Journal Description
Fractal and Fractional
Fractal and Fractional
is an international, scientific, peer-reviewed, open access journal of fractals and fractional calculus and their applications in different fields of science and engineering published monthly online by MDPI.
- Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
- High Visibility: indexed within Scopus, SCIE (Web of Science), Inspec, and other databases.
- Journal Rank: JCR - Q1 (Mathematics, Interdisciplinary Applications) / CiteScore - Q1 (Analysis)
- Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 18.9 days after submission; acceptance to publication is undertaken in 3.5 days (median values for papers published in this journal in the second half of 2023).
- Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Impact Factor:
5.4 (2022);
5-Year Impact Factor:
4.7 (2022)
Latest Articles
Optimizing Variational Problems through Weighted Fractional Derivatives
Fractal Fract. 2024, 8(5), 272; https://doi.org/10.3390/fractalfract8050272 (registering DOI) - 02 May 2024
Abstract
In this article, we explore a variety of problems within the domain of calculus of variations, specifically in the context of fractional calculus. The fractional derivative we consider incorporates the notion of weighted fractional derivatives along with derivatives with respect to another function.
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In this article, we explore a variety of problems within the domain of calculus of variations, specifically in the context of fractional calculus. The fractional derivative we consider incorporates the notion of weighted fractional derivatives along with derivatives with respect to another function. Besides the fractional operators, the Lagrange function depends on extremal points. We examine the fundamental problem, providing the fractional Euler–Lagrange equation and the associated transversality conditions. Both the isoperimetric and Herglotz problems are also explored. Finally, we conclude with an analysis of the variational problem, incorporating fractional derivatives of any positive real order.
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Open AccessArticle
Dynamic Analysis and Field-Programmable Gate Array Implementation of a 5D Fractional-Order Memristive Hyperchaotic System with Multiple Coexisting Attractors
by
Fei Yu, Wuxiong Zhang, Xiaoli Xiao, Wei Yao, Shuo Cai, Jin Zhang, Chunhua Wang and Yi Li
Fractal Fract. 2024, 8(5), 271; https://doi.org/10.3390/fractalfract8050271 - 01 May 2024
Abstract
On the basis of the chaotic system proposed by Wang et al. in 2023, this paper constructs a 5D fractional-order memristive hyperchaotic system (FOMHS) with multiple coexisting attractors through coupling of magnetic control memristors and dimension expansion. Firstly, the divergence, Kaplan–Yorke dimension, and
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On the basis of the chaotic system proposed by Wang et al. in 2023, this paper constructs a 5D fractional-order memristive hyperchaotic system (FOMHS) with multiple coexisting attractors through coupling of magnetic control memristors and dimension expansion. Firstly, the divergence, Kaplan–Yorke dimension, and equilibrium stability of the chaotic model are studied. Subsequently, we explore the construction of the 5D FOMHS, introducing the definitions of the Caputo differential operator and the Riemann–Liouville integral operator and employing the Adomian resolving approach to decompose the linears, the nonlinears, and the constants of the system. The complex dynamic characteristics of the system are analyzed by phase diagrams, Lyapunov exponent spectra, time-domain diagrams, etc. Finally, the hardware circuit of the proposed 5D FOMHS is performed by FPGA, and its randomness is verified using the NIST tool.
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(This article belongs to the Special Issue Advances in Fractional-Order Chaotic and Complex Systems)
Open AccessArticle
On the Controllability of Coupled Nonlocal Partial Integrodifferential Equations Using Fractional Power Operators
by
Hamida Litimein, Zhen-You Huang, Abdelghani Ouahab, Ivanka Stamova and Mohammed Said Souid
Fractal Fract. 2024, 8(5), 270; https://doi.org/10.3390/fractalfract8050270 - 30 Apr 2024
Abstract
In this research paper, we investigate the controllability in the -norm of a coupled system of integrodifferential equations with state-dependent nonlocal conditions in generalized Banach spaces. We establish sufficient conditions for the system’s controllability using resolvent operator theory introduced by Grimmer, fractional
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In this research paper, we investigate the controllability in the -norm of a coupled system of integrodifferential equations with state-dependent nonlocal conditions in generalized Banach spaces. We establish sufficient conditions for the system’s controllability using resolvent operator theory introduced by Grimmer, fractional power operators, and fixed-point theorems associated with generalized measures of noncompactness for condensing operators in vector Banach spaces. Finally, we present an application example to validate the proposed methodology in this research.
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(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional, 2nd Edition)
Open AccessArticle
Lie Symmetries and the Invariant Solutions of the Fractional Black–Scholes Equation under Time-Dependent Parameters
by
Sameerah Jamal, Reginald Champala and Suhail Khan
Fractal Fract. 2024, 8(5), 269; https://doi.org/10.3390/fractalfract8050269 - 29 Apr 2024
Abstract
In this paper, we consider the time-fractional Black–Scholes model with deterministic, time-varying coefficients. These time parametric constituents produce a model with greater flexibility that may capture empirical results from financial markets and their time-series datasets. We make use of transformations to reduce the
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In this paper, we consider the time-fractional Black–Scholes model with deterministic, time-varying coefficients. These time parametric constituents produce a model with greater flexibility that may capture empirical results from financial markets and their time-series datasets. We make use of transformations to reduce the underlying model to the classical heat transfer equation. We show that this transformation procedure is possible for a specific risk-free interest rate and volatility of stock function. Furthermore, we reverse these transformations and apply one-dimensional optimal subalgebras of the infinitesimal symmetry generators to establish invariant solutions.
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(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 2nd Edition)
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Utilizing Cubic B-Spline Collocation Technique for Solving Linear and Nonlinear Fractional Integro-Differential Equations of Volterra and Fredholm Types
by
Ishtiaq Ali, Muhammad Yaseen and Iqra Akram
Fractal Fract. 2024, 8(5), 268; https://doi.org/10.3390/fractalfract8050268 - 29 Apr 2024
Abstract
Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method
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Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method offers a robust and systematic framework for approximating solutions to the FIDEs, facilitating precise representations of complex phenomena. Within this research, we establish the mathematical foundations of the proposed scheme, elucidate its advantages over existing methods, and demonstrate its practical utility through numerical examples. We adopt the Caputo definition for fractional derivatives and conduct a stability analysis to validate the accuracy of the method. The findings showcase the precision and efficiency of the scheme in solving FIDEs, highlighting its potential as a valuable tool for addressing a wide array of practical problems.
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(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 2nd Edition)
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Open AccessArticle
Deep Learning-Based Detection of Human Blastocyst Compartments with Fractal Dimension Estimation
by
Muhammad Arsalan, Adnan Haider, Jin Seong Hong, Jung Soo Kim and Kang Ryoung Park
Fractal Fract. 2024, 8(5), 267; https://doi.org/10.3390/fractalfract8050267 - 28 Apr 2024
Abstract
In vitro fertilization (IVF) is an efficacious form of aided reproduction to deal with infertility. Human embryos are taken from the body, and these are kept in a supervised laboratory atmosphere during the IVF technique until they exhibit blastocyst properties. A human expert
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In vitro fertilization (IVF) is an efficacious form of aided reproduction to deal with infertility. Human embryos are taken from the body, and these are kept in a supervised laboratory atmosphere during the IVF technique until they exhibit blastocyst properties. A human expert manually analyzes the morphometric properties of the blastocyst and its compartments to predict viability through manual microscopic evaluation. A few deep learning-based approaches deal with this task via semantic segmentation, but they are inaccurate and use expensive architecture. To automatically detect the human blastocyst compartments, we propose a parallel stream fusion network (PSF-Net) that performs the semantic segmentation of embryo microscopic images with inexpensive shallow architecture. The PSF-Net has a shallow architecture that combines the benefits of feature aggregation through depth-wise concatenation and element-wise summation, which helps the network to provide accurate detection using 0.7 million trainable parameters only. In addition, we compute fractal dimension estimation for all compartments of the blastocyst, providing medical experts with significant information regarding the distributional characteristics of blastocyst compartments. An open dataset of microscopic images of the human embryo is used to evaluate the proposed approach. The proposed method also demonstrates promising segmentation performance for all compartments of the blastocyst compared with state-of-the-art methods, achieving a mean Jaccard index (MJI) of 87.69%. The effectiveness of PSF-Net architecture is also confirmed with the ablation studies.
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(This article belongs to the Special Issue Advances in Pattern Recognition—Image and Time Series Analyses—through Fractal Geometry and Complexity Theory)
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Open AccessArticle
Extreme Homogeneous and Heterogeneous Multistability in a Novel 5D Memristor-Based Chaotic System with Hidden Attractors
by
Chengwei Dong and Min Yang
Fractal Fract. 2024, 8(5), 266; https://doi.org/10.3390/fractalfract8050266 - 28 Apr 2024
Abstract
This paper proposes a novel five-dimensional (5D) memristor-based chaotic system by introducing a flux-controlled memristor into a 3D chaotic system with two stable equilibrium points, and increases the dimensionality utilizing the state feedback control method. The newly proposed memristor-based chaotic system has line
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This paper proposes a novel five-dimensional (5D) memristor-based chaotic system by introducing a flux-controlled memristor into a 3D chaotic system with two stable equilibrium points, and increases the dimensionality utilizing the state feedback control method. The newly proposed memristor-based chaotic system has line equilibrium points, so the corresponding attractor belongs to a hidden attractor. By using typical nonlinear analysis tools, the complicated dynamical behaviors of the new system are explored, which reveals many interesting phenomena, including extreme homogeneous and heterogeneous multistabilities, hidden transient state and state transition behavior, and offset-boosting control. Meanwhile, the unstable periodic orbits embedded in the hidden chaotic attractor were calculated by the variational method, and the corresponding pruning rules were summarized. Furthermore, the analog and DSP circuit implementation illustrates the flexibility of the proposed memristic system. Finally, the active synchronization of the memristor-based chaotic system was investigated, demonstrating the important engineering application values of the new system.
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(This article belongs to the Special Issue Fractional-Order Chaotic Systems and Circuits: Design, Modeling and Implementation)
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Open AccessArticle
New Study on the Controllability of Non-Instantaneous Impulsive Hilfer Fractional Neutral Stochastic Evolution Equations with Non-Dense Domain
by
Gunasekaran Gokul, Barakah Almarri, Sivajiganesan Sivasankar, Subramanian Velmurugan and Ramalingam Udhayakumar
Fractal Fract. 2024, 8(5), 265; https://doi.org/10.3390/fractalfract8050265 - 27 Apr 2024
Abstract
The purpose of this work is to investigate the controllability of non-instantaneous impulsive (NII) Hilfer fractional (HF) neutral stochastic evolution equations with a non-dense domain. We construct a new set of adequate assumptions for the existence of mild solutions using fractional calculus, semigroup
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The purpose of this work is to investigate the controllability of non-instantaneous impulsive (NII) Hilfer fractional (HF) neutral stochastic evolution equations with a non-dense domain. We construct a new set of adequate assumptions for the existence of mild solutions using fractional calculus, semigroup theory, stochastic analysis, and the fixed point theorem. Then, the discussion is driven by some suitable assumptions, including the Hille–Yosida condition without the compactness of the semigroup of the linear part. Finally, we provide examples to illustrate our main result.
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(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)
Open AccessArticle
Research on Pattern Dynamics Behavior of a Fractional Vegetation-Water Model in Arid Flat Environment
by
Xiao-Long Gao, Hao-Lu Zhang, Yu-Lan Wang and Zhi-Yuan Li
Fractal Fract. 2024, 8(5), 264; https://doi.org/10.3390/fractalfract8050264 - 27 Apr 2024
Abstract
In order to stop and reverse land degradation and curb the loss of biodiversity, the United Nations 2030 Agenda for Sustainable Development proposes to combat desertification. In this paper, a fractional vegetation–water model in an arid flat environment is studied. The pattern behavior
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In order to stop and reverse land degradation and curb the loss of biodiversity, the United Nations 2030 Agenda for Sustainable Development proposes to combat desertification. In this paper, a fractional vegetation–water model in an arid flat environment is studied. The pattern behavior of the fractional model is much more complex than that of the integer order. We study the stability and Turing instability of the system, as well as the Hopf bifurcation of fractional order , and obtain the Turing region in the parameter space. According to the amplitude equation, different types of stationary mode discoveries can be obtained, including point patterns and strip patterns. Finally, the results of the numerical simulation and theoretical analysis are consistent. We find some novel fractal patterns of the fractional vegetation–water model in an arid flat environment. When the diffusion coefficient, d, changes and other parameters remain unchanged, the pattern structure changes from stripes to spots. When the fractional order parameter, , changes, and other parameters remain unchanged, the pattern structure becomes more stable and is not easy to destroy. The research results can provide new ideas for the prevention and control of desertification vegetation patterns.
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(This article belongs to the Section Numerical and Computational Methods)
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Quasi-Projective Synchronization of Discrete-Time Fractional-Order Complex-Valued BAM Fuzzy Neural Networks via Quantized Control
by
Yingying Xu, Hongli Li, Jikai Yang and Long Zhang
Fractal Fract. 2024, 8(5), 263; https://doi.org/10.3390/fractalfract8050263 - 27 Apr 2024
Abstract
In this paper, we ponder a kind of discrete-time fractional-order complex-valued fuzzy BAM neural network. Firstly, in order to guarantee the quasi-projective synchronization of the considered networks, an original quantitative control strategy is designed. Next, by virtue of the relevant definitions and properties
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In this paper, we ponder a kind of discrete-time fractional-order complex-valued fuzzy BAM neural network. Firstly, in order to guarantee the quasi-projective synchronization of the considered networks, an original quantitative control strategy is designed. Next, by virtue of the relevant definitions and properties of the Mittag-Leffler function, we propose a novel discrete-time fractional-order Halanay inequality, which is more efficient for disposing of the discrete-time fractional-order models with time delays. Then, based on the new lemma, fractional-order h-difference theory, and comparison principle, we obtain some easy-to-verify synchronization criteria in terms of algebraic inequalities. Finally, numerical simulations are provided to check the accuracy of the proposed theoretical results.
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(This article belongs to the Special Issue Fractional Order Systems with Time Delay: Theory, Stability Analysis and Applications)
Open AccessArticle
Non-Polynomial Collocation Spectral Scheme for Systems of Nonlinear Caputo–Hadamard Differential Equations
by
Mahmoud A. Zaky, Ibrahem G. Ameen, Mohammed Babatin, Ali Akgül, Magda Hammad and António M. Lopes
Fractal Fract. 2024, 8(5), 262; https://doi.org/10.3390/fractalfract8050262 - 27 Apr 2024
Abstract
In this paper, we provide a collocation spectral scheme for systems of nonlinear Caputo–Hadamard differential equations. Since the Caputo–Hadamard operators contain logarithmic kernels, their solutions can not be well approximated using the usual spectral methods that are classical polynomial-based schemes. Hence, we construct
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In this paper, we provide a collocation spectral scheme for systems of nonlinear Caputo–Hadamard differential equations. Since the Caputo–Hadamard operators contain logarithmic kernels, their solutions can not be well approximated using the usual spectral methods that are classical polynomial-based schemes. Hence, we construct a non-polynomial spectral collocation scheme, describe its effective implementation, and derive its convergence analysis in both and . In addition, we provide numerical results to support our theoretical analysis.
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(This article belongs to the Special Issue Spectral Methods for Fractional Functional Models)
Open AccessArticle
Investigation into the Failure Characteristics and Mechanism of Rock with Single Elliptical Defects under Ultrasonic Vibrations
by
Zhijun Niu, Xufeng Wang, Lei Zhang, Jiyao Wang, Zechao Chang, Chenlong Qian and Xuyang Chen
Fractal Fract. 2024, 8(5), 261; https://doi.org/10.3390/fractalfract8050261 - 27 Apr 2024
Abstract
In order to investigate the effects of elliptical defects on rock failure under ultrasonic vibrations, ultrasonic vibration tests and PFC2D numerical simulations were conducted on rocks with single elliptical defects. The research results indicated that the fracture fractal dimension, axial strain, and
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In order to investigate the effects of elliptical defects on rock failure under ultrasonic vibrations, ultrasonic vibration tests and PFC2D numerical simulations were conducted on rocks with single elliptical defects. The research results indicated that the fracture fractal dimension, axial strain, and crack depth of specimens with elliptical defects at 45° and 90° were the smallest and largest, respectively. The corresponding strain and fractal dimension showed a positive linear and logarithmic function relationship with time. The maximum crack depth of 46.50 mm was observed on the specimens with an elliptical defect angle of 90°. Specimens with elliptical defects at 0°, 30°, 75°, and 90° exhibited more dense and frequent acoustic emission events than those with elliptical defects at 15°, 45°, and 60°. During the ultrasonic vibration process, the maximum total energy (87.86 kJ) and energy consumption coefficient (0.963) were observed on specimens with elliptical defect angles of 30° and 45°, respectively. The difference in the stress field led to varying degrees of plastic strain energy in the specimens, resulting in different forms of crack propagation and triggering differential acoustic emission events, ultimately leading to specimen failure with different crack shapes and depths. The fractal dimensions of elliptical defect specimens under ultrasonic vibration have a high degree of consistency with the changes in axial strain and failure depth, and the fractal dimension of defect specimens is positively correlated with the degree of failure of defect specimens.
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(This article belongs to the Special Issue Applications of Fractal Analysis in Underground Engineering)
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On Theoretical and Numerical Results of Serum Hepatitis Disease Using Piecewise Fractal–Fractional Perspectives
by
Zareen A. Khan, Arshad Ali, Ateeq Ur Rehman Irshad, Burhanettin Ozdemir and Hussam Alrabaiah
Fractal Fract. 2024, 8(5), 260; https://doi.org/10.3390/fractalfract8050260 - 26 Apr 2024
Abstract
In the present research, we consider a biological model of serum hepatitis disease. We carry out a detailed analysis of the mentioned model along with a class with asymptomatic carriers to explore its theoretical and numerical aspects. We initiate the study by using
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In the present research, we consider a biological model of serum hepatitis disease. We carry out a detailed analysis of the mentioned model along with a class with asymptomatic carriers to explore its theoretical and numerical aspects. We initiate the study by using the piecewise fractal–fractional derivative (FFD) by which the crossover effects within the model are examined. We split the time interval into subintervals. In one subinterval, FFD with a power law kernel is taken, while in the second one, FFD with an exponential decay kernel of the proposed model is considered. This model is then studied for its disease-free equilibrium point, existence, and Hyers–Ulam (H-U) stability results. For numerical results of the model and a visual presentation, we apply the Lagrange interpolation method and an extended Adams–Bashforth–Moulton (ABM) method, respectively.
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(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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Impulsive Control of Variable Fractional-Order Multi-Agent Systems
by
Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan
Fractal Fract. 2024, 8(5), 259; https://doi.org/10.3390/fractalfract8050259 - 26 Apr 2024
Abstract
The main goal of the paper is to present and study models of multi-agent systems for which the dynamics of the agents are described by a Caputo fractional derivative of variable order and a kernel that depends on an increasing function. Also, the
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The main goal of the paper is to present and study models of multi-agent systems for which the dynamics of the agents are described by a Caputo fractional derivative of variable order and a kernel that depends on an increasing function. Also, the order of the fractional derivative changes at update times. We study a case for which the exchanged information between agents occurs only at initially given update times. Two types of linear variable-order Caputo fractional models are studied. We consider both multi-agent systems without a leader and multi-agent systems with a leader. In the case of multi-agent systems without a leader, two types of models are studied. The main difference between the models is the fractional derivative describing the dynamics of agents. In the first one, a Caputo fractional derivative with respect to another function and with a continuous variable order is applied. In the second one, the applied fractional derivative changes its constant order at each update time. Mittag–Leffler stability via impulsive control is defined, and sufficient conditions are obtained. In the case of the presence of a leader in the multi-agent system, the dynamic of the agents is described by a Caputo fractional derivative with respect to an increasing function and with a constant order that changes at each update time. The leader-following consensus via impulsive control is defined, and sufficient conditions are derived. The theoretical results are illustrated with examples. We show with an example the leader’s influence on the consensus.
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(This article belongs to the Special Issue Advances in Fractional-Order Multiagent Systems: Theory and Applications)
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Dynamics and Complexity Analysis of Fractional-Order Inventory Management System Model
by
Tengfei Lei, Rita Yi Man Li, Jirawan Deeprasert and Haiyan Fu
Fractal Fract. 2024, 8(5), 258; https://doi.org/10.3390/fractalfract8050258 - 26 Apr 2024
Abstract
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To accurately depict inventory management over time, this paper introduces a fractional inventory management model that builds upon the existing classical inventory management framework. According to the definition of fractional difference equation, the numerical solution and phase diagram of an inventory management system
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To accurately depict inventory management over time, this paper introduces a fractional inventory management model that builds upon the existing classical inventory management framework. According to the definition of fractional difference equation, the numerical solution and phase diagram of an inventory management system are obtained by MATLAB simulation. The influence of parameters on the nonlinear behavior of the system is analyzed by a bifurcation diagram and largest Lyapunov exponent (LLE). Combined with the related indexes of time series, the complex characteristics of a quantization system are analyzed using spectral entropy and C0. This study concluded that the changing law of system complexity is consistent with the LLE of the system. By analyzing the influence of order on the system, it is found that the inventory changes will be periodic in some areas when the system is fractional, which is close to the actual conditions of the company’s inventory situation. The research results of this paper provide useful information for inventory managers to implement inventory and facility management strategies.
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Open AccessArticle
Some Properties of Normalized Tails of Maclaurin Power Series Expansions of Sine and Cosine
by
Tao Zhang, Zhen-Hang Yang, Feng Qi and Wei-Shih Du
Fractal Fract. 2024, 8(5), 257; https://doi.org/10.3390/fractalfract8050257 - 26 Apr 2024
Cited by 1
Abstract
In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and
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In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and concavity of the normalized tails, compute several special values of the Young function, the Lommel function, and a generalized hypergeometric function, recover two inequalities for the tails of the Maclaurin power series expansions of the sine and cosine functions, propose three open problems about the nonnegativity, positivity, decreasing property, and concavity of a newly introduced function which is a generalization of the normalized tails of the Maclaurin power series expansions of the sine and cosine functions. These results are related to the Riemann–Liouville fractional integrals.
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(This article belongs to the Section General Mathematics, Analysis)
Open AccessArticle
Event-Triggered Adaptive Neural Network Control for State-Constrained Pure-Feedback Fractional-Order Nonlinear Systems with Input Delay and Saturation
by
Changhui Wang, Jiaqi Yang and Mei Liang
Fractal Fract. 2024, 8(5), 256; https://doi.org/10.3390/fractalfract8050256 - 26 Apr 2024
Abstract
In this research, the adaptive event-triggered neural network controller design problem is investigated for a class of state-constrained pure-feedback fractional-order nonlinear systems (FONSs) with external disturbances, unknown actuator saturation, and input delay. An auxiliary compensation function based on the integral function of the
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In this research, the adaptive event-triggered neural network controller design problem is investigated for a class of state-constrained pure-feedback fractional-order nonlinear systems (FONSs) with external disturbances, unknown actuator saturation, and input delay. An auxiliary compensation function based on the integral function of the input signal is presented to handle input delay. The barrier Lyapunov function (BLF) is utilized to deal with state constraints, and the event-triggered strategy is applied to overcome the communication burden from the limited communication resources. By the utilization of a backstepping scheme and radial basis function neural network, an adaptive event-triggered neural state-feedback stabilization controller is constructed, in which the fractional-order dynamic surface filters are employed to reduce the computational burden from the recursive procedure. It is proven that with the fractional-order Lyapunov analysis, all the solutions of the closed-loop system are bounded, and the tracking error can converge to a small interval around the zero, while the state constraint is satisfied and the Zeno behavior can be strictly ruled out. Two examples are finally given to show the effectiveness of the proposed control strategy.
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(This article belongs to the Special Issue Fractional Order Systems with Time Delay: Theory, Stability Analysis and Applications)
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Constrained State Regulation Problem of Descriptor Fractional-Order Linear Continuous-Time Systems
by
Hongli Yang, Xindong Si and Ivan G. Ivanov
Fractal Fract. 2024, 8(5), 255; https://doi.org/10.3390/fractalfract8050255 - 25 Apr 2024
Abstract
This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order . The objective is to establish the existence of conditions for a linear feedback control law within state constraints
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This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order . The objective is to establish the existence of conditions for a linear feedback control law within state constraints and to propose a method for solving the CSRP of DFOLCS. First, based on the decomposition and separation method and coordinate transformation, the DFOLCS can be transformed into an equivalent fractional-order reduced system; hence, the CSRP of the DFOLCS is equivalent to the CSRP of the reduced system. By means of positive invariant sets theory, Lyapunov stability theory, and some mathematical techniques, necessary and sufficient conditions for the polyhedral positive invariant set of the equivalent reduced system are presented. Models and corresponding algorithms for solving the CSRP of a linear feedback controller are also presented by the obtained conditions. Under the condition that the resulting closed system is positive, the given model of the CSRP in this paper for the DFOLCS is formulated as nonlinear programming with a linear objective function and quadratic mixed constraints. Two numerical examples illustrate the proposed method.
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(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)
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Representations of Solutions of Time-Fractional Multi-Order Systems of Differential-Operator Equations
by
Sabir Umarov
Fractal Fract. 2024, 8(5), 254; https://doi.org/10.3390/fractalfract8050254 - 25 Apr 2024
Abstract
This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational
[...] Read more.
This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties.
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(This article belongs to the Section General Mathematics, Analysis)
Open AccessArticle
Conformal and Non-Minimal Couplings in Fractional Cosmology
by
Kevin Marroquín, Genly Leon, Alfredo D. Millano, Claudio Michea and Andronikos Paliathanasis
Fractal Fract. 2024, 8(5), 253; https://doi.org/10.3390/fractalfract8050253 - 25 Apr 2024
Abstract
Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus
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Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus must reflect non-local, frequency- and history-dependent properties of power-law phenomena. This tool has various important applications, such as fractional mass conservation, electrochemical analysis, groundwater flow problems, and fractional spatiotemporal diffusion equations. It can also be used in cosmology to explain late-time cosmic acceleration without the need for dark energy. We review some models using fractional differential equations. We look at the Einstein–Hilbert action, which is based on a fractional derivative action, and add a scalar field, , to create a non-minimal interaction theory with the coupling, , between gravity and the scalar field, where is the interaction constant. By employing various mathematical approaches, we can offer precise schemes to find analytical and numerical approximations of the solutions. Moreover, we comprehensively study the modified cosmological equations and analyze the solution space using the theory of dynamical systems and asymptotic expansion methods. This enables us to provide a qualitative description of cosmologies with a scalar field based on fractional calculus formalism.
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(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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Topics
Topic in
Algorithms, Computation, Entropy, Fractal Fract, MCA
Analytical and Numerical Methods for Stochastic Biological Systems
Topic Editors: Mehmet Yavuz, Necati Ozdemir, Mouhcine Tilioua, Yassine SabbarDeadline: 10 May 2024
Topic in
Energies, Environments, Fractal Fract, Materials, Remote Sensing
Geomechanics for Energy and the Environment
Topic Editors: Gan Feng, Ang Liu, Reza Taherdangkoo, Qiao LyuDeadline: 31 May 2024
Topic in
Algorithms, Axioms, Fractal Fract, Mathematics, Symmetry
Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems
Topic Editors: Xiaofeng Wang, Fazlollah SoleymaniDeadline: 30 June 2024
Topic in
Applied Sciences, Energies, Fractal Fract, Polymers, Resources
Petroleum and Gas Engineering
Topic Editors: Xiaochun Wang, Yulong ZhaoDeadline: 31 August 2024
Conferences
Special Issues
Special Issue in
Fractal Fract
New Trends on Generalized Fractional Calculus
Guest Editors: Milton Ferreira, Maria Manuela Fernandes Rodrigues, Nelson Felipe Loureiro VieiraDeadline: 15 May 2024
Special Issue in
Fractal Fract
Fractional Gravity/Cosmology in Classical and Quantum Regimes
Guest Editors: Seyed Meraj Mousavi Rasouli, Shahram JalalzadehDeadline: 31 May 2024
Special Issue in
Fractal Fract
Applications of Fractional-Order Calculus in Robotics
Guest Editors: Abhaya Pal Singh, Kishore BingiDeadline: 15 June 2024
Special Issue in
Fractal Fract
Feature Papers for Numerical and Computational Methods Section
Guest Editor: Riccardo CaponettoDeadline: 25 June 2024