Editor's Choice Articles

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.

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Article
Unified Scale Theorem: A Mathematical Formulation of Scale in the Frame of Earth Observation Image Classification
Fractal Fract. 2021, 5(3), 127; https://doi.org/10.3390/fractalfract5030127 - 17 Sep 2021
Cited by 2 | Viewed by 777
Abstract
In this research, the geographic, observational, functional, and cartographic scale is unified into a single mathematical formulation for the purposes of earth observation image classification. Fractal analysis is used to define functional scales, which then are linked to the other concepts of scale [...] Read more.
In this research, the geographic, observational, functional, and cartographic scale is unified into a single mathematical formulation for the purposes of earth observation image classification. Fractal analysis is used to define functional scales, which then are linked to the other concepts of scale using common equations and conditions. The proposed formulation is called Unified Scale Theorem (UST), and was assessed with Sentinel-2 image covering a variety of land uses from the broad area of Thessaloniki, Greece. Provided as an interactive excel spreadsheet, UST promotes objectivity, rapidity, and accuracy, thus facilitating optimal scale selection for image classification purposes. Full article
(This article belongs to the Special Issue Fractals in Geosciences: Theory and Applications)
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Article
Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach
Fractal Fract. 2021, 5(3), 124; https://doi.org/10.3390/fractalfract5030124 - 15 Sep 2021
Cited by 15 | Viewed by 993
Abstract
It is well established fact that the functional effects, such as relaxation and retardation of materials, can be measured for magnetized permeability based on relative increase or decrease during magnetization. In this context, a mathematical model is formulated based on slippage and non-slippage [...] Read more.
It is well established fact that the functional effects, such as relaxation and retardation of materials, can be measured for magnetized permeability based on relative increase or decrease during magnetization. In this context, a mathematical model is formulated based on slippage and non-slippage assumptions for Oldroyd-B fluid with magnetized permeability. An innovative definition of Caputo-Fabrizio time fractional derivative is implemented to hypothesize the constitutive energy and momentum equations. The exact solutions of presented problem, are determined by using mathematical techniques, namely Laplace transform with slipping boundary conditions have been invoked to tackle governing equations of velocity and temperature. The Nusselt number and limiting solutions have also been persuaded to estimate the heat emission rate through physical interpretation. In order to provide the validation of the problem, the absence of retardation time parameter led the investigated solutions with good agreement in literature. Additionally, comprehensively scrutinize the dynamics of the considered problem with parametric analysis is accomplished, the graphical illustration is depicted for slipping and non-slipping solutions for temperature and velocity. A comparative studies between fractional and non-fractional models describes that the fractional model elucidate the memory effects more efficiently. Full article
(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application)
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Article
On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives
Fractal Fract. 2021, 5(3), 117; https://doi.org/10.3390/fractalfract5030117 - 10 Sep 2021
Cited by 8 | Viewed by 608
Abstract
In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator [...] Read more.
In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work. Full article
Article
Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials
Fractal Fract. 2021, 5(3), 103; https://doi.org/10.3390/fractalfract5030103 - 25 Aug 2021
Cited by 6 | Viewed by 764
Abstract
In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives [...] Read more.
In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost. Full article
(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)
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Article
A Mathematical Study of a Coronavirus Model with the Caputo Fractional-Order Derivative
Fractal Fract. 2021, 5(3), 87; https://doi.org/10.3390/fractalfract5030087 - 03 Aug 2021
Cited by 6 | Viewed by 896
Abstract
In this work, we introduce a minimal model for the current pandemic. It incorporates the basic compartments: exposed, and both symptomatic and asymptomatic infected. The dynamical system is formulated by means of fractional operators. The model equilibria are analyzed. The theoretical results indicate [...] Read more.
In this work, we introduce a minimal model for the current pandemic. It incorporates the basic compartments: exposed, and both symptomatic and asymptomatic infected. The dynamical system is formulated by means of fractional operators. The model equilibria are analyzed. The theoretical results indicate that their stability behavior is the same as for the corresponding system formulated via standard derivatives. This suggests that, at least in this case for the model presented here, the memory effects contained in the fractional operators apparently do not seem to play a relevant role. The numerical simulations instead reveal that the order of the fractional derivative has a definite influence on both the equilibrium population levels and the speed at which they are attained. Full article
(This article belongs to the Special Issue Fractal Functions and Applications)
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Article
Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia
Fractal Fract. 2021, 5(3), 67; https://doi.org/10.3390/fractalfract5030067 - 12 Jul 2021
Cited by 3 | Viewed by 795
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 [...] Read more.
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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Article
An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions
Fractal Fract. 2021, 5(3), 63; https://doi.org/10.3390/fractalfract5030063 - 30 Jun 2021
Cited by 1 | Viewed by 710
Abstract
An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction [...] Read more.
An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
Article
Cole-Impedance Model Representations of Right-Side Segmental Arm, Leg, and Full-Body Bioimpedances of Healthy Adults: Comparison of Fractional-Order
Fractal Fract. 2021, 5(1), 13; https://doi.org/10.3390/fractalfract5010013 - 28 Jan 2021
Cited by 4 | Viewed by 1385
Abstract
The passive electrical properties of a biological tissue, referred to as the tissue bioimpedance, are related to the underlying tissue physiology. These measurements are often well-represented by a fractional-order equivalent circuit model, referred to as the Cole-impedance model. Objective: Identify if there are [...] Read more.
The passive electrical properties of a biological tissue, referred to as the tissue bioimpedance, are related to the underlying tissue physiology. These measurements are often well-represented by a fractional-order equivalent circuit model, referred to as the Cole-impedance model. Objective: Identify if there are differences in the fractional-order (α) of the Cole-impedance parameters that represent the segmental right-body, right-arm, and right-leg of adult participants. Hypothesis: Cole-impedance model parameters often associated with tissue geometry and fluid (R, R1, C) will be different between body segments, but parameters often associated with tissue type (α) will not show any statistical differences. Approach: A secondary analysis was applied to a dataset collected for an agreement study between bioimpedance spectroscopy devices and dual-energy X-ray absoptiometry, identifying the Cole-model parameters of the right-side body segments of N=174 participants using a particle swarm optimization approach. Statistical testing was applied to the different groups of Cole-model parameters to evaluate group differences and correlations of parameters with tissue features. Results: All Cole-impedance model parameters showed statistically significant differences between body segments. Significance: The physiological or geometric features of biological tissues that are linked with the fractional-order (α) of data represented by the Cole-impedance model requires further study to elucidate. Full article
(This article belongs to the Special Issue Fractional-Order Circuits and Systems)
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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 1423
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)
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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 2317
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)
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Article
Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation
Fractal Fract. 2020, 4(3), 41; https://doi.org/10.3390/fractalfract4030041 - 16 Aug 2020
Cited by 47 | Viewed by 1929
Abstract
This manuscript focuses on the application of the (m+1/G)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are [...] Read more.
This manuscript focuses on the application of the (m+1/G)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are derived. Various figures such as 3D and 2D surfaces with the selecting the suitable of parameter values are plotted. Full article
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Article
Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate
Fractal Fract. 2020, 4(3), 35; https://doi.org/10.3390/fractalfract4030035 - 16 Jul 2020
Cited by 68 | Viewed by 2861
Abstract
In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral [...] Read more.
In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense. Full article
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Article
Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation
Fractal Fract. 2020, 4(3), 32; https://doi.org/10.3390/fractalfract4030032 - 08 Jul 2020
Cited by 6 | Viewed by 989
Abstract
The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized [...] Read more.
The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag–Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In addition, two different types of subordination principles are formulated for the two regimes. The Bernstein functions technique is extensively employed in the theoretical proofs. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)
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Article
Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion
Fractal Fract. 2020, 4(2), 28; https://doi.org/10.3390/fractalfract4020028 - 22 Jun 2020
Cited by 9 | Viewed by 1131
Abstract
Nowadays, the stochastic resetting process is an attractive research topic in stochastic process. At the same time, a series of researches on stochastic diffusion in complex structures introduced ways to understand the anomalous diffusion in complex systems. In this work, we propose a [...] Read more.
Nowadays, the stochastic resetting process is an attractive research topic in stochastic process. At the same time, a series of researches on stochastic diffusion in complex structures introduced ways to understand the anomalous diffusion in complex systems. In this work, we propose a non-static stochastic resetting model in the context of comb structure that consists of a structure formed by backbone in x axis and branches in y axis. Then, we find the exact analytical solutions for marginal distribution concerning x and y axis. Moreover, we show the time evolution behavior to mean square displacements (MSD) in both directions. As a consequence, the model revels that until the system reaches the equilibrium, i.e., constant MSD, there is a Brownian diffusion in y direction, i.e., ( Δ y ) 2 t , and a crossover between sub and ballistic diffusion behaviors in x direction, i.e., ( Δ x ) 2 t 1 2 and ( Δ x ) 2 t 2 respectively. For static stochastic resetting, the ballistic regime vanishes. Also, we consider the idealized model according to the memory kernels to investigate the exponential and tempered power-law memory kernels effects on diffusive behaviors. In this way, we expose a rich class of anomalous diffusion process with crossovers among them. The proposal and the techniques applied in this work are useful to describe random walkers with non-static stochastic resetting on comb structure. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)
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Article
Power Law Type Long Memory Behaviors Modeled with Distributed Time Delay Systems
Fractal Fract. 2020, 4(1), 1; https://doi.org/10.3390/fractalfract4010001 - 27 Dec 2019
Cited by 12 | Viewed by 1251
Abstract
This paper studies a class of distributed time delay systems that exhibit power law type long memory behaviors. Such dynamical behaviors are present in multiple domains and it is therefore essential to have tools to model them. The literature is full of examples [...] Read more.
This paper studies a class of distributed time delay systems that exhibit power law type long memory behaviors. Such dynamical behaviors are present in multiple domains and it is therefore essential to have tools to model them. The literature is full of examples in which these behaviors are modeled by means of fractional models. However, several limitations of fractional models have recently been reported and other solutions must be found. In the literature, the analysis of distributed delay models and integro-differential equations in general is older than that of fractional models. In this paper, it is shown that particular delay distributions and conditions on the model coefficients make it possible to obtain power laws. The class of systems considered is then used to model the input-output behavior of a lithium-ion cell. Full article
(This article belongs to the Special Issue 2019 Selected Papers from Fractal Fract’s Editorial Board Members)
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Article
Fractal Logistic Equation
by and
Fractal Fract. 2019, 3(3), 41; https://doi.org/10.3390/fractalfract3030041 - 11 Jul 2019
Cited by 24 | Viewed by 2244
Abstract
In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding [...] Read more.
In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding approximate analytical solutions. Fractal differential equations are solved by using the fractal Euler method. Furthermore, fractal logistic equations and functions are given, which are useful in modeling growth of elements in sciences including biology and economics. Full article
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Article
A Novel Method for Solutions of Fourth-Order Fractional Boundary Value Problems
Fractal Fract. 2019, 3(2), 33; https://doi.org/10.3390/fractalfract3020033 - 18 Jun 2019
Cited by 34 | Viewed by 1756
Abstract
In this paper, we find the solutions of fourth order fractional boundary value problems by using the reproducing kernel Hilbert space method. Firstly, the reproducing kernel Hilbert space method is introduced and then the method is applied to this kind problems. The experiments [...] Read more.
In this paper, we find the solutions of fourth order fractional boundary value problems by using the reproducing kernel Hilbert space method. Firstly, the reproducing kernel Hilbert space method is introduced and then the method is applied to this kind problems. The experiments are discussed and the approximate solutions are obtained to be more correct compared to the other obtained results in the literature. Full article
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Article
Random Variables and Stable Distributions on Fractal Cantor Sets
Fractal Fract. 2019, 3(2), 31; https://doi.org/10.3390/fractalfract3020031 - 11 Jun 2019
Cited by 20 | Viewed by 1920
Abstract
In this paper, we introduce the concept of fractal random variables and their related distribution functions and statistical properties. Fractal calculus is a generalisation of standard calculus which includes function with fractal support. Here we combine this emerging field of study with probability [...] Read more.
In this paper, we introduce the concept of fractal random variables and their related distribution functions and statistical properties. Fractal calculus is a generalisation of standard calculus which includes function with fractal support. Here we combine this emerging field of study with probability theory, defining concepts such as Shannon entropy on fractal thin Cantor-like sets. Stable distributions on fractal sets are suggested and related physical models are presented. Our work is illustrated with graphs for clarity of the results. Full article
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Article
Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces
Fractal Fract. 2019, 3(2), 27; https://doi.org/10.3390/fractalfract3020027 - 16 May 2019
Cited by 39 | Viewed by 2362
Abstract
We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular [...] Read more.
We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results. Full article
Article
Integral Representations and Algebraic Decompositions of the Fox-Wright Type of Special Functions
Fractal Fract. 2019, 3(1), 4; https://doi.org/10.3390/fractalfract3010004 - 25 Jan 2019
Cited by 4 | Viewed by 1465
Abstract
The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under [...] Read more.
The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized Erdélyi-Kober fractional differential and integral operators is demonstrated and discussed. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering 2018)
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Article
Regularized Integral Representations of the Reciprocal Gamma Function
Fractal Fract. 2019, 3(1), 1; https://doi.org/10.3390/fractalfract3010001 - 12 Jan 2019
Cited by 10 | Viewed by 1744
Abstract
This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For [...] Read more.
This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering 2018)
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Review

Review
Fractal Antennas: An Historical Perspective
Fractal Fract. 2020, 4(1), 3; https://doi.org/10.3390/fractalfract4010003 - 19 Jan 2020
Cited by 28 | Viewed by 3327
Abstract
Fractal geometry has been proven to be useful in several disciplines. In the field of antenna engineering, fractal geometry is useful to design small and multiband antenna and arrays, and high-directive elements. A historic overview of the most significant fractal mathematic pioneers is [...] Read more.
Fractal geometry has been proven to be useful in several disciplines. In the field of antenna engineering, fractal geometry is useful to design small and multiband antenna and arrays, and high-directive elements. A historic overview of the most significant fractal mathematic pioneers is presented, at the same time showing how the fractal patterns inspired engineers to design antennas. Full article
(This article belongs to the Special Issue Fractals in Antenna and Microwave Engineering 2019)
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