Fractal and Fractional

16 pages, 342 KiB

Open AccessArticle

A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation

by Ravshan Ashurov, Oqila Mukhiddinova and Sabir Umarov

Fractal Fract. 2023, 7(6), 490; https://doi.org/10.3390/fractalfract7060490 - 20 Jun 2023

Cited by 1 | Viewed by 836

A nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition

u (x, T) = β u (x, 0) + φ (x)

, where [...] Read more.

A nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition

u (x, T) = β u (x, 0) + φ (x)

, where

β

is an arbitrary real number, is proposed instead of the initial condition. If

β = 0

, then we have the inverse problem in time, called the backward problem. It is well-known that the backward problem is ill-posed in the sense of Hadamard. If

β = 1

, then the corresponding non-local problem becomes well-posed in the sense of Hadamard, and moreover, in this case a coercive estimate for the solution can be established. The aim of this work is to find values of the parameter

β

, which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if

β \geq 1,

or

β < 0

, then the problem is well-posed; if

β \in (0, 1)

, then depending on the eigenvalues of the elliptic part of the equation, for the existence of a solution an additional condition on orthogonality of the right-hand side of the equation and the boundary function to some eigenfunctions of the corresponding elliptic operator may emerge. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

11 pages, 310 KiB

Open AccessArticle

Further Generalizations of Some Fractional Integral Inequalities

by Dong Chen, Matloob Anwar, Ghulam Farid and Hafsa Yasmeen

Fractal Fract. 2023, 7(6), 489; https://doi.org/10.3390/fractalfract7060489 - 20 Jun 2023

Viewed by 627

Abstract

This paper aims to establish generalized fractional integral inequalities for operators containing Mittag–Leffler functions. By applying

(α, h - m) - p

-convexity of real valued functions, generalizations of many well-known inequalities are obtained. Hadamard-type inequalities for various classes of functions are given in particular cases. Full article

(This article belongs to the Special Issue Recent Advances in General Integral Operators)

12 pages, 7592 KiB

Open AccessArticle

A Parallel Computational–Statistical Framework for Simulation of Turbulence: Applications to Data-Driven Fractional Modeling

by Ali Akhavan-Safaei and Mohsen Zayernouri

Fractal Fract. 2023, 7(6), 488; https://doi.org/10.3390/fractalfract7060488 - 20 Jun 2023

Viewed by 846

Abstract

In this work, an open-source computational–statistical platform to obtain synthetic homogeneous isotropic turbulent flow and passive scalar transport is presented. A parallel implementation of the well-known pseudo-spectral method in addition to the comprehensive record of the statistical and small-scale quantities of the turbulent [...] Read more.

In this work, an open-source computational–statistical platform to obtain synthetic homogeneous isotropic turbulent flow and passive scalar transport is presented. A parallel implementation of the well-known pseudo-spectral method in addition to the comprehensive record of the statistical and small-scale quantities of the turbulent transport are offered for executing on distributed memory CPU-based supercomputers. The user-friendly workflow and easy-to-run design of the developed package are disclosed through an extensive and step-by-step example. The resulting low- and high-order statistical records vividly verify a well-established and fully developed turbulent state as well as the seamless statistical balance of conservation laws. The post-processing tools provided in this platform would allow the user to easily construct multiple important transport quantities from primitive turbulent fields. The programming codes for this tool are accessible through GitHub (see Data Availability Statement). Full article

(This article belongs to the Section Numerical and Computational Methods)

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21 pages, 5428 KiB

Open AccessArticle

A Time Two-Mesh Finite Difference Numerical Scheme for the Symmetric Regularized Long Wave Equation

by Jingying Gao, Siriguleng He, Qingmei Bai and Jie Liu

Fractal Fract. 2023, 7(6), 487; https://doi.org/10.3390/fractalfract7060487 - 19 Jun 2023

Cited by 1 | Viewed by 905

Abstract

The symmetric regularized long wave (SRLW) equation is a mathematical model used in many areas of physics; the solution of the SRLW equation can accurately describe the behavior of long waves in shallow water. To approximate the solutions of the equation, a time [...] Read more.

The symmetric regularized long wave (SRLW) equation is a mathematical model used in many areas of physics; the solution of the SRLW equation can accurately describe the behavior of long waves in shallow water. To approximate the solutions of the equation, a time two-mesh (TT-M) decoupled finite difference numerical scheme is proposed in this paper to improve the efficiency of solving the SRLW equation. Based on the time two-mesh technique and two time-level finite difference method, the proposed scheme can calculate the velocity

u (x, t)

and density

ρ (x, t)

in the SRLW equation simultaneously. The linearization process involves a modification similar to the Gauss-Seidel method used for linear systems to improve the accuracy of the calculation to obtain solutions. By using the discrete energy and mathematical induction methods, the convergence results with

O (τ_{C}^{2} + τ_{F} + h^{2})

in the discrete

L_{\infty}

-norm for

u (x, t)

and in the discrete

L_{2}

-norm for

ρ (x, t)

are proved, respectively. The stability of the scheme was also analyzed. Finally, some numerical examples, including error estimate, computational time and preservation of conservation laws, are given to verify the efficiency of the scheme. The numerical results show that the new method preserves conservation laws of four quantities successfully. Furthermore, by comparing with the original two-level nonlinear finite difference scheme, the proposed scheme can save the CPU time. Full article

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16 pages, 1102 KiB

Open AccessArticle

Multicorn Sets of

{\bar{z}}^{k} + c^{m}

via S-Iteration with h-Convexity

by Asifa Tassaddiq, Muhammad Tanveer, Khuram Israr, Muhammad Arshad, Khurrem Shehzad and Rekha Srivastava

Fractal Fract. 2023, 7(6), 486; https://doi.org/10.3390/fractalfract7060486 - 18 Jun 2023

Cited by 1 | Viewed by 1130

Abstract

Fractals represent important features of our natural environment, and therefore, several scientific fields have recently begun using fractals that employ fixed-point theory. While many researchers are working on fractals (i.e., Mandelbrot and Julia sets), only a very few have focused on multicorn sets [...] Read more.

Fractals represent important features of our natural environment, and therefore, several scientific fields have recently begun using fractals that employ fixed-point theory. While many researchers are working on fractals (i.e., Mandelbrot and Julia sets), only a very few have focused on multicorn sets and their dynamic nature. In this paper, we study the dynamics of multicorn sets of

{\bar{z}}^{k} + c^{m}

, where

k \geq 2, c \neq 0 \in C

, and

m \in R

, by using S-iteration with h-convexity instead of standard S-iteration. We develop escape criterion

{\bar{z}}^{k} + c^{m}

for S-iteration with h-convexity. We analyse the dynamical behaviour of the proposed conjugate complex function and discuss the variation of iteration parameters along with function parameter m. Moreover, we discuss the effects of input parameters of the proposed iteration and conjugate complex functions of the behaviour of multicorn sets with numerical simulations. Full article

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Multidisciplinary Applications, 2nd Edition)

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15 pages, 4005 KiB

Open AccessArticle

Identification of Fractional Models of an Induction Motor with Errors in Variables

by Dmitriy Ivanov

Fractal Fract. 2023, 7(6), 485; https://doi.org/10.3390/fractalfract7060485 - 18 Jun 2023

Cited by 2 | Viewed by 1001

Abstract

The skin effect in modeling an induction motor can be described by fractional differential equations. The existing methods for identifying the parameters of an induction motor with a rotor skin effect suggest the presence of errors only in the output. The presence of [...] Read more.

The skin effect in modeling an induction motor can be described by fractional differential equations. The existing methods for identifying the parameters of an induction motor with a rotor skin effect suggest the presence of errors only in the output. The presence of errors in measuring currents and voltages leads to errors in both input and output signals. Applying standard methods, such as the ordinary least squares method, leads to biased estimates in these types of problems. The study proposes a new method for identifying the parameters of an induction motor in the presence of a skin effect. Estimates of parameters were determined based on generalized total least squares. The simulation results obtained showed the high accuracy of the obtained estimates. The results of this research can be applied in the development of predictive diagnostic systems. This study shows that ordinary least squares parameter estimates can lead to incorrect operation of the fault diagnosis system. Full article

(This article belongs to the Special Issue Application of Fractional Calculus as an Interdisciplinary Modeling Framework, 2nd Edition)

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19 pages, 5095 KiB

Open AccessArticle

Robust Adaptive Fuzzy Fractional Control for Nonlinear Chaotic Systems with Uncertainties

by Masoud S. Bahraini, Mohammad Javad Mahmoodabadi and Niels Lohse

Fractal Fract. 2023, 7(6), 484; https://doi.org/10.3390/fractalfract7060484 - 18 Jun 2023

Cited by 3 | Viewed by 919

Abstract

The control of nonlinear chaotic systems with uncertainties is a challenging problem that has attracted the attention of researchers in recent years. In this paper, we propose a robust adaptive fuzzy fractional control strategy for stabilizing nonlinear chaotic systems with uncertainties. The proposed [...] Read more.

The control of nonlinear chaotic systems with uncertainties is a challenging problem that has attracted the attention of researchers in recent years. In this paper, we propose a robust adaptive fuzzy fractional control strategy for stabilizing nonlinear chaotic systems with uncertainties. The proposed strategy combined a fuzzy logic controller with fractional-order calculus to accurately model the system’s behavior and adapt to uncertainties in real-time. The proposed controller was based on a supervised sliding mode controller and an optimal robust adaptive fractional PID controller subjected to fuzzy rules. The stability of the closed-loop system was guaranteed using Lyapunov theory. To evaluate the performance of the proposed controller, we applied it to the Duffing–Holmes oscillator. Simulation results demonstrated that the proposed control method outperformed a recently introduced controller in the literature. The response of the system was significantly improved, highlighting the effectiveness and robustness of the proposed approach. The presented results provide strong evidence of the potential of the proposed strategy in a range of applications involving nonlinear chaotic systems with uncertainties. Full article

(This article belongs to the Section Engineering)

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17 pages, 344 KiB

Open AccessArticle

Fractional Telegraph Equation with the Caputo Derivative

by Ravshan Ashurov and Rajapboy Saparbayev

Fractal Fract. 2023, 7(6), 483; https://doi.org/10.3390/fractalfract7060483 - 17 Jun 2023

Cited by 1 | Viewed by 778

Abstract

The Cauchy problem for the telegraph equation

{(D_{t}^{ρ})}^{2} u (t) + 2 α D_{t}^{ρ} u (t) + A u (t) = f (t)

( [...] Read more.

The Cauchy problem for the telegraph equation

{(D_{t}^{ρ})}^{2} u (t) + 2 α D_{t}^{ρ} u (t) + A u (t) = f (t)

(

0 < t \leq T, 0 < ρ < 1

,

α > 0

), with the Caputo derivative is considered. Here, A is a selfadjoint positive operator, acting in a Hilbert space, H;

D_{t}

is the Caputo fractional derivative. Conditions are found for the initial functions and the right side of the equation that guarantee both the existence and uniqueness of the solution of the Cauchy problem. It should be emphasized that these conditions turned out to be less restrictive than expected in a well-known paper by R. Cascaval et al. where a similar problem for a homogeneous equation with some restriction on the spectrum of the operator, A, was considered. We also prove stability estimates important for the application. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes, 2nd Edition)

12 pages, 690 KiB

Open AccessArticle

Interpolated Coefficient Mixed Finite Elements for Semilinear Time Fractional Diffusion Equations

by Xiaowu Li and Yuelong Tang

Fractal Fract. 2023, 7(6), 482; https://doi.org/10.3390/fractalfract7060482 - 16 Jun 2023

Cited by 4 | Viewed by 818

Abstract

In this paper, we consider a fully discrete interpolated coefficient mixed finite element method for semilinear time fractional reaction–diffusion equations. The classic

L 1

scheme based on graded meshes and new mixed finite element based on triangulation is used for the temporal and [...] Read more.

In this paper, we consider a fully discrete interpolated coefficient mixed finite element method for semilinear time fractional reaction–diffusion equations. The classic

L 1

scheme based on graded meshes and new mixed finite element based on triangulation is used for the temporal and spatial discretization, respectively. The interpolation coefficient technique is used to deal with the semilinear term, and the discrete nonlinear system is solved by a Newton-like iterative method. Stability and convergence results for both the original variable and its flux are derived. Numerical experiments confirm our theoretical analysis. Full article

(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)

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25 pages, 424 KiB

Open AccessArticle

Scale-Invariant General Fractional Calculus: Mellin Convolution Operators

by Vasily E. Tarasov

Fractal Fract. 2023, 7(6), 481; https://doi.org/10.3390/fractalfract7060481 - 16 Jun 2023

Cited by 5 | Viewed by 887

Abstract

General fractional calculus (GFC) of operators that is defined through the Mellin convolution instead of Laplace convolution is proposed. This calculus of Mellin convolution operators can be considered as an analogue of the Luchko GFC for the Laplace convolution operators. The proposed general [...] Read more.

General fractional calculus (GFC) of operators that is defined through the Mellin convolution instead of Laplace convolution is proposed. This calculus of Mellin convolution operators can be considered as an analogue of the Luchko GFC for the Laplace convolution operators. The proposed general fractional differential operators are generalizations of scaling (dilation) differential operator for the case of general form of nonlocality. Semi-group and scale-invariant properties of these operators are proven. The Hadamard and Hadamard-type fractional operators are special case of the proposed operators. The fundamental theorems for the scale-invariant general fractional operators are proven. The proposed GFC can be applied in the study of dynamics, which is characterized by nonlocality and scale invariance. Full article

16 pages, 1170 KiB

Open AccessArticle

Analysis of a High-Accuracy Numerical Method for Time-Fractional Integro-Differential Equations

by Ziyang Luo, Xindong Zhang and Leilei Wei

Fractal Fract. 2023, 7(6), 480; https://doi.org/10.3390/fractalfract7060480 - 16 Jun 2023

Viewed by 979

Abstract

A high-order finite difference numerical scheme based on the compact difference operator is proposed in this paper for time-fractional partial integro-differential equations with a weakly singular kernel, where the time-fractional derivative term is defined in the Riemann-Liouville sense. Here, the stability and convergence [...] Read more.

A high-order finite difference numerical scheme based on the compact difference operator is proposed in this paper for time-fractional partial integro-differential equations with a weakly singular kernel, where the time-fractional derivative term is defined in the Riemann-Liouville sense. Here, the stability and convergence of the constructed compact finite difference scheme are proved in

L^{\infty}

norm, with the accuracy order

O (τ^{2} + h^{4})

, where

τ

and h are temporal and spatial step sizes, respectively. The advantage of this numerical scheme is that arbitrary parameters can be applied to achieve the desired accuracy. Some numerical examples are presented to support the theoretical analysis. Full article

(This article belongs to the Section Numerical and Computational Methods)

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16 pages, 359 KiB

Open AccessArticle

Asymptotics for Time-Fractional Venttsel’ Problems in Fractal Domains

by Raffaela Capitanelli, Simone Creo and Maria Rosaria Lancia

Fractal Fract. 2023, 7(6), 479; https://doi.org/10.3390/fractalfract7060479 - 16 Jun 2023

Viewed by 691

Abstract

In this study, we consider fractional-in-time Venttsel’ problems in fractal domains of the Koch type. Well-posedness and regularity results are given. In view of numerical approximation, we consider the associated approximating pre-fractal problems. Our main result is the convergence of the solutions of [...] Read more.

In this study, we consider fractional-in-time Venttsel’ problems in fractal domains of the Koch type. Well-posedness and regularity results are given. In view of numerical approximation, we consider the associated approximating pre-fractal problems. Our main result is the convergence of the solutions of such problems towards the solution of the fractional-in-time Venttsel’ problem in the corresponding fractal domain. This is achieved via the convergence (in the Mosco–Kuwae–Shioya sense) of the approximating energy forms in varying Hilbert spaces. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

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31 pages, 18905 KiB

Open AccessArticle

Comparing Market Efficiency in Developed, Emerging, and Frontier Equity Markets: A Multifractal Detrended Fluctuation Analysis

by Min-Jae Lee and Sun-Yong Choi

Fractal Fract. 2023, 7(6), 478; https://doi.org/10.3390/fractalfract7060478 - 15 Jun 2023

Cited by 2 | Viewed by 1961

Abstract

In this article, we investigate the market efficiency of global stock markets using the multifractal detrended fluctuation analysis methodology and analyze the results by dividing them into developed, emerging, and frontier groups. The static analysis results reveal that financially advanced countries, such as [...] Read more.

In this article, we investigate the market efficiency of global stock markets using the multifractal detrended fluctuation analysis methodology and analyze the results by dividing them into developed, emerging, and frontier groups. The static analysis results reveal that financially advanced countries, such as Switzerland, the UK, and the US, have more efficient stock markets than other countries. Rolling window analysis shows that global issues dominate the developed country group, while emerging markets are vulnerable to foreign capital movements and political risks. In the frontier group, intensive domestic market issues vary, making it difficult to distinguish similar dynamics. Our findings have important implications for international investors and policymakers. International investors can establish investment strategies based on the degree of market efficiency of individual stock markets. Policymakers in countries with significant fluctuations in market efficiency should consider implementing new regulations to enhance market efficiency. Overall, this study provides valuable insights into the market efficiency of global stock markets and highlights the need for careful consideration by international investors and policymakers. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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20 pages, 355 KiB

Open AccessArticle

Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler Function

by Ricardo Almeida

Fractal Fract. 2023, 7(6), 477; https://doi.org/10.3390/fractalfract7060477 - 15 Jun 2023

Viewed by 667

Abstract

In this paper, we investigate the necessary conditions to optimize a given functional, involving a generalization of the tempered fractional derivative. The exponential function is replaced by the Mittag–Leffler function, and the kernel depends on an arbitrary increasing function. The Lagrangian depends on [...] Read more.

In this paper, we investigate the necessary conditions to optimize a given functional, involving a generalization of the tempered fractional derivative. The exponential function is replaced by the Mittag–Leffler function, and the kernel depends on an arbitrary increasing function. The Lagrangian depends on time, the state function, its fractional derivative, and we add a terminal cost function to the formulation of the problem. Since this new fractional derivative is presented in a general form, some previous works are our own particular cases. In addition, for different choices of the kernel, new results can be deduced. Using variational techniques, the fractional Euler–Lagrange equation is proved, as are its associated transversality conditions. The variational problem with additional constraints is also considered. Then, the question of minimizing functionals with an infinite interval of integration is addressed. To end, we study the case of the Herglotz variational problem, which generalizes the previous one. With this work, several optimization conditions are proven that can be useful for different optimization problems dealing with various fractional derivatives. Full article

(This article belongs to the Section General Mathematics, Analysis)

18 pages, 2860 KiB

Open AccessArticle

Generalized Thermoelastic Infinite Annular Cylinder under the Hyperbolic Two-Temperature Fractional-Order Strain Theory

by Eman A. N. Al-Lehaibi

Fractal Fract. 2023, 7(6), 476; https://doi.org/10.3390/fractalfract7060476 - 15 Jun 2023

Cited by 1 | Viewed by 685

Abstract

This work introduces a new thermoelastic model of an isotropic and homogeneous annular cylinder. The cylinder’s bounding inner surface is shocked thermally, and the bounding outer surface has no temperature increment and volumetric strain. The governing equations in the context of the hyperbolic [...] Read more.

This work introduces a new thermoelastic model of an isotropic and homogeneous annular cylinder. The cylinder’s bounding inner surface is shocked thermally, and the bounding outer surface has no temperature increment and volumetric strain. The governing equations in the context of the hyperbolic two-temperature generalized thermoelasticity with fractional-order strain theory have been derived. The numerical solutions of the conductive temperature, dynamic temperature, displacement, strain, and stress are illustrated in figures that use various values of fractional-order and two-temperature parameters to stand on their effects on the thermal and mechanical waves. The fractional-order parameter has significant impacts on the displacement, strain, and stress distributions. However, it does not affect dynamic or conductive temperatures. The hyperbolic two-temperature model is a successful model for making thermal and mechanical waves propagate at limited speeds. Full article

(This article belongs to the Section Mathematical Physics)

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13 pages, 4985 KiB

Open AccessArticle

Thermal Conductivity of Fractal-Textured Foamed Concrete

by Guosheng Xiang, Huajian Li, Yinkang Zhou and Zhe Huang

Fractal Fract. 2023, 7(6), 475; https://doi.org/10.3390/fractalfract7060475 - 15 Jun 2023

Cited by 1 | Viewed by 907

Abstract

To provide scientific guidance for the use of foamed concrete (FC) in construction engineering, a thermal conductivity calculation method, based on the fractal model of FC, has been developed. The thermal conductivity (TC) of FC has been tested by the transient planar heat [...] Read more.

To provide scientific guidance for the use of foamed concrete (FC) in construction engineering, a thermal conductivity calculation method, based on the fractal model of FC, has been developed. The thermal conductivity (TC) of FC has been tested by the transient planar heat source method in order to verify the reliability of the proposed calculation model. The FC was made of cement, fly ash, and ore powder, and cured under natural conditions for 7 d, 14 d, 28 d, and 42 d, respectively. The TC of FC gradually decreases with the increase in age. The fractal dimension of FC can be determined by both the box-counting method and compressive strength test, and the dimensions determined by both methods are similar. The TC of FC at different porosities and curing ages can be calculated by the fractal dimension, and the estimated values are basically consistent with the test data. Full article

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16 pages, 6869 KiB

Open AccessArticle

Dynamical Analysis of the Incommensurate Fractional-Order Hopfield Neural Network System and Its Digital Circuit Realization

by Miao Wang, Yuru Wang and Ran Chu

Fractal Fract. 2023, 7(6), 474; https://doi.org/10.3390/fractalfract7060474 - 15 Jun 2023

Cited by 2 | Viewed by 924

Abstract

Dynamical analysis of the incommensurate fractional-order neural network is a novel topic in the field of chaos research. This article investigates a Hopfield neural network (HNN) system in view of incommensurate fractional orders. Using the Adomian decomposition method (ADM) algorithm, the solution of [...] Read more.

Dynamical analysis of the incommensurate fractional-order neural network is a novel topic in the field of chaos research. This article investigates a Hopfield neural network (HNN) system in view of incommensurate fractional orders. Using the Adomian decomposition method (ADM) algorithm, the solution of the incommensurate fractional-order Hopfield neural network (FOHNN) system is solved. The equilibrium point of the system is discussed, and the dissipative characteristics are verified and discussed. By varying the order values of the proposed system, different dynamical behaviors of the incommensurate FOHNN system are explored and discussed via bifurcation diagrams, the Lyapunov exponent spectrum, complexity, etc. Finally, using the DSP platform to implement the system, the results are in good agreement with those of the simulation. The actual results indicate that the system shows many complex and interesting phenomena, such as attractor coexistence and an inversion property, with dynamic changes of the order of q₀, q₁, and q₂. These phenomena provide important insights for simulating complex neural system states in pathological conditions and provide the theoretical basis for the later study of incommensurate fractional-order neural network systems. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)

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21 pages, 4951 KiB

Open AccessArticle

Investigation of Egyptian Banks’ Competition through a Riesz–Caputo Fractional Model

by Othman A. M. Omar, Hamdy M. Ahmed and Walid Hamdy

Fractal Fract. 2023, 7(6), 473; https://doi.org/10.3390/fractalfract7060473 - 14 Jun 2023

Cited by 1 | Viewed by 842

Abstract

In this paper, a four-dimensional competition model, driven by the Riesz–Caputo operator, is established. Then, the presented model’s uniqueness, existence, and stability are discussed. After that, the model is applied to describe the profit competition between Egyptian banks. Using the Hamiltonian definition of [...] Read more.

In this paper, a four-dimensional competition model, driven by the Riesz–Caputo operator, is established. Then, the presented model’s uniqueness, existence, and stability are discussed. After that, the model is applied to describe the profit competition between Egyptian banks. Using the Hamiltonian definition of optimal control, we set a control strategy for banks’ profits during crises as a pre-step measure to counteract their negative impact. Finally, the model is solved numerically using an extended Adams–Bashford scheme. The valid data between the end of 2016 and 2020 are used for numerical simulations, while the data between the end of 2021 and 2022 are used for prediction measurements. From the results, it can be seen that the COVID-19 crisis resulted in a sudden decrease in banks’ profits during 2020, and if there were a control system enabled, it could have compensated for this decrease. Full article

(This article belongs to the Section General Mathematics, Analysis)

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16 pages, 7016 KiB

Open AccessArticle

Design and Analysis of Fractal-Shaped High-Impedance Surface Unit Cell Characteristics

by Akash Kumar Gupta, Harish Chandra Mohanta, P. Satish Rama Chowdary, M. Vamshi Krishna and Heba G. Mohamed

Fractal Fract. 2023, 7(6), 472; https://doi.org/10.3390/fractalfract7060472 - 14 Jun 2023

Cited by 2 | Viewed by 1569

Abstract

Fractal geometries consistently provide solutions to several electromagnetic design problems. In this paper, fractal geometries such as Hilbert and Moore curves are used to design efficient High-Impedance Surfaces. Modern communication devices have many sensors that are needed to communicate wirelessly. The critical component [...] Read more.

Fractal geometries consistently provide solutions to several electromagnetic design problems. In this paper, fractal geometries such as Hilbert and Moore curves are used to design efficient High-Impedance Surfaces. Modern communication devices have many sensors that are needed to communicate wirelessly. The critical component of wireless communications is antennas. Planar microstrip patch antennas are popular due to their low profile, compactness, and good radiation characteristics. The structural disadvantages of microstrip antennas are that they have surface waves that propagate over the ground plane. High-Impedance Surface (HIS) planes are a prominent solution to minimize and eliminate surface waves. The HIS structures behave as active LC filters that suppress surface waves at their resonance frequency. The resonance frequency of the structure is obtained by its LC equivalent or by analyzing the reflection phase characteristics. This work presents conventional HIS structures similar to mushroom HIS and fractal HIS such as Hilbert curve and Moore curve HIS. The HIS reflection phase characteristics are obtained by applying periodic boundary conditions with plane wave illumination. The results were obtained in terms of the reflection phase angle. The conventional mushroom structures show narrow band characteristics at given dimensions of 10 mm × 10 mm and 20 mm × 20 mm. These structures are helpful in the replacement of PEC ground planes for patch antennas under sub-6 GHz. The Hilbert and Moore fractals are also designed and have a multiband response that can be useful for L, S, and C band applications. Another design challenge of HIS is protrusions, which make design difficult. The work also presents the effect of having vias and the absence of vias on reflection phase characteristics. The response shows the least and no significant effect of vias under the x-band operation. Full article

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18 pages, 496 KiB

Open AccessArticle

Block-Centered Finite-Difference Methods for Time-Fractional Fourth-Order Parabolic Equations

by Taixiu Zhang, Zhe Yin and Ailing Zhu

Fractal Fract. 2023, 7(6), 471; https://doi.org/10.3390/fractalfract7060471 - 14 Jun 2023

Viewed by 996

Abstract

The block-centered finite-difference method has many advantages, and the time-fractional fourth-order equation is widely used in physics and engineering science. In this paper, we consider variable-coefficient fourth-order parabolic equations of fractional-order time derivatives with Neumann boundary conditions. The fractional-order time derivatives are approximated [...] Read more.

The block-centered finite-difference method has many advantages, and the time-fractional fourth-order equation is widely used in physics and engineering science. In this paper, we consider variable-coefficient fourth-order parabolic equations of fractional-order time derivatives with Neumann boundary conditions. The fractional-order time derivatives are approximated by

L 1

interpolation. We propose the block-centered finite-difference scheme for fourth-order parabolic equations with fractional-order time derivatives. We prove the stability of the block-centered finite-difference scheme and the second-order convergence of the discrete

L 2

norms of the approximate solution and its derivatives of every order. Numerical examples are provided to verify the effectiveness of the block-centered finite-difference scheme. Full article

(This article belongs to the Special Issue Fractional Derivatives and Their Applications to Fractional Diffusion Problems)

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17 pages, 10832 KiB

Open AccessArticle

Circuit Realization of the Fractional-Order Sprott K Chaotic System with Standard Components

by Abdullah Gokyildirim

Fractal Fract. 2023, 7(6), 470; https://doi.org/10.3390/fractalfract7060470 - 13 Jun 2023

Cited by 7 | Viewed by 1432

Abstract

Interest in studies on fractional calculus and its applications has greatly increased in recent years. Fractional-order analysis has the potential to enhance the dynamic structure of chaotic systems. This study presents the implementation of a lower-order fractional electronic circuit using standard components for [...] Read more.

Interest in studies on fractional calculus and its applications has greatly increased in recent years. Fractional-order analysis has the potential to enhance the dynamic structure of chaotic systems. This study presents the implementation of a lower-order fractional electronic circuit using standard components for the Sprott K system. To our knowledge, there are no chaotic circuit realizations in the literature where the value of a fractional-order parameter is approximately 0.8, making this study pioneering in this aspect. Additionally, various numerical analyses of the system are conducted, including chaotic time series and phase planes, Lyapunov exponents, spectral entropy (SE), and bifurcation diagrams, in order to examine its dynamic characteristics and complexity. As anticipated, the voltage outputs obtained from the oscilloscope demonstrated good agreement with both the numerical analysis and PSpice simulations. Full article

(This article belongs to the Special Issue Fractional-Order Chaotic Systems and Circuits: Design, Modeling and Implementation)

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8 pages, 292 KiB

Open AccessBrief Report

On a Quadratic Nonlinear Fractional Equation

by Iván Area and Juan J. Nieto

Fractal Fract. 2023, 7(6), 469; https://doi.org/10.3390/fractalfract7060469 - 12 Jun 2023

Cited by 3 | Viewed by 781

Abstract

In this paper, we study a quadratic nonlinear equation from the fractional point of view. An explicit solution is given in terms of the Lambert special function. A new phenomenon appears involving the collapsing of the solution and the blow-up of the derivative. [...] Read more.

In this paper, we study a quadratic nonlinear equation from the fractional point of view. An explicit solution is given in terms of the Lambert special function. A new phenomenon appears involving the collapsing of the solution and the blow-up of the derivative. The explicit representation of the solution reveals the non-elementary nature of the solution. Full article

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Multidisciplinary Applications, 2nd Edition)

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30 pages, 649 KiB

Open AccessArticle

Factorized Doubling Algorithm for Large-Scale High-Ranked Riccati Equations in Fractional System

by Bo Yu and Ning Dong

Fractal Fract. 2023, 7(6), 468; https://doi.org/10.3390/fractalfract7060468 - 10 Jun 2023

Viewed by 850

Abstract

In real-life control problems, such as power systems, there are large-scale high-ranked discrete-time algebraic Riccati equations (DAREs) from fractional systems that require stabilizing solutions. However, these solutions are no longer numerically low-rank, which creates difficulties in computation and storage. Fortunately, the potential structures [...] Read more.

In real-life control problems, such as power systems, there are large-scale high-ranked discrete-time algebraic Riccati equations (DAREs) from fractional systems that require stabilizing solutions. However, these solutions are no longer numerically low-rank, which creates difficulties in computation and storage. Fortunately, the potential structures of the state matrix in these systems (e.g., being banded-plus-low-rank) could be beneficial for large-scale computation. In this paper, a factorized structure-preserving doubling algorithm (FSDA) is developed under the assumptions that the non-linear and constant terms are positive semidefinite and banded-plus-low-rank. The detailed iteration scheme and a deflation process for FSDA are analyzed. Additionally, a technique of partial truncation and compression is introduced to reduce the dimensions of the low-rank factors. The computation of residual and the termination condition of the structured version are also redesigned. Illustrative numerical examples show that the proposed FSDA outperforms SDA with hierarchical matrices toolbox (SDA_HODLR) on CPU time for large-scale problems. Full article

(This article belongs to the Special Issue Feature Papers for the 'Complexity' Section)

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12 pages, 3980 KiB

Open AccessArticle

Investigation of Fractal Characteristics of Karman Vortex for NACA0009 Hydrofoil

by Fangfang Zhang, Yaju Zuo, Di Zhu, Ran Tao and Ruofu Xiao

Fractal Fract. 2023, 7(6), 467; https://doi.org/10.3390/fractalfract7060467 - 10 Jun 2023

Viewed by 868

Abstract

A Karman vortex is a phenomenon of fluid flow that can cause fluctuation and vibration. As a result, it leads to fatigue damage to structures and induces safety accidents. Therefore, the analysis of the shedding law and strength of the Karman vortex is [...] Read more.

A Karman vortex is a phenomenon of fluid flow that can cause fluctuation and vibration. As a result, it leads to fatigue damage to structures and induces safety accidents. Therefore, the analysis of the shedding law and strength of the Karman vortex is significant. To further understand the laws of turbulent Karman vortex shedding and strength, this study conducts a numerical vorticity simulation of a Karman vortex at the trailing edge of a hydrofoil based on the two-dimensional simplified model of the NACA0009 hydrofoil under different Reynolds numbers. Combined with image segmentation technology, the fractal characteristics of a turbulent Karman vortex at the trailing edge of a hydrofoil are extracted, the number and total area of vortex cores are calculated, and the fractal dimension of the vortex is obtained. The results show that the fractal dimension can characterize the change in vortex shape and strength under different Reynolds numbers, and that the fractal analysis method is feasible and effective for the shedding analysis of a turbulent Karman vortex. Full article

(This article belongs to the Special Issue Turbulence Structure and Fractal Characteristics in Turbomachinery)

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18 pages, 3198 KiB

Open AccessArticle

Discerning Xylella fastidiosa-Infected Olive Orchards in the Time Series of MODIS Terra Satellite Evapotranspiration Data by Using the Fisher–Shannon Analysis and the Multifractal Detrended Fluctuation Analysis

by Luciano Telesca, Nicodemo Abate, Farid Faridani, Michele Lovallo and Rosa Lasaponara

Fractal Fract. 2023, 7(6), 466; https://doi.org/10.3390/fractalfract7060466 - 09 Jun 2023

Cited by 4 | Viewed by 1375

Abstract

Xylella fastidiosa is a phytobacterium able to provoke severe diseases in many species. When it infects olive trees, it induces the olive quick decline syndrome that leads the tree to a rapid desiccation and then to the death. This phytobacterium has been recently [...] Read more.

Xylella fastidiosa is a phytobacterium able to provoke severe diseases in many species. When it infects olive trees, it induces the olive quick decline syndrome that leads the tree to a rapid desiccation and then to the death. This phytobacterium has been recently detected in olive groves in southern Italy, representing an important threat to the olive growing of the area. In this paper, in order to identify patterns revealing the presence of Xylella fastidiosa, several hundreds pixels of MODIS satellite evapostranspiration covering infected and healthy olive groves in southern Italy were analyzed by means of the Fisher–Shannon method and the multifractal detrended fluctuation analysis. The analysis of the receiver operating characteric curve indicates that the two informational quantities (the Fisher information measure and the Shannon entropy) and the three multifractal parameters (the range of generalized Hurst exponents and the width and the maximum of the multifractal spectrum) are well suited to discriminate between infected and healthy sites, although the maximum of the multifractal spectrum performs better than the others. These results could suggest the use of both the methods as an operational tool for early detection of plant diseases. Full article

(This article belongs to the Section Life Science, Biophysics)

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23 pages, 8504 KiB

Open AccessArticle

Overlap Functions-Based Fuzzy Mathematical Morphological Operators and Their Applications in Image Edge Extraction

by Xiaohong Zhang, Mengyuan Li and Hui Liu

Fractal Fract. 2023, 7(6), 465; https://doi.org/10.3390/fractalfract7060465 - 09 Jun 2023

Cited by 2 | Viewed by 1213

Abstract

As special aggregation functions, overlap functions have been widely used in the soft computing field. In this work, with the aid of overlap functions, two new groups of fuzzy mathematical morphology (FMM) operators were proposed and applied to image processing, and they obtained [...] Read more.

As special aggregation functions, overlap functions have been widely used in the soft computing field. In this work, with the aid of overlap functions, two new groups of fuzzy mathematical morphology (FMM) operators were proposed and applied to image processing, and they obtained better results than existing algorithms. First, based on overlap functions and structuring elements, the first group of new FMM operators (called OSFMM operators) was proposed, and their properties were systematically analyzed. With the implementation of OSFMM operators and the fuzzy C-means (FCM) algorithm, a new image edge extraction algorithm (called the OS-FCM algorithm) was proposed. Then, the second group of new FMM operators (called ORFMM operators) was proposed based on overlap functions and fuzzy relations. Another new image edge extraction algorithm (called OR-FCM algorithm) was proposed by using ORFMM operators and FCM algorithm. Finally, through the edge segmentation experiments of multiple standard images, the actual segmentation effects of the above-mentioned two algorithms and relevant algorithms were compared. The acquired results demonstrate that the image edge extraction algorithms proposed in this work can extract the complete edge of foreground objects on the basis of introducing the least noise. Full article

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15 pages, 345 KiB

Open AccessArticle

Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

by Vladimir E. Fedorov, Marina V. Plekhanova and Daria V. Melekhina

Fractal Fract. 2023, 7(6), 464; https://doi.org/10.3390/fractalfract7060464 - 07 Jun 2023

Cited by 1 | Viewed by 719

Abstract

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order [...] Read more.

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan–Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan–Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan–Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin–Voigt viscoelastic media. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

16 pages, 335 KiB

Open AccessArticle

Several Quantum Hermite–Hadamard-Type Integral Inequalities for Convex Functions

by Loredana Ciurdariu and Eugenia Grecu

Fractal Fract. 2023, 7(6), 463; https://doi.org/10.3390/fractalfract7060463 - 07 Jun 2023

Cited by 5 | Viewed by 995

Abstract

The aim of this study was to present several improved quantum Hermite–Hadamard-type integral inequalities for convex functions using a parameter. Thus, a new quantum identity is proven to be used as the main tool in the proof of our results. Consequently, in some [...] Read more.

The aim of this study was to present several improved quantum Hermite–Hadamard-type integral inequalities for convex functions using a parameter. Thus, a new quantum identity is proven to be used as the main tool in the proof of our results. Consequently, in some special cases several new quantum estimations for q-midpoints and q-trapezoidal-type inequalities are derived with an example. The results obtained could be applied in the optimization of several economic geology problems. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)

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14 pages, 313 KiB

Open AccessArticle

Approximate Controllability of Fractional Stochastic Evolution Inclusions with Non-Local Conditions

by Kinda Abuasbeh, Azmat Ullah Khan Niazi, Hafiza Maria Arshad, Muath Awadalla and Salma Trabelsi

Fractal Fract. 2023, 7(6), 462; https://doi.org/10.3390/fractalfract7060462 - 07 Jun 2023

Cited by 2 | Viewed by 834

Abstract

This article investigates the approximate controllability of non-linear fractional stochastic differential inclusions with non-local conditions. We establish a set of sufficient conditions for their approximate controllability and provide results in terms of controllability for the fractional stochastic control system. Our approach relies on [...] Read more.

This article investigates the approximate controllability of non-linear fractional stochastic differential inclusions with non-local conditions. We establish a set of sufficient conditions for their approximate controllability and provide results in terms of controllability for the fractional stochastic control system. Our approach relies on using fractional calculus and the fixed-point theorem for multiple-valued operators. Finally, we present an illustrative example to support our findings. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Evolution Equations and Related Topics)

13 pages, 2990 KiB

Open AccessArticle

Characteristics of New Stochastic Solitonic Solutions for the Chiral Type of Nonlinear Schrödinger Equation

by H. G. Abdelwahed, A. F. Alsarhana, E. K. El-Shewy and Mahmoud A. E. Abdelrahman

Fractal Fract. 2023, 7(6), 461; https://doi.org/10.3390/fractalfract7060461 - 05 Jun 2023

Cited by 3 | Viewed by 987

Abstract

The Wiener process was used to explore the (2 + 1)-dimensional chiral nonlinear Schrödinger equation (CNLSE). This model outlines the energy characteristics of quantum physics’ fractional Hall effect edge states. The sine-Gordon expansion technique (SGET) was implemented to extract stochastic solutions for the [...] Read more.

The Wiener process was used to explore the (2 + 1)-dimensional chiral nonlinear Schrödinger equation (CNLSE). This model outlines the energy characteristics of quantum physics’ fractional Hall effect edge states. The sine-Gordon expansion technique (SGET) was implemented to extract stochastic solutions for the CNLSE through multiplicative noise effects. This method accurately described a variety of solitary behaviors, including bright solitons, dark periodic envelopes, solitonic forms, and dissipative and dissipative–soliton-like waves, showing how the solutions changed as the values of the studied system’s physical parameters were changed. The stochastic parameter was shown to affect the damping, growth, and conversion effects on the bright (dark) envelope and shock-forced oscillatory wave energy, amplitudes, and frequencies. In addition, the intensity of noise resulted in enormous periodic envelope stochastic structures and shock-forced oscillatory behaviors. The proposed technique is applicable to various energy equations in the nonlinear applied sciences. Full article

(This article belongs to the Special Issue Mathematical Modelling of Real Phenomena Based on Fractional Derivatives)

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Fractal Fract., Volume 7, Issue 6 (June 2023) – 75 articles

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