Research

25 pages, 4789 KiB

Open AccessArticle

Further Generalization and Approximation of Fractional-Order Filters and Their Inverse Functions of the Second-Order Limiting Form

by Shibendu Mahata, Norbert Herencsar and David Kubanek

Fractal Fract. 2022, 6(4), 209; https://doi.org/10.3390/fractalfract6040209 - 08 Apr 2022

Cited by 8 | Viewed by 1850

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

Open AccessArticle

Electronically Controlled Power-Law Filters Realizations

by Errikos Tsouvalas, Stavroula Kapoulea, Costas Psychalinos, Ahmed S. Elwakil and Dražen Jurišić

Fractal Fract. 2022, 6(2), 111; https://doi.org/10.3390/fractalfract6020111 - 14 Feb 2022

Cited by 8 | Viewed by 1752

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

μ

m 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

Open AccessArticle

A New Approach to Fractional Kinetic Evolutions

by Vassili N. Kolokoltsov and Marianna Troeva

Fractal Fract. 2022, 6(2), 49; https://doi.org/10.3390/fractalfract6020049 - 18 Jan 2022

Cited by 8 | Viewed by 1683

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

Open AccessArticle

A Langevin-Type q-Variant System of Nonlinear Fractional Integro-Difference Equations with Nonlocal Boundary Conditions

by Ravi P. Agarwal, Hana Al-Hutami and Bashir Ahmad

Fractal Fract. 2022, 6(1), 45; https://doi.org/10.3390/fractalfract6010045 - 14 Jan 2022

Cited by 5 | Viewed by 1773

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

Open AccessArticle

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

by Christoph Bandt and Dmitry Mekhontsev

Fractal Fract. 2022, 6(1), 39; https://doi.org/10.3390/fractalfract6010039 - 12 Jan 2022

Viewed by 3474

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

Open AccessArticle

Shifted Fractional-Order Jacobi Collocation Method for Solving Variable-Order Fractional Integro-Differential Equation with Weakly Singular Kernel

by Mohamed A. Abdelkawy, Ahmed Z. M. Amin, António M. Lopes, Ishak Hashim and Mohammed M. Babatin

Fractal Fract. 2022, 6(1), 19; https://doi.org/10.3390/fractalfract6010019 - 30 Dec 2021

Cited by 8 | Viewed by 1723

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

Open AccessArticle

Lyapunov Approach for Almost Periodicity in Impulsive Gene Regulatory Networks of Fractional Order with Time-Varying Delays

by Ivanka Stamova and Gani Stamov

Fractal Fract. 2021, 5(4), 268; https://doi.org/10.3390/fractalfract5040268 - 09 Dec 2021

Cited by 8 | Viewed by 2116

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

Open AccessArticle

Fox H-Functions in Self-Consistent Description of a Free-Electron Laser

by Alexander Iomin

Fractal Fract. 2021, 5(4), 263; https://doi.org/10.3390/fractalfract5040263 - 07 Dec 2021

Cited by 3 | Viewed by 1941

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

Open AccessArticle

Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects

by Yaswanth Sai Jetti and Martin Ostoja-Starzewski

Fractal Fract. 2021, 5(4), 255; https://doi.org/10.3390/fractalfract5040255 - 04 Dec 2021

Cited by 2 | Viewed by 1913

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

Open AccessArticle

An Entropy Paradox Free Fractional Diffusion Equation

by Manuel Duarte Ortigueira

Fractal Fract. 2021, 5(4), 236; https://doi.org/10.3390/fractalfract5040236 - 21 Nov 2021

Cited by 4 | Viewed by 1191

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

- \infty

. 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

Open AccessArticle

Applications of a Fixed Point Result for Solving Nonlinear Fractional and Integral Differential Equations

by Liliana Guran, Zoran D. Mitrović, G. Sudhaamsh Mohan Reddy, Abdelkader Belhenniche and Stojan Radenović

Fractal Fract. 2021, 5(4), 211; https://doi.org/10.3390/fractalfract5040211 - 11 Nov 2021

Cited by 14 | Viewed by 1699

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

Open AccessArticle

On the Design of Power Law Filters and Their Inverse Counterparts

by Shibendu Mahata, Norbert Herencsar and David Kubanek

Fractal Fract. 2021, 5(4), 197; https://doi.org/10.3390/fractalfract5040197 - 04 Nov 2021

Cited by 13 | Viewed by 2149

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

Open AccessArticle

Generalized Bessel Quasilinearization Technique Applied to Bratu and Lane–Emden-Type Equations of Arbitrary Order

by Mohammad Izadi and Hari M. Srivastava

Fractal Fract. 2021, 5(4), 179; https://doi.org/10.3390/fractalfract5040179 - 22 Oct 2021

Cited by 17 | Viewed by 1385

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

Open AccessArticle

Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule

by Samaneh Soradi-Zeid, Mehdi Mesrizadeh and Carlo Cattani

Fractal Fract. 2021, 5(3), 111; https://doi.org/10.3390/fractalfract5030111 - 07 Sep 2021

Cited by 3 | Viewed by 1935

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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