2021 Feature Papers by Fractal Fract's Editorial Board Members

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (26 December 2021) | Viewed by 26191

Special Issue Editor

Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; 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 2021. 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 first impact factor of 3.313, 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 as 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 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.

Published Papers (14 papers)

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Research

25 pages, 4789 KiB  
Article
Further Generalization and Approximation of Fractional-Order Filters and Their Inverse Functions of the Second-Order Limiting Form
Fractal Fract. 2022, 6(4), 209; https://doi.org/10.3390/fractalfract6040209 - 08 Apr 2022
Cited by 8 | Viewed by 1699
Abstract
This paper proposes a further generalization of the fractional-order filters whose limiting form is that of the second-order filter. This new filter class can also be regarded as a superset of the recently reported power-law filters. An optimal approach incorporating constraints that restricts [...] Read more.
This paper proposes a further generalization of the fractional-order filters whose limiting form is that of the second-order filter. This new filter class can also be regarded as a superset of the recently reported power-law filters. An optimal approach incorporating constraints that restricts the real part of the roots of the numerator and denominator polynomials of the proposed rational approximant to negative values is formulated. Consequently, stable inverse filter characteristics can also be achieved using the suggested method. Accuracy of the proposed low-pass, high-pass, band-pass, and band-stop filters for various combinations of design parameters is evaluated using the absolute relative magnitude/phase error metrics. Current feedback operational amplifier-based circuit simulations validate the efficacy of the four types of designed filters and their inverse functions. Experimental results for the frequency and time-domain performances of the proposed fractional-order band-pass filter and its inverse counterpart are also presented. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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18 pages, 1393 KiB  
Article
Electronically Controlled Power-Law Filters Realizations
Fractal Fract. 2022, 6(2), 111; https://doi.org/10.3390/fractalfract6020111 - 14 Feb 2022
Cited by 8 | Viewed by 1640
Abstract
A generalized structure that is capable of implementing power-law filters derived from 1st and 2nd-order mother filter functions is presented in this work. This is achieved thanks to the employment of Operational Transconductance Amplifiers (OTAs) as active elements, because of the electronic tuning [...] Read more.
A generalized structure that is capable of implementing power-law filters derived from 1st and 2nd-order mother filter functions is presented in this work. This is achieved thanks to the employment of Operational Transconductance Amplifiers (OTAs) as active elements, because of the electronic tuning capability of their transconductance parameter. Appropriate design examples are provided and the performance of the introduced structure is evaluated through simulation results using the Cadence Integrated Circuits (IC) design suite and Metal Oxide Semiconductor (MOS) transistors models available from the Austria Mikro Systeme (AMS) 0.35 μm Complementary Metal Oxide Semiconductor (CMOS) process. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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19 pages, 360 KiB  
Article
A New Approach to Fractional Kinetic Evolutions
Fractal Fract. 2022, 6(2), 49; https://doi.org/10.3390/fractalfract6020049 - 18 Jan 2022
Cited by 8 | Viewed by 1597
Abstract
Kinetic equations describe the limiting deterministic evolution of properly scaled systems of interacting particles. A rather universal extension of the classical evolutions, that aims to take into account the effects of memory, suggests the generalization of these evolutions obtained by changing the standard [...] Read more.
Kinetic equations describe the limiting deterministic evolution of properly scaled systems of interacting particles. A rather universal extension of the classical evolutions, that aims to take into account the effects of memory, suggests the generalization of these evolutions obtained by changing the standard time derivative with a fractional one. In the present paper, extending some previous notes of the authors related to models with a finite state space, we develop systematically the idea of CTRW (continuous time random walk) modelling of the Markovian evolution of interacting particle systems, which leads to a more nontrivial class of fractional kinetic measure-valued evolutions, with the mixed fractional order derivatives varying with the change of the state of the particle system, and with variational derivatives with respect to the measure variable. We rigorously justify the limiting procedure, prove the well-posedness of the new equations, and present a probabilistic formula for their solutions. As the most basic examples we present the fractional versions of the Smoluchovski coagulation and Boltzmann collision models. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
27 pages, 394 KiB  
Article
A Langevin-Type q-Variant System of Nonlinear Fractional Integro-Difference Equations with Nonlocal Boundary Conditions
Fractal Fract. 2022, 6(1), 45; https://doi.org/10.3390/fractalfract6010045 - 14 Jan 2022
Cited by 5 | Viewed by 1689
Abstract
We introduce a new class of boundary value problems consisting of a q-variant system of Langevin-type nonlinear coupled fractional integro-difference equations and nonlocal multipoint boundary conditions. We make use of standard fixed-point theorems to derive the existence and uniqueness results for the [...] Read more.
We introduce a new class of boundary value problems consisting of a q-variant system of Langevin-type nonlinear coupled fractional integro-difference equations and nonlocal multipoint boundary conditions. We make use of standard fixed-point theorems to derive the existence and uniqueness results for the given problem. Illustrative examples for the obtained results are also presented. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
23 pages, 10450 KiB  
Article
Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations
Fractal Fract. 2022, 6(1), 39; https://doi.org/10.3390/fractalfract6010039 - 12 Jan 2022
Viewed by 3302
Abstract
Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and [...] Read more.
Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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19 pages, 518 KiB  
Article
Shifted Fractional-Order Jacobi Collocation Method for Solving Variable-Order Fractional Integro-Differential Equation with Weakly Singular Kernel
Fractal Fract. 2022, 6(1), 19; https://doi.org/10.3390/fractalfract6010019 - 30 Dec 2021
Cited by 7 | Viewed by 1579
Abstract
We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving [...] Read more.
We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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17 pages, 480 KiB  
Article
Lyapunov Approach for Almost Periodicity in Impulsive Gene Regulatory Networks of Fractional Order with Time-Varying Delays
Fractal Fract. 2021, 5(4), 268; https://doi.org/10.3390/fractalfract5040268 - 09 Dec 2021
Cited by 8 | Viewed by 2005
Abstract
This paper investigates a class of fractional-order delayed impulsive gene regulatory networks (GRNs). The proposed model is an extension of some existing integer-order GRNs using fractional derivatives of Caputo type. The existence and uniqueness of an almost periodic state of the model are [...] Read more.
This paper investigates a class of fractional-order delayed impulsive gene regulatory networks (GRNs). The proposed model is an extension of some existing integer-order GRNs using fractional derivatives of Caputo type. The existence and uniqueness of an almost periodic state of the model are investigated and new criteria are established by the Lyapunov functions approach. The effects of time-varying delays and impulsive perturbations at fixed times on the almost periodicity are considered. In addition, sufficient conditions for the global Mittag–Leffler stability of the almost periodic solutions are proposed. To justify our findings a numerical example is also presented. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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6 pages, 263 KiB  
Article
Fox H-Functions in Self-Consistent Description of a Free-Electron Laser
Fractal Fract. 2021, 5(4), 263; https://doi.org/10.3390/fractalfract5040263 - 07 Dec 2021
Cited by 3 | Viewed by 1857
Abstract
A fractional calculus concept is considered in the framework of a Volterra type integro-differential equation, which is employed for the self-consistent description of the high-gain free-electron laser (FEL). It is shown that the Fox H-function is the Laplace image of the kernel [...] Read more.
A fractional calculus concept is considered in the framework of a Volterra type integro-differential equation, which is employed for the self-consistent description of the high-gain free-electron laser (FEL). It is shown that the Fox H-function is the Laplace image of the kernel of the integro-differential equation, which is also known as a fractional FEL equation with Caputo–Fabrizio type fractional derivative. Asymptotic solutions of the equation are analyzed as well. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
13 pages, 24660 KiB  
Article
Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects
Fractal Fract. 2021, 5(4), 255; https://doi.org/10.3390/fractalfract5040255 - 04 Dec 2021
Cited by 2 | Viewed by 1810
Abstract
The scale dependence of the effective anti-plane shear modulus response in microstructures with statistical ergodicity and spatial wide-sense stationarity is investigated. In particular, Cauchy and Dagum autocorrelation functions which can decouple the fractal and the Hurst effects are used to describe the random [...] Read more.
The scale dependence of the effective anti-plane shear modulus response in microstructures with statistical ergodicity and spatial wide-sense stationarity is investigated. In particular, Cauchy and Dagum autocorrelation functions which can decouple the fractal and the Hurst effects are used to describe the random shear modulus fields. The resulting stochastic boundary value problems (BVPs) are set up in line with the Hill–Mandel condition of elastostatics for different sizes of statistical volume elements (SVEs). These BVPs are solved using a physics-based cellular automaton (CA) method that is applicable for anti-plane elasticity to study the scaling of SVEs towards a representative volume element (RVE). This progression from SVE to RVE is described through a scaling function, which is best approximated by the same form as the Cauchy and Dagum autocorrelation functions. The scaling function is obtained by fitting the scaling data from simulations conducted over a large number of random field realizations. The numerical simulation results show that the scaling function is strongly dependent on the fractal dimension D, the Hurst parameter H, and the mesoscale δ, and is weakly dependent on the autocorrelation function. Specifically, it is found that a larger D and a smaller H results in a higher rate of convergence towards an RVE with respect to δ. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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21 pages, 441 KiB  
Article
An Entropy Paradox Free Fractional Diffusion Equation
Fractal Fract. 2021, 5(4), 236; https://doi.org/10.3390/fractalfract5040236 - 21 Nov 2021
Cited by 3 | Viewed by 1098
Abstract
A new look at the fractional diffusion equation was done. Using the unified fractional derivative, a new formulation was proposed, and the equation was solved for three different order cases: neutral, dominant time, and dominant space. The solutions were expressed by generalizations of [...] Read more.
A new look at the fractional diffusion equation was done. Using the unified fractional derivative, a new formulation was proposed, and the equation was solved for three different order cases: neutral, dominant time, and dominant space. The solutions were expressed by generalizations of classic formulae used for the stable distributions. The entropy paradox problem was studied and clarified through the Rényi entropy: in the extreme wave regime the entropy is . In passing, Tsallis and Rényi entropies for stable distributions are introduced and exemplified. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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13 pages, 373 KiB  
Article
Applications of a Fixed Point Result for Solving Nonlinear Fractional and Integral Differential Equations
Fractal Fract. 2021, 5(4), 211; https://doi.org/10.3390/fractalfract5040211 - 11 Nov 2021
Cited by 13 | Viewed by 1570
Abstract
In this article, we apply one fixed point theorem in the setting of b-metric-like spaces to prove the existence of solutions for one type of Caputo fractional differential equation as well as the existence of solutions for one integral equation created in [...] Read more.
In this article, we apply one fixed point theorem in the setting of b-metric-like spaces to prove the existence of solutions for one type of Caputo fractional differential equation as well as the existence of solutions for one integral equation created in mechanical engineering. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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23 pages, 2848 KiB  
Article
On the Design of Power Law Filters and Their Inverse Counterparts
Fractal Fract. 2021, 5(4), 197; https://doi.org/10.3390/fractalfract5040197 - 04 Nov 2021
Cited by 12 | Viewed by 2037
Abstract
This paper presents the optimal modeling of Power Law Filters (PLFs) with the low-pass (LP), high-pass (HP), band-pass (BP), and band-stop (BS) responses by means of rational approximants. The optimization is performed for three different objective functions and second-order filter mother functions. The [...] Read more.
This paper presents the optimal modeling of Power Law Filters (PLFs) with the low-pass (LP), high-pass (HP), band-pass (BP), and band-stop (BS) responses by means of rational approximants. The optimization is performed for three different objective functions and second-order filter mother functions. The formulated design constraints help avoid placement of the zeros and poles on the right-half s-plane, thus, yielding stable PLF and inverse PLF (IPLF) models. The performances of the approximants exhibiting the fractional-step magnitude and phase responses are evaluated using various statistical indices. At the cost of higher computational complexity, the proposed approach achieved improved accuracy with guaranteed stability when compared to the published literature. The four types of optimal PLFs and IPLFs with an exponent α of 0.5 are implemented using the follow-the-leader feedback topology employing AD844AN current feedback operational amplifiers. The experimental results demonstrate that the Total Harmonic Distortion achieved for all the practical PLF and IPLF circuits was equal or lower than 0.21%, whereas the Spurious-Free Dynamic Range also exceeded 57.23 and 54.72 dBc, respectively. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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27 pages, 964 KiB  
Article
Generalized Bessel Quasilinearization Technique Applied to Bratu and Lane–Emden-Type Equations of Arbitrary Order
Fractal Fract. 2021, 5(4), 179; https://doi.org/10.3390/fractalfract5040179 - 22 Oct 2021
Cited by 17 | Viewed by 1303
Abstract
The ultimate goal of this study is to develop a numerically effective approximation technique to acquire numerical solutions of the integer and fractional-order Bratu and the singular Lane–Emden-type problems especially with exponential nonlinearity. Both the initial and boundary conditions were considered and the [...] Read more.
The ultimate goal of this study is to develop a numerically effective approximation technique to acquire numerical solutions of the integer and fractional-order Bratu and the singular Lane–Emden-type problems especially with exponential nonlinearity. Both the initial and boundary conditions were considered and the fractional derivative being considered in the Liouville–Caputo sense. In the direct approach, the generalized Bessel matrix method based on collocation points was utilized to convert the model problems into a nonlinear fundamental matrix equation. Then, the technique of quasilinearization was employed to tackle the nonlinearity that arose in our considered model problems. Consequently, the quasilinearization method was utilized to transform the original nonlinear problems into a sequence of linear equations, while the generalized Bessel collocation scheme was employed to solve the resulting linear equations iteratively. In particular, to convert the Neumann initial or boundary condition into a matrix form, a fast algorithm for computing the derivative of the basis functions is presented. The error analysis of the quasilinear approach is also discussed. The effectiveness of the present linearized approach is illustrated through several simulations with some test examples. Comparisons with existing well-known schemes revealed that the presented technique is an easy-to-implement method while being very effective and convenient for the nonlinear Bratu and Lane–Emden equations. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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14 pages, 424 KiB  
Article
Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule
Fractal Fract. 2021, 5(3), 111; https://doi.org/10.3390/fractalfract5030111 - 07 Sep 2021
Cited by 3 | Viewed by 1807
Abstract
This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional [...] Read more.
This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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