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
On Theoretical and Numerical Results of Serum Hepatitis Disease Using Piecewise Fractal–Fractional Perspectives
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.
Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
Open AccessArticle
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 (registering DOI) - 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.
Full article
(This article belongs to the Special Issue Advances in Fractional-Order Multiagent Systems: Theory and Applications)
Open AccessArticle
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
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.
Full article
(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.
Full article
(This article belongs to the Special Issue Fractional Order Systems with Time Delay: Theory, Stability Analysis and Applications)
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Open AccessArticle
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.
Full article
(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)
Open AccessArticle
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
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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.
Full article
(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 (registering DOI) - 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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Analysis of the Fractal Dimension, b-value, Slip Ratio, and Decay Rate of Aftershock Seismicity Following the 6 February 2023 (Mw 7.8 and 7.5) Türkiye Earthquakes
by
Sherif M. Ali and Kamal Abdelrahman
Fractal Fract. 2024, 8(5), 252; https://doi.org/10.3390/fractalfract8050252 - 25 Apr 2024
Abstract
On 6 February 2023, Türkiye experienced a pair of consecutive earthquakes with magnitudes of Mw 7.8 and 7.5, and accompanied by an intense aftershock sequence. These seismic events were particularly impactful on the segments of the East Anatolian Fault Zone (EAFZ), causing extensive
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On 6 February 2023, Türkiye experienced a pair of consecutive earthquakes with magnitudes of Mw 7.8 and 7.5, and accompanied by an intense aftershock sequence. These seismic events were particularly impactful on the segments of the East Anatolian Fault Zone (EAFZ), causing extensive damage to both human life and urban centers in Türkiye and Syria. This study explores the analysis of a dataset spanning almost one year following the Turkiye mainshocks, including 471 events with a magnitude of completeness (Mc) ≥ 4.4. We employed the maximum likelihood approach to estimate the b-value and Omori-Utsu parameters (K, c, and p-values). The estimated b-value is 1.21 ± 0.1, indicating that the mainshocks occurred in a region characterized by elevated stress levels, leading to a sequence of aftershocks of larger magnitudes due to notable irregularities in the rupture zone. The aftershock decay rate (p-value = 1.1 ± 0.04) indicates a rapid decrease in stress levels following the main shocks. However, the c-value of 0.204 ± 0.058 would indicate a relatively moderate or low initial productivity of aftershocks. Furthermore, the k-value of 76.75 ± 8.84 suggests that the decay of aftershock activity commenced within a range of approximately 68 to 86 days following the mainshocks. The fractal dimension (Dc) was assessed using the correlation integral method, yielding a value of 0.99 ± 0.03. This implies a tendency toward clustering in the aftershock seismicity and a linear configuration of the epicenters. The slip ratio during the aftershock activity was determined to be 0.75, signifying that 75% of the total slip occurred in the primary rupture, with the remaining fraction distributed among secondary faults. The methodologies and insights acquired in this research can be extended to assist in forecasting aftershock occurrences for future earthquakes, thus offering crucial data for future risk assessment.
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(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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A Novel Application of Fractional Order Derivative Moth Flame Optimization Algorithm for Solving the Problem of Optimal Coordination of Directional Overcurrent Relays
by
Abdul Wadood and Herie Park
Fractal Fract. 2024, 8(5), 251; https://doi.org/10.3390/fractalfract8050251 - 25 Apr 2024
Abstract
The proper coordination of directional overcurrent relays (DOCRs) is crucial in electrical power systems. The coordination of DOCRs in a multi-loop power system is expressed as an optimization problem. The aim of this study focuses on improving the protection system’s performance by minimizing
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The proper coordination of directional overcurrent relays (DOCRs) is crucial in electrical power systems. The coordination of DOCRs in a multi-loop power system is expressed as an optimization problem. The aim of this study focuses on improving the protection system’s performance by minimizing the total operating time of DOCRs via effective coordination with main and backup DOCRs while keeping the coordination constraints within allowable limits. The coordination problem of DOCRs is solved by developing a new application strategy called Fractional Order Derivative Moth Flame Optimizer (FODMFO). This approach involves incorporating the ideas of fractional calculus (FC) into the mathematical model of the conventional moth flame algorithm to improve the characteristics of the optimizer. The FODMFO approach is then tested on the coordination problem of DOCRs in standard power systems, specifically the IEEE 3, 8, and 15 bus systems as well as in 11 benchmark functions including uni- and multimodal functions. The results obtained from the proposed method, as well as its comparison with other recently developed algorithms, demonstrate that the combination of FOD and MFO improves the overall efficiency of the optimizer by utilizing the individual strengths of these tools and identifying the globally optimal solution and minimize the total operating time of DOCRs up to an optimal value. The reliability, strength, and dependability of FODMFO are supported by a thorough statistics study using the box-plot, histograms, empirical cumulative distribution function demonstrations, and the minimal fitness evolution seen in each distinct simulation. Based on these data, it is evident that FODMFO outperforms other modern nature-inspired and conventional algorithms.
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(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)
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Existence and Uniqueness Result for Fuzzy Fractional Order Goursat Partial Differential Equations
by
Muhammad Sarwar, Noor Jamal, Kamaleldin Abodayeh, Chanon Promsakon and Thanin Sitthiwirattham
Fractal Fract. 2024, 8(5), 250; https://doi.org/10.3390/fractalfract8050250 - 25 Apr 2024
Abstract
In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s -differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s -differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert
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In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s -differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s -differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert Goursat problems to equivalent systems fuzzy integral equations to deal properly with the mixed derivative term and two types of Caputo’s -differentiability. In this study, we utilize the concept of metric fixed point theory to discuss the existence of a unique solution of fractional fuzzy Goursat problems. For the useability of established theoretical work, we provide some numerical problems. We also discuss the solutions to numerical problems by conformable double Laplace transform. To show the validity of the solutions we provide 3D plots. We discuss, as an application, why fractional partial fuzzy differential equations are the generalization of usual partial fuzzy differential equations by providing a suitable reason. Moreover, we show the advantages of the proposed fractional transform over the usual Laplace transform.
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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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Finite-Time Adaptive Event-Triggered Control for Full States Constrained FONSs with Uncertain Parameters and Disturbances
by
Changhui Wang, Wencheng Li and Mei Liang
Fractal Fract. 2024, 8(5), 249; https://doi.org/10.3390/fractalfract8050249 - 25 Apr 2024
Abstract
This article focuses the event-triggered adaptive finite-time control scheme for the states constrained fractional-order nonlinear systems (FONSs) under uncertain parameters and external disturbances. The backstepping scheme is employed to construct the finite-time controller via a series of barrier Lyapunov function (BLF) to solve
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This article focuses the event-triggered adaptive finite-time control scheme for the states constrained fractional-order nonlinear systems (FONSs) under uncertain parameters and external disturbances. The backstepping scheme is employed to construct the finite-time controller via a series of barrier Lyapunov function (BLF) to solve that all the state constraints are not violated. Different from the trigger condition with fixed value, the event-triggered strategy is applied to overcome the communication burden of controller caused by the limited communication resources. By utilizing fractional-order Lyapunov analysis, all variables in the resulted system are proven to be bounded, and the tracking error converges to the small neighborhood around origin in finite time and without the Zeno behavior. Finally, the effectiveness of the proposed control scheme is verified by the simulation analysis of a bus power system.
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(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control, 2nd Edition)
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Impressive Exact Solitons to the Space-Time Fractional Mathematical Physics Model via an Effective Method
by
Abdulaziz Khalid Alsharidi and Moin-ud-Din Junjua
Fractal Fract. 2024, 8(5), 248; https://doi.org/10.3390/fractalfract8050248 - 24 Apr 2024
Abstract
A new class of truncated M-fractional exact soliton solutions for a mathematical physics model known as a truncated M-fractional (1+1)-dimensional nonlinear modified mixed-KdV model are achieved. We obtain these solutions by using a modified extended direct algebraic method. The obtained results consist of
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A new class of truncated M-fractional exact soliton solutions for a mathematical physics model known as a truncated M-fractional (1+1)-dimensional nonlinear modified mixed-KdV model are achieved. We obtain these solutions by using a modified extended direct algebraic method. The obtained results consist of trigonometric, hyperbolic trigonometric and mixed functions. We also discuss the effect of fractional order derivative. To validate our results, we utilized the Mathematica software. Additionally, we depict some of the obtained kink, periodic, singular, and kink-singular wave solitons, using two and three dimensional graphs. The obtained results are useful in the fields of fluid dynamics, nonlinear optics, ocean engineering and others. Furthermore, these employed techniques are not only straightforward, but also highly effective when used to solve non-linear fractional partial differential equations (FPDEs).
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(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)
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Multi-Scale Research on the Mechanisms of Soil Arching Development and Degradation in Granular Materials with Different Relative Density
by
Luju Liang, Yi Pik Cheng, Xiaozhen Fan, Zhi Ding and Changjie Xu
Fractal Fract. 2024, 8(5), 247; https://doi.org/10.3390/fractalfract8050247 - 24 Apr 2024
Abstract
Soil arching is significantly influenced by relative density, while its mechanisms have barely been analyzed. A series of DEM numerical simulations of the classical trapdoor test were carried out to investigate the multi-scale mechanisms of arching development and degradation in granular materials with
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Soil arching is significantly influenced by relative density, while its mechanisms have barely been analyzed. A series of DEM numerical simulations of the classical trapdoor test were carried out to investigate the multi-scale mechanisms of arching development and degradation in granular materials with different relative density. For analysis, the granular assembly was divided into three zones according to the particle vertical displacement normalized by the trapdoor displacement δ. The results show that before the maximum arching state (corresponding to the minimum arching ratio), contact forces between particles in a specific zone (where the vertical displacement of particles is larger than 0.1δ but less than 0.9δ) increase rapidly and robust arched force chains with large particle contact forces are generated. The variation in contact forces and force chains becomes more obvious as the sample porosity decreases. As a result, soil arching generated in a denser particle assembly is stronger, and the minimum value of the arching ratio is increased with the sample porosity. After the maximum arching state, the force chains in this zone are degenerated gradually, leading to a decrease in particle contact forces in microscale and an increase in the arching ratio in macroscale. The recovery of the arching ratio after the minimum value is also more significant in simulations with a larger relative density, as the degeneration of contact force chains is more obvious in denser samples. These results indicate the importance of contact force chain stabilities in specific zones for improving soil arching in engineering practice.
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(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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Leader-Following Formation Control for Discrete-Time Fractional Stochastic Multi-Agent Systems by Event-Triggered Strategy
by
Jiawei Wu, Yongguang Yu and Guojian Ren
Fractal Fract. 2024, 8(5), 246; https://doi.org/10.3390/fractalfract8050246 - 23 Apr 2024
Abstract
Fractional differential equations, which are non-local and can better describe memory and genetic properties, are widely used to describe various physical, chemical, and biological phenomena. Therefore, the multi-agent systems based on discrete-time fractional stochastic models are established. First, some followers are selected for
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Fractional differential equations, which are non-local and can better describe memory and genetic properties, are widely used to describe various physical, chemical, and biological phenomena. Therefore, the multi-agent systems based on discrete-time fractional stochastic models are established. First, some followers are selected for pinning control. In order to save resources and energy, an event-triggered based control mechanism is proposed. Second, under this control mechanism, sufficient conditions on the interaction graph and the fractional derivative order such that formation control can be achieved are given. Additionally, influenced by noise, the multi-agent system completes formation control in the mean square. In addition to that, these results are equally applicable to the discrete-time fractional formation problem without noise. Finally, the example of numerical simulation is given to prove the correctness of the results.
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(This article belongs to the Special Issue Synchronization and Adaptive Control for Fractional-Order Network Systems)
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Heterogeneity of Micro- and Nanopore Structure of Lacustrine Shales with Complex Lamina Structure
by
Shuning Liu, Juncheng Qiao, Jianhui Zeng, Chuanming Li, Yazhou Liu, Zheng Kong and Xinlong Liu
Fractal Fract. 2024, 8(4), 245; https://doi.org/10.3390/fractalfract8040245 - 22 Apr 2024
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Thin sections, AIM-SEM, MICP, and nitrogen adsorption were performed on laminated and layered shales to characterize their complex pore and fracture structure. Combining the MICP model with the FHH model, this work proposes a new fractal method for lacustrine shales with complex lamina
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Thin sections, AIM-SEM, MICP, and nitrogen adsorption were performed on laminated and layered shales to characterize their complex pore and fracture structure. Combining the MICP model with the FHH model, this work proposes a new fractal method for lacustrine shales with complex lamina structure. The fractal characteristics presented four zones, representing the heterogeneity of fractures, macropores, mesopores, and micropores. The pores and fractures of shale have strong heterogeneity. Laminated shale has strong heterogeneity in mesopores and moderate heterogeneity in micropores. Layered shale has strong heterogeneity in fractures and moderate heterogeneity in micropores. The lamina structure and content of organic and mineral composition has a great influence on heterogeneity. The mineral laminae in laminated shale change frequently; lamellation fractures are mainly developed, and the structures are similar. Layered shales develop fractures between layers and structural fractures; the structural differences are significant. Macropores are mostly interparticle pores between quarts with similar structures. The wider lamina thickness of layered shale provides sufficient crystallization space for minerals, so the mesopores of layered shale are more homogeneous. Micropores are less developed, mainly consisting of intraparticle pores between clay minerals, which are complex but similar in structure in the two types of shale. The heterogeneity of mesopores and micropores is not conducive to hydrocarbon migration. Fractures and macropores need to be connected with meso–micropores to form a transport system. So, mesopores and micropores play decisive roles in hydrocarbon migration. Based on the above understanding, this paper points out that hydrocarbon in laminated shale with more carbonate minerals and a high thermal evolution degree has better availability.
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Open AccessArticle
Robust Fractional-Order PI/PD Controllers for a Cascade Control Structure of Servo Systems
by
Vo Lam Chuong, Ngo Hong Nam, Le Hieu Giang and Truong Nguyen Luan Vu
Fractal Fract. 2024, 8(4), 244; https://doi.org/10.3390/fractalfract8040244 - 22 Apr 2024
Abstract
In this paper, a cascade control structure is suggested to control servo systems that normally include a servo motor in coupling with two kinds of mechanism elements, a translational or rotational movement. These kinds of systems have high demands for performance in terms
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In this paper, a cascade control structure is suggested to control servo systems that normally include a servo motor in coupling with two kinds of mechanism elements, a translational or rotational movement. These kinds of systems have high demands for performance in terms of fastest response and no overshoot/oscillation to a ramp function input. The fractional-order proportional integral (FOPI) and proportional derivative (FOPD) controllers are addressed to deal with those control problems due to their flexibility in tuning rules and robustness. The tuning rules are designed in the frequency domain based on the concept of the direct synthesis method and also ensure the robust stability of controlled systems by using the maximum sensitivity function. The M-Δ structure, using multiplicative output uncertainties for both control loops simultaneously, is addressed to justify the robustness of the controlled systems. Simulation studies are considered for two kinds of plants that prove the effectiveness of the proposed method, with good tracking of the ramp function input under the effects of the disturbances. In addition, the robustness of the controlled system is illustrated by a structured singular value (µ) plot in which its value is less than 1 over the frequency range.
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(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)
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Fixed Point Theorems: Exploring Applications in Fractional Differential Equations for Economic Growth
by
Afrah Ahmad Noman Abdou
Fractal Fract. 2024, 8(4), 243; https://doi.org/10.3390/fractalfract8040243 - 22 Apr 2024
Abstract
The aim of this research is to introduce two new notions, -( ,h)-contraction and rational ( , - -interpolative contraction, in the setting of -metric space and to establish corresponding fixed point theorems. To
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The aim of this research is to introduce two new notions, -( ,h)-contraction and rational ( , - -interpolative contraction, in the setting of -metric space and to establish corresponding fixed point theorems. To reinforce understanding and highlight the novelty of our findings, we provide a non-trivial example that not only supports the obtained results but also illuminates the established theory. Finally, we apply our main result to discuss the existence and uniqueness of solutions for a fractional differential equation describing an economic growth model.
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(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)
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Using Fractal Theory to Study the Influence of Movable Oil on the Pore Structure of Different Types of Shale: A Case Study of the Fengcheng Formation Shale in Well X of Mahu Sag, Junggar Basin, China
by
Hong Zhang, Zhengchen Zhang, Zhenlin Wang, Yamin Wang, Rui Yang, Tao Zhu, Feifei Luo and Kouqi Liu
Fractal Fract. 2024, 8(4), 242; https://doi.org/10.3390/fractalfract8040242 - 20 Apr 2024
Abstract
This study investigated the influence of movable oil on the pore structure of various shale types, analyzing 19 shale samples from Well X in the Mahu Sag of the Junggar Basin. Initially, X-ray diffraction (XRD) analysis classified the shale samples. Subsequently, the geochemical
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This study investigated the influence of movable oil on the pore structure of various shale types, analyzing 19 shale samples from Well X in the Mahu Sag of the Junggar Basin. Initially, X-ray diffraction (XRD) analysis classified the shale samples. Subsequently, the geochemical properties and pore structures of the samples, both pre and post oil Soxhlet extraction, were comparatively analyzed through Total Organic Carbon (TOC) content measurement, Rock-Eval pyrolysis, and nitrogen adsorption experiments. Additionally, fractal theory quantitatively described the impact of movable oil on the pore structure of different shale types. Results indicated higher movable oil content in siliceous shale compared to calcareous shale. Oil extraction led to a significant increase in specific surface area and pore volume in all samples, particularly in siliceous shale. Calcareous shale predominantly displays H2–H3 type hysteresis loops, indicating a uniform pore structure with ink-bottle-shaped pores. Conversely, siliceous shale exhibited diverse hysteresis loops, reflecting its complex pore structure. The fractal dimension in calcareous shale correlated primarily with pore structure, exhibiting no significant correlation with TOC content before or after oil extraction. Conversely, the fractal dimension changes in siliceous shale samples do not have a clear correlation with either TOC content or pore structure, suggesting variations may result from both TOC and pore structure.
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(This article belongs to the Special Issue Flow and Transport in Fractal Models of Rock Mechanics)
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Open AccessArticle
Unravelling the Fractal Complexity of Temperature Datasets across Indian Mainland
by
Adarsh Sankaran, Thomas Plocoste, Arathy Nair Geetha Raveendran Nair and Meera Geetha Mohan
Fractal Fract. 2024, 8(4), 241; https://doi.org/10.3390/fractalfract8040241 - 20 Apr 2024
Abstract
Studying atmospheric temperature characteristics is crucial under climate change, as it helps us to understand the changing patterns in temperature that have significant implications for the environment, ecosystems, and human well-being. This study presents the comprehensive analysis of the spatiotemporal variability of scaling
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Studying atmospheric temperature characteristics is crucial under climate change, as it helps us to understand the changing patterns in temperature that have significant implications for the environment, ecosystems, and human well-being. This study presents the comprehensive analysis of the spatiotemporal variability of scaling behavior of daily temperature series across the whole Indian mainland, using a Multifractal Detrended Fluctuation Analysis (MFDFA). The analysis considered 1° × 1° datasets of maximum temperature (Tmax), minimum temperature (Tmin), mean temperature (Tmean), and diurnal temperature range (DTR) (TDTR = Tmax − Tmin) from 1951 to 2016 to compare their scaling behavior for the first time. Our results indicate that the Tmin series exhibits the highest persistence (with the Hurst exponent ranging from 0.849 to unity, and a mean of 0.971), and all four-temperature series display long-term persistence and multifractal characteristics. The variability of the multifractal characteristics is less significant in North–Central India, while it is highest along the western coast of India. Moreover, the assessment of multifractal characteristics of different temperature series during the pre- and post-1976–1977 period of the Pacific climate shift reveals a notable decrease in multifractal strength and persistence in the post-1976–1977 series across all regions. Moreover, for the detection of climate change and its dominant driver, we propose a new rolling window multifractal (RWM) framework by evaluating the temporal evolution of the spectral exponents and the Hurst exponent. This study successfully captured the regime shifts during the periods of 1976–1977 and 1997–1998. Interestingly, the earlier climatic shift primarily mitigated the persistence of the Tmax series, whereas the latter shift significantly influenced the persistence of the Tmean series in the majority of temperature-homogeneous regions in India.
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(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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