentials propagation in myelinated neurons is proposed. The impact of the ionic anomalous diffusion on the distribution of the membrane potential is investigated using numerical simulations. The results show spatially narrower action potentials at the nodes of Ranvier when using spatial derivatives of the fractional order only and delayed or lack of action potentials when adding a temporal derivative of the fractional order. These findings could reveal the pathological patterns of brain diseases such as epilepsy, multiple sclerosis, and Alzheimer’s disease, which have become more prevalent in the latest years. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
Show Figures

Figure 1

Article
Boundary Value Problem for Fractional Order Generalized Hilfer-Type Fractional Derivative with Non-Instantaneous Impulses
Fractal Fract. 2021, 5(1), 1; https://doi.org/10.3390/fractalfract5010001 - 22 Dec 2020
Cited by 8 | Viewed by 1759
Abstract
This manuscript is devoted to proving some results concerning the existence of solutions to a class of boundary value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivatives. The results are based on Banach’s contraction principle and [...] Read more.
This manuscript is devoted to proving some results concerning the existence of solutions to a class of boundary value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivatives. The results are based on Banach’s contraction principle and Krasnosel’skii’s fixed point theorem. To illustrate the results, an example is provided. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
Article
An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations
Fractal Fract. 2020, 4(4), 57; https://doi.org/10.3390/fractalfract4040057 - 14 Dec 2020
Cited by 4 | Viewed by 1933
Abstract
A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction–diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a [...] Read more.
A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction–diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a new, fractional version of the Alternating Direction Implicit (ADI) method, where the source term is balanced, in that its effect is split in the three space directions, and it may be relevant, especially in the case of anisotropy. The method is unconditionally stable, second-order in space, and third-order in time. A strategy is devised in order to improve its speed of convergence by means of an extrapolation method that is coupled to the PageRank algorithm. Some numerical examples are given. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
Show Figures

Figure 1

Article
Fractional Diffusion to a Cantor Set in 2D
Fractal Fract. 2020, 4(4), 52; https://doi.org/10.3390/fractalfract4040052 - 07 Nov 2020
Cited by 2 | Viewed by 1801
Abstract
A random walk on a two dimensional square in R2 space with a hidden absorbing fractal set Fμ is considered. This search-like problem is treated in the framework of a diffusion–reaction equation, when an absorbing term is included inside a Fokker–Planck [...] Read more.
A random walk on a two dimensional square in R2 space with a hidden absorbing fractal set Fμ is considered. This search-like problem is treated in the framework of a diffusion–reaction equation, when an absorbing term is included inside a Fokker–Planck equation as a reaction term. This macroscopic approach for the 2D transport in the R2 space corresponds to the comb geometry, when the random walk consists of 1D movements in the x and y directions, respectively, as a direct-Cartesian product of the 1D movements. The main value in task is the first arrival time distribution (FATD) to sink points of the fractal set, where travelling particles are absorbed. Analytical expression for the FATD is obtained in the subdiffusive regime for both the fractal set of sinks and for a single sink. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
Show Figures

Figure 1

Article
Adaptive Neural Network Sliding Mode Control for Nonlinear Singular Fractional Order Systems with Mismatched Uncertainties
Fractal Fract. 2020, 4(4), 50; https://doi.org/10.3390/fractalfract4040050 - 22 Oct 2020
Cited by 18 | Viewed by 1853
Abstract
This paper focuses on the sliding mode control (SMC) problem for a class of uncertain singular fractional order systems (SFOSs). The uncertainties occur in both state and derivative matrices. A radial basis function (RBF) neural network strategy was utilized to estimate the nonlinear [...] Read more.
This paper focuses on the sliding mode control (SMC) problem for a class of uncertain singular fractional order systems (SFOSs). The uncertainties occur in both state and derivative matrices. A radial basis function (RBF) neural network strategy was utilized to estimate the nonlinear terms of SFOSs. Firstly, by expanding the dimension of the SFOS, a novel sliding surface was constructed. A necessary and sufficient condition was given to ensure the admissibility of the SFOS while the system state moves on the sliding surface. The obtained results are linear matrix inequalities (LMIs), which are more general than the existing research. Then, the adaptive control law based on the RBF neural network was organized to guarantee that the SFOS reaches the sliding surface in a finite time. Finally, a simulation example is proposed to verify the validity of the designed procedures. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
Show Figures

Figure 1

Article
Numerical Simulation of the Fractal-Fractional Ebola Virus
Fractal Fract. 2020, 4(4), 49; https://doi.org/10.3390/fractalfract4040049 - 29 Sep 2020
Cited by 20 | Viewed by 2975
Abstract
In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function [...] Read more.
In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function by using the concepts of the fractal differentiation and fractional differentiation. These operators have two parameters: The first parameter ρ is considered as the fractal dimension and the second parameter k is the fractional order. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions. In the case of ρ=k=1, all of the numerical solutions based on the power kernel, the exponential kernel and the generalized Mittag-Leffler kernel are found to be close to each other and, therefore, one of the kernels is compared with such numerical methods as the finite difference methods. This has led to an excellent agreement. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values of ρ and k. All calculations in this work are accomplished by using the Mathematica package. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
Show Figures

Figure 1

Article
Integral Representation of Fractional Derivative of Delta Function
by
Fractal Fract. 2020, 4(3), 47; https://doi.org/10.3390/fractalfract4030047 - 20 Sep 2020
Cited by 6 | Viewed by 1574
Abstract
Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the [...] Read more.
Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the delta function. Moreover, we provide its application in representing the fractional Gaussian noise. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
Show Figures

Figure 1

Article
Functional Differential Equations Involving the ψ-Caputo Fractional Derivative
Fractal Fract. 2020, 4(2), 29; https://doi.org/10.3390/fractalfract4020029 - 23 Jun 2020
Cited by 18 | Viewed by 1718
Abstract
This paper is devoted to the study of existence and uniqueness of solutions for fractional functional differential equations, whose derivative operator depends on an arbitrary function. The introduction of such function allows generalization of some known results, and others can be also obtained. [...] Read more.
This paper is devoted to the study of existence and uniqueness of solutions for fractional functional differential equations, whose derivative operator depends on an arbitrary function. The introduction of such function allows generalization of some known results, and others can be also obtained. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

Other

Jump to: Research

Viewpoint
Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive?
Fractal Fract. 2020, 4(3), 40; https://doi.org/10.3390/fractalfract4030040 - 11 Aug 2020
Cited by 17 | Viewed by 2280
Abstract
In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are [...] Read more.
In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
Back to TopTop