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Fractal Fract., Volume 5, Issue 4 (December 2021) – 151 articles

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14 pages, 1358 KiB  
Article
An Implicit Numerical Approach for 2D Rayleigh Stokes Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative
by Anam Naz, Umair Ali, Ashraf Elfasakhany, Khadiga Ahmed Ismail, Abdullah G. Al-Sehemi and Ahmed A. Al-Ghamdi
Fractal Fract. 2021, 5(4), 283; https://doi.org/10.3390/fractalfract5040283 - 20 Dec 2021
Cited by 5 | Viewed by 2103
Abstract
In this research work, our aim is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) involving Riemann–Liouville time fractional derivative. We suggest the modified implicit scheme formulated in the Riemann–Liouville integral sense and the scheme [...] Read more.
In this research work, our aim is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) involving Riemann–Liouville time fractional derivative. We suggest the modified implicit scheme formulated in the Riemann–Liouville integral sense and the scheme can be applied to the fractional RSP-HGSGF. Numerical experiments will be conducted, to show that the scheme is stress-free to implement, and the outcomes reveal the ideal execution of the suggested technique. The Fourier series will be used to examine the proposed scheme stability and convergence. The technique is stable, and the approximation solution converges to the exact result. To demonstrate the applicability and viability of the suggested strategy, a numerical demonstration will be provided. Full article
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11 pages, 289 KiB  
Article
Further Integral Inequalities through Some Generalized Fractional Integral Operators
by Abd-Allah Hyder, Mohamed A. Barakat, Ashraf Fathallah and Clemente Cesarano
Fractal Fract. 2021, 5(4), 282; https://doi.org/10.3390/fractalfract5040282 - 20 Dec 2021
Cited by 12 | Viewed by 2087
Abstract
In this article, we utilize recent generalized fractional operators to establish some fractional inequalities in Hermite–Hadamard and Minkowski settings. It is obvious that many previously published inequalities can be derived as particular cases from our outcomes. Moreover, we articulate some flaws in the [...] Read more.
In this article, we utilize recent generalized fractional operators to establish some fractional inequalities in Hermite–Hadamard and Minkowski settings. It is obvious that many previously published inequalities can be derived as particular cases from our outcomes. Moreover, we articulate some flaws in the proofs of recently affiliated formulas by revealing the weak points and introducing more rigorous proofs amending and expanding the results. Full article
2 pages, 167 KiB  
Editorial
The Craft of Fractional Modelling in Science and Engineering: II and III
by Jordan Hristov
Fractal Fract. 2021, 5(4), 281; https://doi.org/10.3390/fractalfract5040281 - 20 Dec 2021
Cited by 1 | Viewed by 1655
Abstract
A comprehensive understanding of fractional systems plays a pivotal role in practical applications [...] Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)
25 pages, 346 KiB  
Article
Capacity and the Corresponding Heat Semigroup Characterization from Dunkl-Bounded Variation
by Xiangling Meng, Yu Liu and Xiangyun Xie
Fractal Fract. 2021, 5(4), 280; https://doi.org/10.3390/fractalfract5040280 - 18 Dec 2021
Viewed by 1700
Abstract
In this paper, we study some important basic properties of Dunkl-bounded variation functions. In particular, we derive a way of approximating Dunkl-bounded variation functions by smooth functions and establish a version of the Gauss–Green Theorem. We also establish the Dunkl BV capacity and [...] Read more.
In this paper, we study some important basic properties of Dunkl-bounded variation functions. In particular, we derive a way of approximating Dunkl-bounded variation functions by smooth functions and establish a version of the Gauss–Green Theorem. We also establish the Dunkl BV capacity and investigate some measure theoretic properties, moreover, we show that the Dunkl BV capacity and the Hausdorff measure of codimension one have the same null sets. Finally, we develop the characterization of a heat semigroup of the Dunkl-bounded variation space, thereby giving its relation to the functions of Dunkl-bounded variation. Full article
(This article belongs to the Section General Mathematics, Analysis)
19 pages, 330 KiB  
Article
A Study on Controllability of a Class of Impulsive Fractional Nonlinear Evolution Equations with Delay in Banach Spaces
by Daliang Zhao
Fractal Fract. 2021, 5(4), 279; https://doi.org/10.3390/fractalfract5040279 - 17 Dec 2021
Cited by 7 | Viewed by 1752
Abstract
Under a new generalized definition of exact controllability we introduced and with a appropriately constructed time delay term in a special complete space to overcome the delay-induced-difficulty, we establish the sufficient conditions of the exact controllability for a class of impulsive fractional nonlinear [...] Read more.
Under a new generalized definition of exact controllability we introduced and with a appropriately constructed time delay term in a special complete space to overcome the delay-induced-difficulty, we establish the sufficient conditions of the exact controllability for a class of impulsive fractional nonlinear evolution equations with delay by using the resolvent operator theory and the theory of nonlinear functional analysis. Nonlinearity in the system is only supposed to be continuous rather than Lipschitz continuous by contrast. The results obtained in the present work are generalizations and continuations of the recent results on this issue. Further, an example is presented to show the effectiveness of the new results. Full article
16 pages, 9452 KiB  
Article
A Measurement of Visual Complexity for Heterogeneity in the Built Environment Based on Fractal Dimension and Its Application in Two Gardens
by Lan Ma, Shaoying He and Mingzhen Lu
Fractal Fract. 2021, 5(4), 278; https://doi.org/10.3390/fractalfract5040278 - 17 Dec 2021
Cited by 9 | Viewed by 2815
Abstract
In this study, a fractal dimension-based method has been developed to compute the visual complexity of the heterogeneity in the built environment. The built environment is a very complex combination, structurally consisting of both natural and artificial elements. Its fractal dimension computation is [...] Read more.
In this study, a fractal dimension-based method has been developed to compute the visual complexity of the heterogeneity in the built environment. The built environment is a very complex combination, structurally consisting of both natural and artificial elements. Its fractal dimension computation is often disturbed by the homogenous visual redundancy, which is textured but needs less attention to process, so that it leads to a pseudo-evaluation of visual complexity in the built environment. Based on human visual perception, the study developed a method: fractal dimension of heterogeneity in the built environment, which includes Potts segmentation and Canny edge detection as image preprocessing procedure and fractal dimension as computation procedure. This proposed method effectively extracts perceptually meaningful edge structures in the visual image and computes its visual complexity which is consistent with human visual characteristics. In addition, an evaluation system combining the proposed method and the traditional method has been established to classify and assess the visual complexity of the scenario more comprehensively. Two different gardens had been computed and analyzed to demonstrate that the proposed method and the evaluation system provide a robust and accurate way to measure the visual complexity in the built environment. Full article
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17 pages, 1952 KiB  
Article
Swarm Intelligence Procedures Using Meyer Wavelets as a Neural Network for the Novel Fractional Order Pantograph Singular System
by Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Juan L. G. Guirao and Tareq Saeed
Fractal Fract. 2021, 5(4), 277; https://doi.org/10.3390/fractalfract5040277 - 17 Dec 2021
Cited by 7 | Viewed by 1844
Abstract
The purpose of the current investigation is to find the numerical solutions of the novel fractional order pantograph singular system (FOPSS) using the applications of Meyer wavelets as a neural network. The FOPSS is presented using the standard form of the Lane–Emden equation [...] Read more.
The purpose of the current investigation is to find the numerical solutions of the novel fractional order pantograph singular system (FOPSS) using the applications of Meyer wavelets as a neural network. The FOPSS is presented using the standard form of the Lane–Emden equation and the detailed discussions of the singularity, shape factor terms along with the fractional order forms. The numerical discussions of the FOPSS are described based on the fractional Meyer wavelets (FMWs) as a neural network (NN) with the optimization procedures of global/local search procedures of particle swarm optimization (PSO) and interior-point algorithm (IPA), i.e., FMWs-NN-PSOIPA. The FMWs-NN strength is pragmatic and forms a merit function based on the differential system and the initial conditions of the FOPSS. The merit function is optimized, using the integrated capability of PSOIPA. The perfection, verification and substantiation of the FOPSS using the FMWs is pragmatic for three cases through relative investigations from the true results in terms of stability and convergence. Additionally, the statics’ descriptions further authorize the presentation of the FMWs-NN-PSOIPA in terms of reliability and accuracy. Full article
(This article belongs to the Special Issue Numerical Methods and Simulations in Fractal and Fractional Problems)
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24 pages, 4562 KiB  
Article
Lower and Upper Bounds of Fractional Metric Dimension of Connected Networks
by Muhammad Javaid, Muhammad Kamran Aslam, Muhammad Imran Asjad, Bander N. Almutairi, Mustafa Inc and Bandar Almohsen
Fractal Fract. 2021, 5(4), 276; https://doi.org/10.3390/fractalfract5040276 - 15 Dec 2021
Viewed by 1837
Abstract
The distance centric parameter in the theory of networks called by metric dimension plays a vital role in encountering the distance-related problems for the monitoring of the large-scale networks in the various fields of chemistry and computer science such as navigation, image processing, [...] Read more.
The distance centric parameter in the theory of networks called by metric dimension plays a vital role in encountering the distance-related problems for the monitoring of the large-scale networks in the various fields of chemistry and computer science such as navigation, image processing, pattern recognition, integer programming, optimal transportation models and drugs discovery. In particular, it is used to find the locations of robots with respect to shortest distance among the destinations, minimum consumption of time, lesser number of the utilized nodes, and to characterize the chemical compounds, having unique presentations in molecular networks. After the arrival of its weighted version, known as fractional metric dimension, the rectification of distance-related problems in the aforementioned fields has revived to a great extent. In this article, we compute fractional as well as local fractional metric dimensions of web-related networks called by subdivided QCL, 2-faced web, 3-faced web, and antiprism web networks. Moreover, we analyse their final results using 2D and 3D plots. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
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18 pages, 339 KiB  
Article
Some Double Generalized Weighted Fractional Integral Inequalities Associated with Monotone Chebyshev Functionals
by Gauhar Rahman, Saud Fahad Aldosary, Muhammad Samraiz and Kottakkaran Sooppy Nisar
Fractal Fract. 2021, 5(4), 275; https://doi.org/10.3390/fractalfract5040275 - 15 Dec 2021
Cited by 2 | Viewed by 1610
Abstract
In this manuscript, we study the unified integrals recently defined by Rahman et al. and present some new double generalized weighted type fractional integral inequalities associated with increasing, positive, monotone and measurable function F. Also, we establish some new double-weighted inequalities, which [...] Read more.
In this manuscript, we study the unified integrals recently defined by Rahman et al. and present some new double generalized weighted type fractional integral inequalities associated with increasing, positive, monotone and measurable function F. Also, we establish some new double-weighted inequalities, which are particular cases of the main result and are represented by corollaries. These inequalities are further refinement of all other inequalities associated with increasing, positive, monotone and measurable function existing in literature. The existing inequalities associated with increasing, positive, monotone and measurable function are also restored by applying specific conditions as given in Remarks. Many other types of fractional integral inequalities can be obtained by applying certain conditions on F and Ψ given in the literature. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
14 pages, 846 KiB  
Article
A Mixed Element Algorithm Based on the Modified L1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model
by Jinfeng Wang, Baoli Yin, Yang Liu, Hong Li and Zhichao Fang
Fractal Fract. 2021, 5(4), 274; https://doi.org/10.3390/fractalfract5040274 - 14 Dec 2021
Cited by 3 | Viewed by 1939
Abstract
In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with [...] Read more.
In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified L1-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal L2 error estimates are performed and the feasibility is validated by the calculated data. Full article
(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)
13 pages, 583 KiB  
Article
Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel
by Iván Area and Juan J. Nieto
Fractal Fract. 2021, 5(4), 273; https://doi.org/10.3390/fractalfract5040273 - 14 Dec 2021
Cited by 16 | Viewed by 2810
Abstract
In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated [...] Read more.
In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series. Full article
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17 pages, 1524 KiB  
Article
Fractals as Julia Sets of Complex Sine Function via Fixed Point Iterations
by Swati Antal, Anita Tomar, Darshana J. Prajapati and Mohammad Sajid
Fractal Fract. 2021, 5(4), 272; https://doi.org/10.3390/fractalfract5040272 - 14 Dec 2021
Cited by 11 | Viewed by 3335
Abstract
We explore some new variants of the Julia set by developing the escape criteria for a function sin(zn)+az+c, where a,cC, n2, and z is a [...] Read more.
We explore some new variants of the Julia set by developing the escape criteria for a function sin(zn)+az+c, where a,cC, n2, and z is a complex variable, utilizing four distinct fixed point iterative methods. Furthermore, we examine the impact of parameters on the deviation of dynamics, color, and appearance of fractals. Some of these fractals represent the stunning art on glass, and Rangoli (made in different parts of India, especially during the festive season) which are useful in interior decoration. Some fractals are similar to beautiful objects found in our surroundings like flowers (to be specific Hibiscus and Catharanthus Roseus), and ants. Full article
(This article belongs to the Special Issue Convergence and Dynamics of Iterative Methods: Chaos and Fractals)
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14 pages, 4638 KiB  
Article
A Comprehensive Mathematical Model for SARS-CoV-2 in Caputo Derivative
by Yu Gu, Muhammad Altaf Khan, Y. S. Hamed and Bassem F. Felemban
Fractal Fract. 2021, 5(4), 271; https://doi.org/10.3390/fractalfract5040271 - 14 Dec 2021
Cited by 13 | Viewed by 1926
Abstract
In the present work, we study the COVID-19 infection through a new mathematical model using the Caputo derivative. The model has all the possible interactions that are responsible for the spread of disease in the community. We first formulate the model in classical [...] Read more.
In the present work, we study the COVID-19 infection through a new mathematical model using the Caputo derivative. The model has all the possible interactions that are responsible for the spread of disease in the community. We first formulate the model in classical differential equations and then extend it into fractional differential equations using the definition of the Caputo derivative. We explore in detail the stability results for the model of the disease-free case when R0<1. We show that the model is stable locally when R0<1. We give the result that the model is globally asymptotically stable whenever R01. Further, to estimate the model parameters, we consider the real data of the fourth wave from Pakistan and provide a reasonable fitting to the data. We estimate the basic reproduction number for the proposed data to be R0=1.0779. Moreover, using the real parameters, we present the numerical solution by first giving a reliable scheme that can numerically handle the solution of the model. In our simulation, we give the graphical results for some sensitive parameters that have a large impact on disease elimination. Our results show that taking into consideration all the possible interactions can describe COVID-19 infection. Full article
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20 pages, 865 KiB  
Article
Nonlocal Boundary Value Problems of Nonlinear Fractional (p,q)-Difference Equations
by Pheak Neang, Kamsing Nonlaopon, Jessada Tariboon, Sotiris K. Ntouyas and Bashir Ahmad
Fractal Fract. 2021, 5(4), 270; https://doi.org/10.3390/fractalfract5040270 - 10 Dec 2021
Cited by 7 | Viewed by 1975
Abstract
In this paper, we study nonlinear fractional (p,q)-difference equations equipped with separated nonlocal boundary conditions. The existence of solutions for the given problem is proven by applying Krasnoselskii’s fixed-point theorem and the Leray–Schauder alternative. In contrast, the uniqueness [...] Read more.
In this paper, we study nonlinear fractional (p,q)-difference equations equipped with separated nonlocal boundary conditions. The existence of solutions for the given problem is proven by applying Krasnoselskii’s fixed-point theorem and the Leray–Schauder alternative. In contrast, the uniqueness of the solutions is established by employing Banach’s contraction mapping principle. Examples illustrating the main results are also presented. Full article
17 pages, 344 KiB  
Article
Hermite–Jensen–Mercer-Type Inequalities via Caputo–Fabrizio Fractional Integral for h-Convex Function
by Miguel Vivas-Cortez, Muhammad Shoaib Saleem, Sana Sajid, Muhammad Sajid Zahoor and Artion Kashuri
Fractal Fract. 2021, 5(4), 269; https://doi.org/10.3390/fractalfract5040269 - 10 Dec 2021
Cited by 15 | Viewed by 2242
Abstract
Integral inequalities involving many fractional integral operators are used to solve various fractional differential equations. In the present paper, we will generalize the Hermite–Jensen–Mercer-type inequalities for an h-convex function via a Caputo–Fabrizio fractional integral. We develop some novel Caputo–Fabrizio fractional integral inequalities. [...] Read more.
Integral inequalities involving many fractional integral operators are used to solve various fractional differential equations. In the present paper, we will generalize the Hermite–Jensen–Mercer-type inequalities for an h-convex function via a Caputo–Fabrizio fractional integral. We develop some novel Caputo–Fabrizio fractional integral inequalities. We also present Caputo–Fabrizio fractional integral identities for differentiable mapping, and these will be used to give estimates for some fractional Hermite–Jensen–Mercer-type inequalities. Some familiar results are recaptured as special cases of our results. Full article
(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)
17 pages, 480 KiB  
Article
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 - 9 Dec 2021
Cited by 8 | Viewed by 2163
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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23 pages, 842 KiB  
Article
Rational Approximations of Arbitrary Order: A Survey
by José Daniel Colín-Cervantes, Carlos Sánchez-López, Rocío Ochoa-Montiel, Delia Torres-Muñoz, Carlos Manuel Hernández-Mejía, Luis Abraham Sánchez-Gaspariano and Hugo Gustavo González-Hernández
Fractal Fract. 2021, 5(4), 267; https://doi.org/10.3390/fractalfract5040267 - 8 Dec 2021
Cited by 24 | Viewed by 3124
Abstract
This paper deals with the study and analysis of several rational approximations to approach the behavior of arbitrary-order differentiators and integrators in the frequency domain. From the Riemann–Liouville, Grünwald–Letnikov and Caputo basic definitions of arbitrary-order calculus until the reviewed approximation methods, each of [...] Read more.
This paper deals with the study and analysis of several rational approximations to approach the behavior of arbitrary-order differentiators and integrators in the frequency domain. From the Riemann–Liouville, Grünwald–Letnikov and Caputo basic definitions of arbitrary-order calculus until the reviewed approximation methods, each of them is coded in a Maple 18 environment and their behaviors are compared. For each approximation method, an application example is explained in detail. The advantages and disadvantages of each approximation method are discussed. Afterwards, two model order reduction methods are applied to each rational approximation and assist a posteriori during the synthesis process using analog electronic design or reconfigurable hardware. Examples for each reduction method are discussed, showing the drawbacks and benefits. To wrap up, this survey is very useful for beginners to get started quickly and learn arbitrary-order calculus and then to select and tune the best approximation method for a specific application in the frequency domain. Once the approximation method is selected and the rational transfer function is generated, the order can be reduced by applying a model order reduction method, with the target of facilitating the electronic synthesis. Full article
(This article belongs to the Special Issue Fractional-Order Circuit Theory and Applications)
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17 pages, 338 KiB  
Article
Qualitative Analysis of Langevin Integro-Fractional Differential Equation under Mittag–Leffler Functions Power Law
by Mohammed A. Almalahi, F. Ghanim, Thongchai Botmart, Omar Bazighifan and Sameh Askar
Fractal Fract. 2021, 5(4), 266; https://doi.org/10.3390/fractalfract5040266 - 8 Dec 2021
Cited by 13 | Viewed by 1951
Abstract
This research paper intends to investigate some qualitative analysis for a nonlinear Langevin integro-fractional differential equation. We investigate the sufficient conditions for the existence and uniqueness of solutions for the proposed problem using Banach’s and Krasnoselskii’s fixed point theorems. Furthermore, we discuss different [...] Read more.
This research paper intends to investigate some qualitative analysis for a nonlinear Langevin integro-fractional differential equation. We investigate the sufficient conditions for the existence and uniqueness of solutions for the proposed problem using Banach’s and Krasnoselskii’s fixed point theorems. Furthermore, we discuss different types of stability results in the frame of Ulam–Hyers by using a mathematical analysis approach. The obtained results are illustrated by presenting a numerical example. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Fractional Differential Problems)
20 pages, 495 KiB  
Article
Non-Debye Relaxations: The Ups and Downs of the Stretched Exponential vs. Mittag–Leffler’s Matchings
by Katarzyna Górska, Andrzej Horzela and Karol A. Penson
Fractal Fract. 2021, 5(4), 265; https://doi.org/10.3390/fractalfract5040265 - 7 Dec 2021
Cited by 6 | Viewed by 1999
Abstract
Experimental data collected to provide us with information on the course of dielectric relaxation phenomena are obtained according to two distinct schemes: one can measure either the time decay of depolarization current or use methods of the broadband dielectric spectroscopy. Both sets of [...] Read more.
Experimental data collected to provide us with information on the course of dielectric relaxation phenomena are obtained according to two distinct schemes: one can measure either the time decay of depolarization current or use methods of the broadband dielectric spectroscopy. Both sets of data are usually fitted by time or frequency dependent functions which, in turn, may be analytically transformed among themselves using the Laplace transform. This leads to the question on comparability of results obtained using just mentioned experimental procedures. If we would like to do that in the time domain we have to go beyond widely accepted Kohlrausch–Williams–Watts approximation and become acquainted with description using the Mittag–Leffler functions. To convince the reader that the latter is not difficult to understand we propose to look at the problem from the point of view of objects which appear in the stochastic processes approach to relaxation. These are the characteristic exponents which are read out from the standard non-Debye frequency dependent patterns. Characteristic functions appear to be expressed in terms of elementary functions whose asymptotics is simple. This opens new possibility to compare behavior of functions used to describe non-Debye relaxations. It turnes out that the use of Mittag-Leffler function proves very convenient for such a comparison. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
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23 pages, 9260 KiB  
Article
Dual Penta-Compound Combination Anti-Synchronization with Analysis and Application to a Novel Fractional Chaotic System
by Lone Seth Jahanzaib, Pushali Trikha, Rajaa T. Matoog, Shabbir Muhammad, Ahmed Al-Ghamdi and Mahmoud Higazy
Fractal Fract. 2021, 5(4), 264; https://doi.org/10.3390/fractalfract5040264 - 7 Dec 2021
Cited by 5 | Viewed by 2024
Abstract
This paper studies a fractional-order chaotic system with sine non-linearities and highlights its dynamics using the Lyapunov spectrum, bifurcation analysis, stagnation points, the solution of the system, the impact of the fractional order on the system, etc. The system considering uncertainties and disturbances [...] Read more.
This paper studies a fractional-order chaotic system with sine non-linearities and highlights its dynamics using the Lyapunov spectrum, bifurcation analysis, stagnation points, the solution of the system, the impact of the fractional order on the system, etc. The system considering uncertainties and disturbances was synchronized using dual penta-compound combination anti-synchronization among four master systems and twenty slave systems by non-linear control and the adaptive sliding mode technique. The estimates of the disturbances and uncertainties were also obtained using the sliding mode technique. The application of the achieved synchronization in secure communication is illustrated with the help of an example. Full article
(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application)
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6 pages, 263 KiB  
Article
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 - 7 Dec 2021
Cited by 3 | Viewed by 1962
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)
9 pages, 1133 KiB  
Article
The Influence of Noise on the Exact Solutions of the Stochastic Fractional-Space Chiral Nonlinear Schrödinger Equation
by Wael W. Mohammed, Omar Bazighifan, Mohammed M. Al-Sawalha, A. Othman Almatroud and Elkhateeb S. Aly
Fractal Fract. 2021, 5(4), 262; https://doi.org/10.3390/fractalfract5040262 - 7 Dec 2021
Cited by 19 | Viewed by 2330
Abstract
In this paper, we consider the stochastic fractional-space Chiral nonlinear Schrödinger equation (S-FS-CNSE) derived via multiplicative noise. We obtain the exact solutions of the S-FS-CNSE by using the Riccati equation method. The obtained solutions are extremely important in the development of nuclear medicine, [...] Read more.
In this paper, we consider the stochastic fractional-space Chiral nonlinear Schrödinger equation (S-FS-CNSE) derived via multiplicative noise. We obtain the exact solutions of the S-FS-CNSE by using the Riccati equation method. The obtained solutions are extremely important in the development of nuclear medicine, the entire computer industry and quantum mechanics, especially in the quantum hall effect. Moreover, we discuss how the multiplicative noise affects the exact solutions of the S-FS-CNSE. This equation has never previously been studied using a combination of multiplicative noise and fractional space. Full article
(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application)
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16 pages, 786 KiB  
Article
Fractional-Order Modelling and Optimal Control of Cholera Transmission
by Silvério Rosa and Delfim F. M. Torres
Fractal Fract. 2021, 5(4), 261; https://doi.org/10.3390/fractalfract5040261 - 7 Dec 2021
Cited by 15 | Viewed by 2531
Abstract
A Caputo-type fractional-order mathematical model for “metapopulation cholera transmission” was recently proposed in [Chaos Solitons Fractals 117 (2018), 37–49]. A sensitivity analysis of that model is done here to show the accuracy relevance of parameter estimation. Then, a fractional optimal control (FOC) problem [...] Read more.
A Caputo-type fractional-order mathematical model for “metapopulation cholera transmission” was recently proposed in [Chaos Solitons Fractals 117 (2018), 37–49]. A sensitivity analysis of that model is done here to show the accuracy relevance of parameter estimation. Then, a fractional optimal control (FOC) problem is formulated and numerically solved. A cost-effectiveness analysis is performed to assess the relevance of studied control measures. Moreover, such analysis allows us to assess the cost and effectiveness of the control measures during intervention. We conclude that the FOC system is more effective only in part of the time interval. For this reason, we propose a system where the derivative order varies along the time interval, being fractional or classical when more advantageous. Such variable-order fractional model, that we call a FractInt system, shows to be the most effective in the control of the disease. Full article
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15 pages, 619 KiB  
Article
Quasi-Projective Synchronization of Distributed-Order Recurrent Neural Networks
by Xiao Liu, Kelin Li, Qiankun Song and Xujun Yang
Fractal Fract. 2021, 5(4), 260; https://doi.org/10.3390/fractalfract5040260 - 6 Dec 2021
Cited by 5 | Viewed by 2060
Abstract
In this paper, the quasi-projective synchronization of distributed-order recurrent neural networks is investigated. Firstly, based on the definition of the distributed-order derivative and metric space theory, two distributed-order differential inequalities are obtained. Then, by employing the Lyapunov method, Laplace transform, Laplace final value [...] Read more.
In this paper, the quasi-projective synchronization of distributed-order recurrent neural networks is investigated. Firstly, based on the definition of the distributed-order derivative and metric space theory, two distributed-order differential inequalities are obtained. Then, by employing the Lyapunov method, Laplace transform, Laplace final value theorem, and some inequality techniques, the quasi-projective synchronization sufficient conditions for distributed-order recurrent neural networks are established in cases of feedback control and hybrid control schemes, respectively. Finally, two numerical examples are given to verify the effectiveness of the theoretical results. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
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9 pages, 277 KiB  
Article
Asymptotic and Oscillatory Properties of Noncanonical Delay Differential Equations
by Osama Moaaz, Clemente Cesarano and Sameh Askar
Fractal Fract. 2021, 5(4), 259; https://doi.org/10.3390/fractalfract5040259 - 6 Dec 2021
Cited by 2 | Viewed by 1713
Abstract
In this work, by establishing new asymptotic properties of non-oscillatory solutions of the even-order delay differential equation, we obtain new criteria for oscillation. The new criteria provide better results when determining the values of coefficients that correspond to oscillatory solutions. To explain the [...] Read more.
In this work, by establishing new asymptotic properties of non-oscillatory solutions of the even-order delay differential equation, we obtain new criteria for oscillation. The new criteria provide better results when determining the values of coefficients that correspond to oscillatory solutions. To explain the significance of our results, we apply them to delay differential equation of Euler-type. Full article
20 pages, 353 KiB  
Article
Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary
by Areej Bin Sultan, Mohamed Jleli and Bessem Samet
Fractal Fract. 2021, 5(4), 258; https://doi.org/10.3390/fractalfract5040258 - 6 Dec 2021
Cited by 3 | Viewed by 2002
Abstract
We first consider the damped wave inequality [...] Read more.
We first consider the damped wave inequality 2ut22ux2+utxσ|u|p,t>0,x(0,L), where L>0, σR, and p>1, under the Dirichlet boundary conditions (u(t,0),u(t,L))=(f(t),g(t)),t>0. We establish sufficient conditions depending on σ, p, the initial conditions, and the boundary conditions, under which the considered problem admits no global solution. Two cases of boundary conditions are investigated: g0 and g(t)=tγ, γ>1. Next, we extend our study to the time-fractional analogue of the above problem, namely, the time-fractional damped wave inequality αutα2ux2+βutβxσ|u|p,t>0,x(0,L), where α(1,2), β(0,1), and τtτ is the time-Caputo fractional derivative of order τ, τ{α,β}. Our approach is based on the test function method. Namely, a judicious choice of test functions is made, taking in consideration the boundedness of the domain and the boundary conditions. Comparing with previous existing results in the literature, our results hold without assuming that the initial values are large with respect to a certain norm. Full article
27 pages, 25650 KiB  
Article
Chaos Control for a Fractional-Order Jerk System via Time Delay Feedback Controller and Mixed Controller
by Changjin Xu, Maoxin Liao, Peiluan Li, Lingyun Yao, Qiwen Qin and Youlin Shang
Fractal Fract. 2021, 5(4), 257; https://doi.org/10.3390/fractalfract5040257 - 5 Dec 2021
Cited by 36 | Viewed by 2549
Abstract
In this study, we propose a novel fractional-order Jerk system. Experiments show that, under some suitable parameters, the fractional-order Jerk system displays a chaotic phenomenon. In order to suppress the chaotic behavior of the fractional-order Jerk system, we design two control strategies. Firstly, [...] Read more.
In this study, we propose a novel fractional-order Jerk system. Experiments show that, under some suitable parameters, the fractional-order Jerk system displays a chaotic phenomenon. In order to suppress the chaotic behavior of the fractional-order Jerk system, we design two control strategies. Firstly, we design an appropriate time delay feedback controller to suppress the chaos of the fractional-order Jerk system. The delay-independent stability and bifurcation conditions are established. Secondly, we design a suitable mixed controller, which includes a time delay feedback controller and a fractional-order PDσ controller, to eliminate the chaos of the fractional-order Jerk system. The sufficient condition ensuring the stability and the creation of Hopf bifurcation for the fractional-order controlled Jerk system is derived. Finally, computer simulations are executed to verify the feasibility of the designed controllers. The derived results of this study are absolutely new and possess potential application value in controlling chaos in physics. Moreover, the research approach also enriches the chaos control theory of fractional-order dynamical system. Full article
(This article belongs to the Special Issue Fractional Order Systems and Their Applications)
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16 pages, 339 KiB  
Article
Asymptotic Separation of Solutions to Fractional Stochastic Multi-Term Differential Equations
by Arzu Ahmadova and Nazim I. Mahmudov
Fractal Fract. 2021, 5(4), 256; https://doi.org/10.3390/fractalfract5040256 - 4 Dec 2021
Cited by 2 | Viewed by 1692
Abstract
In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs). Our goal in this paper is to establish results of the global existence and uniqueness and continuity dependence of the initial [...] Read more.
In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs). Our goal in this paper is to establish results of the global existence and uniqueness and continuity dependence of the initial values of the solutions to Caputo SMTDEs with non-permutable matrices of order α(12,1) and β(0,1) whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show the asymptotic separation property between two different solutions of Caputo SMTDEs with a more general condition based on λ. Furthermore, the asymptotic separation rate for the two distinct mild solutions reveals that our asymptotic results are general. Full article
13 pages, 24660 KiB  
Article
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 - 4 Dec 2021
Cited by 2 | Viewed by 1955
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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11 pages, 804 KiB  
Article
Random Times for Markov Processes with Killing
by Yuri G. Kondratiev and José Luís da Silva
Fractal Fract. 2021, 5(4), 254; https://doi.org/10.3390/fractalfract5040254 - 4 Dec 2021
Viewed by 1775
Abstract
We consider random time changes in Markov processes with killing potentials. We study how random time changes may be introduced in these Markov processes with killing potential and how these changes may influence their time behavior. As applications, we study the parabolic Anderson [...] Read more.
We consider random time changes in Markov processes with killing potentials. We study how random time changes may be introduced in these Markov processes with killing potential and how these changes may influence their time behavior. As applications, we study the parabolic Anderson problem, the non-local Schrödinger operators as well as the generalized Anderson problem. Full article
(This article belongs to the Special Issue Probabilistic Method in Fractional Calculus)
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