Fractal and Fractional

3 pages, 193 KiB

Open AccessCorrection

Correction: Gürel Yılmaz, Ö., et al. On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative. Fractal Fract. 2020, 4, 48

by Övgü Gürel Yılmaz, Rabia Aktaş and Fatma Taşdelen

Fractal Fract. 2020, 4(4), 60; https://doi.org/10.3390/fractalfract4040060 - 21 Dec 2020

Viewed by 1972

Abstract

The authors wish to make the following corrections to this paper [...] Full article

11 pages, 456 KiB

Open AccessArticle

Fractional-Fractal Modeling of Filtration-Consolidation Processes in Saline Saturated Soils

by Vsevolod Bohaienko and Volodymyr Bulavatsky

Fractal Fract. 2020, 4(4), 59; https://doi.org/10.3390/fractalfract4040059 - 16 Dec 2020

Cited by 13 | Viewed by 1868

Abstract

To study the peculiarities of anomalous consolidation processes in saturated porous (soil) media in the conditions of salt transfer, we present a new mathematical model developed on the base of the fractional-fractal approach that allows considering temporal non-locality of transfer processes in media [...] Read more.

To study the peculiarities of anomalous consolidation processes in saturated porous (soil) media in the conditions of salt transfer, we present a new mathematical model developed on the base of the fractional-fractal approach that allows considering temporal non-locality of transfer processes in media of fractal structure. For the case of the finite thickness domain with permeable boundaries, a finite-difference technique for numerical solution of the corresponding one-dimensional non-linear boundary value problem is developed. The paper also presents a fractional-fractal model of a filtration-consolidation process in clay soils of fractal structure saturated with salt solutions. An analytical solution is found for the corresponding one-dimensional boundary value problem in the domain of finite thickness with permeable upper and impermeable lower boundaries. Full article

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11 pages, 994 KiB

Open AccessArticle

LMI Criteria for Admissibility and Robust Stabilization of Singular Fractional-Order Systems Possessing Poly-Topic Uncertainties

by Xuefeng Zhang and Jia Dong

Fractal Fract. 2020, 4(4), 58; https://doi.org/10.3390/fractalfract4040058 - 15 Dec 2020

Cited by 8 | Viewed by 1791

Abstract

The issue of robust admissibility and control for singular fractional-order systems (FOSs) with polytopic uncertainties is investigated in this paper. Firstly, a new method based on linear matrix inequalities (LMIs) is presented to solve the admissibility problems of uncertain linear systems. Then, a [...] Read more.

The issue of robust admissibility and control for singular fractional-order systems (FOSs) with polytopic uncertainties is investigated in this paper. Firstly, a new method based on linear matrix inequalities (LMIs) is presented to solve the admissibility problems of uncertain linear systems. Then, a solid criterion of robust admissibility and a corresponding state feedback controller are derived, which overcome the conservatism of the existing results. Finally, for the sake of demonstrating the validity of proposed results, some relevant examples are provided. Full article

(This article belongs to the Special Issue Fractional-Order Circuits and Systems)

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21 pages, 1479 KiB

Open AccessArticle

An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations

by Moreno Concezzi and Renato Spigler

Fractal Fract. 2020, 4(4), 57; https://doi.org/10.3390/fractalfract4040057 - 14 Dec 2020

Cited by 7 | Viewed by 3027

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)

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17 pages, 342 KiB

Open AccessArticle

Generalized Differentiability of Continuous Functions

by Dimiter Prodanov

Fractal Fract. 2020, 4(4), 56; https://doi.org/10.3390/fractalfract4040056 - 10 Dec 2020

Viewed by 1970

Abstract

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an [...] Read more.

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated. Full article

(This article belongs to the Special Issue New Aspects of Local Fractional Calculus)

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17 pages, 817 KiB

Open AccessArticle

Optimal Modelling of (1 + α) Order Butterworth Filter under the CFE Framework

by Shibendu Mahata, Rajib Kar and Durbadal Mandal

Fractal Fract. 2020, 4(4), 55; https://doi.org/10.3390/fractalfract4040055 - 3 Dec 2020

Cited by 9 | Viewed by 2946

Abstract

This paper presents the optimal rational approximation of (1+α) order Butterworth filter, where α ∊ (0,1) under the continued fraction expansion framework, by employing a new cost function. Two simple techniques based on the constrained optimization and the optimal pole-zero placements [...] Read more.

This paper presents the optimal rational approximation of (1+α) order Butterworth filter, where α ∊ (0,1) under the continued fraction expansion framework, by employing a new cost function. Two simple techniques based on the constrained optimization and the optimal pole-zero placements are proposed to model the magnitude-frequency response of the fractional-order lowpass Butterworth filter (FOLBF). The third-order FOLBF approximants achieve good agreement to the ideal characteristic for six decades of design bandwidth. Circuit realization using the current feedback operational amplifier is presented, and the modelling efficacy is validated in the OrCAD PSPICE platform. Full article

(This article belongs to the Special Issue Fractional-Order Circuits and Systems)

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15 pages, 4851 KiB

Open AccessArticle

Realization of Cole–Davidson Function-Based Impedance Models: Application on Plant Tissues

by Stavroula Kapoulea, Costas Psychalinos and Ahmed S. Elwakil

Fractal Fract. 2020, 4(4), 54; https://doi.org/10.3390/fractalfract4040054 - 30 Nov 2020

Cited by 10 | Viewed by 2597

Abstract

The Cole–Davidson function is an efficient tool for describing the tissue behavior, but the conventional methods of approximation are not applicable due the form of this function. In order to overcome this problem, a novel scheme for approximating the Cole–Davidson function, based on [...] Read more.

The Cole–Davidson function is an efficient tool for describing the tissue behavior, but the conventional methods of approximation are not applicable due the form of this function. In order to overcome this problem, a novel scheme for approximating the Cole–Davidson function, based on the utilization of a curve fitting procedure offered by the MATLAB software, is introduced in this work. The derived rational transfer function is implemented using the conventional Cauer and Foster RC networks. As an application example, the impedance model of the membrane of mesophyll cells is realized, with simulation results verifying the validity of the introduced procedure. Full article

(This article belongs to the Special Issue Fractional-Order Circuits and Systems)

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13 pages, 791 KiB

Open AccessArticle

A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems

by Mehmet Giyas Sakar and Onur Saldır

Fractal Fract. 2020, 4(4), 53; https://doi.org/10.3390/fractalfract4040053 - 24 Nov 2020

Cited by 6 | Viewed by 1855

Abstract

In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, [...] Read more.

In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to

α

and

β

are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)

12 pages, 333 KiB

Open AccessEditor’s ChoiceArticle

Fractional Diffusion to a Cantor Set in 2D

by Alexander Iomin and Trifce Sandev

Fractal Fract. 2020, 4(4), 52; https://doi.org/10.3390/fractalfract4040052 - 7 Nov 2020

Cited by 4 | Viewed by 2500

Abstract

A random walk on a two dimensional square in

R^{2}

space with a hidden absorbing fractal set

F_{μ}

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

R^{2}

space with a hidden absorbing fractal set

F_{μ}

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

R^{2}

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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29 pages, 554 KiB

Open AccessArticle

Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

by Thomas M. Michelitsch, Federico Polito and Alejandro P. Riascos

Fractal Fract. 2020, 4(4), 51; https://doi.org/10.3390/fractalfract4040051 - 31 Oct 2020

Cited by 12 | Viewed by 2165

Abstract

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we [...] Read more.

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs. Full article

(This article belongs to the Special Issue Fractional Behavior in Nature 2019)

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13 pages, 400 KiB

Open AccessArticle

Adaptive Neural Network Sliding Mode Control for Nonlinear Singular Fractional Order Systems with Mismatched Uncertainties

by Xuefeng Zhang and Wenkai Huang

Fractal Fract. 2020, 4(4), 50; https://doi.org/10.3390/fractalfract4040050 - 22 Oct 2020

Cited by 28 | Viewed by 2773

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)

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13 pages, 1186 KiB

Open AccessEditor’s ChoiceArticle

Numerical Simulation of the Fractal-Fractional Ebola Virus

by H. M. Srivastava and Khaled M. Saad

Fractal Fract. 2020, 4(4), 49; https://doi.org/10.3390/fractalfract4040049 - 29 Sep 2020

Cited by 39 | Viewed by 3924

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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20 pages, 347 KiB

Open AccessArticle

On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative

by Övgü Gürel Yılmaz, Rabia Aktaş and Fatma Taşdelen

Fractal Fract. 2020, 4(4), 48; https://doi.org/10.3390/fractalfract4040048 - 24 Sep 2020

Cited by 7 | Viewed by 2580 | Correction

Abstract

Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol in Diaz et al. 2007, which are one of the vital generalizations of hypergeometric functions. In this study, we focus on [...] Read more.

Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol in Diaz et al. 2007, which are one of the vital generalizations of hypergeometric functions. In this study, we focus on the k-analogues of

F_{1}

Appell function introduced by Mubeen et al. 2015 and the k-generalizations of F₂ and F₃ Appell functions indicated in Kıymaz et al. 2017. we present some important transformation formulas and some reduction formulas which show close relation not only with k-Appell functions but also with k-hypergeometric functions. Employing the theory of Riemann–Liouville k-fractional derivative from Rahman et al. 2020, and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for k-analogue of hypergeometric functions and Appell functions. Full article

(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)

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Fractal Fract., Volume 4, Issue 4 (December 2020) – 13 articles

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