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
Investigation of Well-Posedness for a Direct Problem for a Nonlinear Fractional Diffusion Equation and an Inverse Problem
Fractal Fract. 2024, 8(6), 315; https://doi.org/10.3390/fractalfract8060315 (registering DOI) - 26 May 2024
Abstract
In this paper, we consider a direct problem and an inverse problem involving a nonlinear fractional diffusion equation, which can be applied to many physical situations. The equation contains a Caputo fractional derivative, a symmetric uniformly elliptic operator and a source term consisting
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In this paper, we consider a direct problem and an inverse problem involving a nonlinear fractional diffusion equation, which can be applied to many physical situations. The equation contains a Caputo fractional derivative, a symmetric uniformly elliptic operator and a source term consisting of the sum of two terms, one of which is linear and the other is nonlinear. The well-posedness of the direct problem is examined and the results are used to investigate the stability of an inverse problem of determining a function in the linear part of the source. The main tools in our study are the generalized eigenfunction expansions theory for nonlinear fractional diffusion equations, contraction mapping, Young’s convolution and generalized Grönwall’s inequalities. We present a stability estimate for the solution of the inverse source problem by means of observation data at a given point in the domain.
Full article
(This article belongs to the Special Issue Recent Advances in the Equation with Nonlinear Fractional Diffusion)
Open AccessArticle
Correlation between Agglomerates Hausdorff Dimension and Mechanical Properties of Denture Poly(methyl methacrylate)-Based Composites
by
Houda Taher Elhmali, Cristina Serpa, Vesna Radojevic, Aleksandar Stajcic, Milos Petrovic, Ivona Jankovic-Castvan and Ivana Stajcic
Fractal Fract. 2024, 8(6), 314; https://doi.org/10.3390/fractalfract8060314 (registering DOI) - 26 May 2024
Abstract
The microstructure–property relationship in poly(methyl methacrylate) PMMA composites is very important for understanding interface phenomena and the future prediction of properties that further help in designing improved materials. In this research, field emission scanning electron microscopy (FESEM) images of denture PMMA composites with
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The microstructure–property relationship in poly(methyl methacrylate) PMMA composites is very important for understanding interface phenomena and the future prediction of properties that further help in designing improved materials. In this research, field emission scanning electron microscopy (FESEM) images of denture PMMA composites with SrTiO3, MnO2 and SrTiO3/MnO2 were used for fractal reconstructions of particle agglomerates in the polymer matrix. Fractal analysis represents a valuable mathematical tool for the characterization of the microstructure and finding correlation between microstructural features and mechanical properties. Utilizing the mathematical affine fractal regression model, the Fractal Real Finder software was employed to reconstruct agglomerate shapes and estimate the Hausdorff dimensions (HD). Controlled energy impact and tensile tests were used to evaluate the mechanical performance of PMMA-MnO2, PMMA-SrTiO3 and PMMA-SrTiO3/MnO2 composites. It was determined that PMMA-SrTiO3/MnO2 had the highest total absorbed energy value (Etot), corresponding to the lowest HD value of 1.03637 calculated for SrTiO3/MnO2 agglomerates. On the other hand, the highest HD value of 1.21521 was calculated for MnO2 agglomerates, while the PMMA-MnO2 showed the lowest Etot. The linear correlation between the total absorbed impact energy of composites and the HD of the corresponding agglomerates was determined, with an R2 value of 0.99486, showing the potential use of this approach in the optimization of composite materials’ microstructure–property relationship.
Full article
(This article belongs to the Special Issue Advanced Research in Fractal Properties of Nanoparticle and Its Application)
Open AccessArticle
New Perturbation–Iteration Algorithm for Nonlinear Heat Transfer of Fractional Order
by
Mohammad Abdel Aal
Fractal Fract. 2024, 8(6), 313; https://doi.org/10.3390/fractalfract8060313 (registering DOI) - 25 May 2024
Abstract
Ordinary differential equations have recently been extended to fractional equations that are transformed using fractional differential equations. These fractional equations are believed to have high accuracy and low computational cost compared to ordinary differential equations. For the first time, this paper focuses on
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Ordinary differential equations have recently been extended to fractional equations that are transformed using fractional differential equations. These fractional equations are believed to have high accuracy and low computational cost compared to ordinary differential equations. For the first time, this paper focuses on extending the nonlinear heat equations to a fractional order in a Caputo order. A new perturbation iteration algorithm (PIA) of the fractional order is applied to solve the nonlinear heat equations. Solving numerical problems that involve fractional differential equations can be challenging due to their inherent complexity and high computational cost. To overcome these challenges, there is a need to develop numerical schemes such as the PIA method. This method can provide approximate solutions to problems that involve classical fractional derivatives. The results obtained from this algorithm are compared with those obtained from the perturbation iteration method (PIM), the variational iteration method (VIM), and the Bezier curve method (BCM). All solutions are tested with numerical simulations. The study found that the new PIA algorithm performs better than the PIM, VIM, and BCM, achieving high accuracy and low computational cost. One significant advantage of this algorithm is that the solutions obtained have established that the fractional values of alpha, specifically , significantly influencing the accuracy of the outcome and the associated computational cost.
Full article
(This article belongs to the Section Mathematical Physics)
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Multifractal Properties of Human Chromosome Sequences
by
J. P. Correia, R. Silva, D. H. A. L. Anselmo, M. S. Vasconcelos and L. R. da Silva
Fractal Fract. 2024, 8(6), 312; https://doi.org/10.3390/fractalfract8060312 - 24 May 2024
Abstract
The intricacy and fractal properties of human DNA sequences are examined in this work. The core of this study is to discern whether complete DNA sequences present distinct complexity and fractal attributes compared with sequences containing exclusively exon regions. In this regard, the
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The intricacy and fractal properties of human DNA sequences are examined in this work. The core of this study is to discern whether complete DNA sequences present distinct complexity and fractal attributes compared with sequences containing exclusively exon regions. In this regard, the entire base pair sequences of DNA are extracted from the NCBI (National Center for Biotechnology Information) database. In order to create a time series representation for the base pair sequence , we use the Chaos Game Representation (CGR) approach and a mapping rule f, which enables us to apply the metric known as the Complexity–Entropy Plane (CEP) and multifractal detrended fluctuation analysis (MF-DFA). To carry out our investigation, we divided human DNA into two groups: the first is composed of the 24 chromosomes, which comprises all the base pairs that form the DNA sequence, and another group that also includes the 24 chromosomes, but the DNA sequences rely only on the exons’ presence. The results show that both sets provide fractal patterns in their structure, as obtained by the CGR approach. Complete DNA sequences show a sharper visual fractal pattern than sequences composed only of exons. Moreover, the sequences occupy distinct areas of the complexity–entropy plane, and the complete DNA sequences lead to greater statistical complexity and lower entropy than the exon sequences. Also, we observed that different fractal parameters between chromosomes indicate diversity in genomic sequences. All these results occur in different scales for all chromosomes.
Full article
(This article belongs to the Special Issue Modern Methods for Fractal and Multifractal Analysis of Time Series: Theoretical Frameworks and Practical Applications)
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Open AccessArticle
The Existence and Ulam Stability Analysis of a Multi-Term Implicit Fractional Differential Equation with Boundary Conditions
by
Peiguang Wang, Bing Han and Junyan Bao
Fractal Fract. 2024, 8(6), 311; https://doi.org/10.3390/fractalfract8060311 - 24 May 2024
Abstract
In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the
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In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the given equation, and the solution is unique. Afterwards, we give the criteria of Ulam–Hyers stability and Ulam–Hyers–Rassias stability. Additionally, we present an example to illustrate the practical application and effectiveness of the results.
Full article
(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)
Open AccessArticle
Multiple Normalized Solutions to a Choquard Equation Involving Fractional p-Laplacian in RN
by
Xin Zhang and Sihua Liang
Fractal Fract. 2024, 8(6), 310; https://doi.org/10.3390/fractalfract8060310 - 23 May 2024
Abstract
In this paper, we study the existence of multiple normalized solutions for a Choquard equation involving fractional p-Laplacian in . With the help of variational methods, minimization techniques, and the Lusternik–Schnirelmann category, the existence of multiple normalized solutions is obtained
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In this paper, we study the existence of multiple normalized solutions for a Choquard equation involving fractional p-Laplacian in . With the help of variational methods, minimization techniques, and the Lusternik–Schnirelmann category, the existence of multiple normalized solutions is obtained for the above problem.
Full article
(This article belongs to the Special Issue Variational Problems and Fractional Differential Equations)
Open AccessArticle
Fractional Second-Grade Fluid Flow over a Semi-Infinite Plate by Constructing the Absorbing Boundary Condition
by
Jingyu Yang, Lin Liu, Siyu Chen, Libo Feng and Chiyu Xie
Fractal Fract. 2024, 8(6), 309; https://doi.org/10.3390/fractalfract8060309 - 23 May 2024
Abstract
The modified second-grade fluid flow across a plate of semi-infinite extent, which is initiated by the plate’s movement, is considered herein. The relaxation parameters and fractional parameters are introduced to express the generalized constitutive relation. A convolution-based absorbing boundary condition (ABC) is developed
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The modified second-grade fluid flow across a plate of semi-infinite extent, which is initiated by the plate’s movement, is considered herein. The relaxation parameters and fractional parameters are introduced to express the generalized constitutive relation. A convolution-based absorbing boundary condition (ABC) is developed based on the artificial boundary method (ABM), addressing issues related to the semi-infinite boundary. We adopt the finite difference method (FDM) for deriving the numerical solution by employing the L1 scheme to approximate the fractional derivative. To confirm the precision of this method, a source term is added to establish an exact solution for verification purposes. A comparative evaluation of the ABC versus the direct truncated boundary condition (DTBC) is conducted, with their effectiveness and soundness being visually scrutinized and assessed. This study investigates the impact of the motion of plates at different fluid flow velocities, focusing on the effects of dynamic elements influencing flow mechanisms and velocity. This research’s primary conclusion is that a higher fractional parameter correlates with the fluid flow. As relaxation parameters decrease, the delay effect intensifies and the fluid velocity decreases.
Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
Open AccessArticle
Fuzzy Subordination Results for Meromorphic Functions Connected with a Linear Operator
by
Ekram E. Ali, Miguel Vivas-Cortez, Rabha M. El-Ashwah and Abeer M. Albalahi
Fractal Fract. 2024, 8(6), 308; https://doi.org/10.3390/fractalfract8060308 - 23 May 2024
Abstract
The concept of subordination is expanded in this study from the fuzzy sets theory to the geometry theory of analytic functions with a single complex variable. This work aims to clarify fuzzy subordination as a notion and demonstrate its primary attributes. With this
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The concept of subordination is expanded in this study from the fuzzy sets theory to the geometry theory of analytic functions with a single complex variable. This work aims to clarify fuzzy subordination as a notion and demonstrate its primary attributes. With this work’s assistance, new fuzzy differential subordinations will be presented. The first theorems lead to intriguing corollaries for specific aspects chosen to exhibit fuzzy best dominance. The work introduces a new integral operator for meromorphic functions and uses the newly developed integral operator, which is starlike and convex, respectively, to obtain conclusions on fuzzy differential subordination.
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(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
Open AccessArticle
Dynamic Analysis and Sliding Mode Synchronization Control of Chaotic Systems with Conditional Symmetric Fractional-Order Memristors
by
Huaigu Tian, Mingwei Zhao, Jindong Liu, Qiao Wang, Xiong Yu and Zhen Wang
Fractal Fract. 2024, 8(6), 307; https://doi.org/10.3390/fractalfract8060307 - 23 May 2024
Abstract
In this paper, the characteristics of absolute value memristors are verified through the circuit implementation and construction of a chaotic system with a conditional symmetric fractional-order memristor. The dynamic behavior of fractional-order memristor systems is explored using fractional-order calculus theory and the Adomian
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In this paper, the characteristics of absolute value memristors are verified through the circuit implementation and construction of a chaotic system with a conditional symmetric fractional-order memristor. The dynamic behavior of fractional-order memristor systems is explored using fractional-order calculus theory and the Adomian Decomposition Method (ADM). Concurrently, the investigation probes into the existence of coexisting symmetric attractors, multiple coexisting bifurcation diagrams, and Lyapunov exponent spectra (LEs) utilizing system parameters as variables. Additionally, the system demonstrates an intriguing phenomenon known as offset boosting, where the embedding of an offset can adjust the position and size of the system’s attractors. To ensure the practical applicability of these findings, a fractional-order sliding mode synchronization control scheme, inspired by integer-order sliding mode theory, is designed. The rationality and feasibility of this scheme are validated through a theoretical analysis and numerical simulation.
Full article
(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing, 2nd Edition)
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Non-Extensive Statistical Analysis of Seismicity on the West Coastline of Mexico
by
Elsa Leticia Flores-Márquez, Alejandro Ramírez-Rojas and Leonardo Di G. Sigalotti
Fractal Fract. 2024, 8(6), 306; https://doi.org/10.3390/fractalfract8060306 - 22 May 2024
Abstract
Mexico is a well-known seismically active country, which is primarily affected by several tectonic plate interactions along the southern Pacific coastline and by active structures in the Gulf of California. In this paper, we investigate this seismicity using the classical Gutenberg–Richter (GR) law
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Mexico is a well-known seismically active country, which is primarily affected by several tectonic plate interactions along the southern Pacific coastline and by active structures in the Gulf of California. In this paper, we investigate this seismicity using the classical Gutenberg–Richter (GR) law and a non-extensive statistical approach based on Tsallis entropy. The analysis is performed using data from the corrected Mexican seismic catalog provided by the National Seismic Service, spanning the period from January 2000 to October 2023, and unlike previous work, it includes six different regions along the entire west coastline of Mexico. The Gutenberg–Richter law fitting to the earthquake sub-catalogs for all six regions studied indicates magnitudes of completeness between 3.30 and 3.76, implying that the majority of seismic movements occur for magnitudes less than 4. The cumulative distribution of earthquakes as derived from the Tsallis entropy was fitted to the corrected catalog data to estimate the q-entropic index for all six regions, which for values greater than one is a measure of the non-extensivity (i.e., non-equilibrium) of the system. All regions display values of the entropic index in the range , slightly lower than previously estimated ( ) using catalog data from 1988 to 2010. The reason for this difference is related to the use of modern recording devices, which are sensitive to the detection of a larger number of low-magnitude events compared to older instrumentation.
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(This article belongs to the Section Probability and Statistics)
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New Multiplicity Results for a Boundary Value Problem Involving a ψ-Caputo Fractional Derivative of a Function with Respect to Another Function
by
Yankai Li, Dongping Li, Fangqi Chen and Xiangjing Liu
Fractal Fract. 2024, 8(6), 305; https://doi.org/10.3390/fractalfract8060305 - 22 May 2024
Abstract
This paper considers a nonlinear impulsive fractional boundary value problem, which involves a -Caputo-type fractional derivative and integral. Combining critical point theory and fractional calculus properties, such as the semigroup laws, and relationships between the fractional integration and differentiation, new multiplicity results
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This paper considers a nonlinear impulsive fractional boundary value problem, which involves a -Caputo-type fractional derivative and integral. Combining critical point theory and fractional calculus properties, such as the semigroup laws, and relationships between the fractional integration and differentiation, new multiplicity results of infinitely many solutions are established depending on some simple algebraic conditions. Finally, examples are also presented, which show that Caputo-type fractional models can be more accurate by selecting different kernels for the fractional integral and derivative.
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(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)
Open AccessReview
Fractal Analysis of Cement-Based Composite Microstructure and Its Application in Evaluation of Macroscopic Performance of Cement-Based Composites: A Review
by
Peng Zhang, Junyao Ding, Jinjun Guo and Fei Wang
Fractal Fract. 2024, 8(6), 304; https://doi.org/10.3390/fractalfract8060304 - 21 May 2024
Abstract
Cement-based composites’, as the most widely used building material, macroscopic performance significantly influences the safety of engineering structures. Meanwhile, the macroscopic properties of cement-based composites are tightly related to their microscopic structure. The complexity of cement-based composites’ microscopic structure is challenging to describe
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Cement-based composites’, as the most widely used building material, macroscopic performance significantly influences the safety of engineering structures. Meanwhile, the macroscopic properties of cement-based composites are tightly related to their microscopic structure. The complexity of cement-based composites’ microscopic structure is challenging to describe geometrically, so fractal theory is extensively applied to quantify the microscopic structure of cement-based composites. However, existing studies have not clearly defined the quantification methods for various microscopic structures in CCs, nor have they provided a comprehensive evaluation of the correlation between the fractal dimensions of different microscopic structures and macroscopic performance. So, this study categorizes the commonly used testing methods in fractal theory into three categories: particle distribution (laser granulometry, etc.), pore structure (mercury intrusion porosity, etc.), and fracture (computed tomography, etc.). It systematically establishes a detailed process for the application of testing methods, the processing of test results, model building, and fractal dimension calculation. The applicability of different fractal dimension calculation models and the range of the same fractal dimension established by different models are compared and discussed, and the advantages and disadvantages of different models are analyzed. Finally, the research delves into an in-depth analysis of the relationship between the fractal dimension of cement-based composites’ microscopic structure and its macroscopic properties, such as compressive strength, corrosion resistance, impermeability, and high-temperature resistance. The principle that affects the positive and negative correlation between fractal dimension and macroscopic performance is discussed and revealed in this study. The comprehensive review in this paper provides scholars with methods and models for quantitative research on the microscopic structural parameters of cement-based composites and offers a pathway for the non-destructive assessment of the macroscopic performance of cement-based composites.
Full article
(This article belongs to the Special Issue Fractal and Fractional in Construction Materials)
Open AccessArticle
Parameter Sensitivity Analysis for Long-Term Nuclide Migration in Granite Barriers Considering a 3D Discrete Fracture–Matrix System
by
Yingtao Hu, Wenjie Xu, Ruiqi Chen, Liangtong Zhan, Shenbo He and Zhi Ding
Fractal Fract. 2024, 8(6), 303; https://doi.org/10.3390/fractalfract8060303 - 21 May 2024
Abstract
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As a geological barrier for high-level radioactive waste (HLW) disposal in China, granite is crucial for blocking nuclide migration into the biosphere. However, the high uncertainty associated with the 3D geological system, such as the stochastic discrete fracture networks in granite, significantly impedes
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As a geological barrier for high-level radioactive waste (HLW) disposal in China, granite is crucial for blocking nuclide migration into the biosphere. However, the high uncertainty associated with the 3D geological system, such as the stochastic discrete fracture networks in granite, significantly impedes practical safety assessments of HLW disposal. This study proposes a Monte Carlo simulation (MCS)-based simulation framework for evaluating the long-term barrier performance of nuclide migration in fractured rocks. Statistical data on fracture geometric parameters, on-site hydrogeological conditions, and relevant migration parameters are obtained from a research site in Northwestern China. The simulation models consider the migration of three key nuclides, Cs-135, Se-79, and Zr-93, in fractured granite, with mechanisms including adsorption, advection, diffusion, dispersion, and decay considered as factors. Subsequently, sixty MCS realizations are performed to conduct a sensitivity analysis using the open-source software OpenGeoSys-5 (OGS-5). The results reveal the maximum and minimum values of the nuclide breakthrough time Tt (12,000 and 3600 years, respectively) and the maximum and minimum values of the nuclide breakthrough concentration Cmax (4.26 × 10−4 mSv/a and 2.64 × 10−5 mSv/a, respectively). These significant differences underscore the significant effect of the uncertainty in the discrete fracture network model on long-term barrier performance. After the failure of the waste tank (1000 years), nuclides are estimated to reach the outlet boundary 6480 years later. The individual effective dose in the biosphere initially increases and then decreases, reaching a peak value of Cmax = 4.26 × 10−4 mSv/a around 350,000 years, which is below the critical dose of 0.01 mSv/a. These sensitivity analysis results concerning nuclide migration in discrete fractured granite can enhance the simulation and prediction accuracy for risk evaluation of HLW disposal.
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Novel Estimations of Hadamard-Type Integral Inequalities for Raina’s Fractional Operators
by
Merve Coşkun, Çetin Yildiz and Luminiţa-Ioana Cotîrlă
Fractal Fract. 2024, 8(5), 302; https://doi.org/10.3390/fractalfract8050302 - 20 May 2024
Abstract
In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using
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In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., , and . In conclusion, the methodology described in this article is expected to stimulate further research in this area.
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(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
Open AccessArticle
Time-Varying Function Matrix Projection Synchronization of Caputo Fractional-Order Uncertain Memristive Neural Networks with Multiple Delays via Mixed Open Loop Feedback Control and Impulsive Control
by
Hongguang Fan, Yue Rao, Kaibo Shi and Hui Wen
Fractal Fract. 2024, 8(5), 301; https://doi.org/10.3390/fractalfract8050301 - 20 May 2024
Abstract
This paper shows solicitude for the generalized projective synchronization of Caputo fractional-order uncertain memristive neural networks (FOUMNNs) with multiple delays. By extending the constant scale factor to the time-varying function matrix, we establish an extraordinary synchronization mode called time-varying function matrix projection synchronization
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This paper shows solicitude for the generalized projective synchronization of Caputo fractional-order uncertain memristive neural networks (FOUMNNs) with multiple delays. By extending the constant scale factor to the time-varying function matrix, we establish an extraordinary synchronization mode called time-varying function matrix projection synchronization (TFMPS), which is a generalized version of traditional matrix projection synchronization, modified projection synchronization, complete synchronization, and anti-synchronization. To achieve the goal of TFMPS, we design a novel mixed controller including the open loop feedback control and impulsive control, which employs the state information in the time-delayed interval and the sampling information at the impulse instants. It has a prominent advantage that impulse intervals are not restricted by time delays. To establish the connection between the error system and the auxiliary system, a generalized fractional-order comparison theorem with time-varying coefficients and impulses is established. Applying the stability theory, the comparison theorem, and the Laplace transform, new synchronization criteria of FOUMNNs are acquired under the mixed impulsive control schemes, and the derived synchronization theorem and corollary can effectively expand the correlative synchronization achievements of fractional-order systems.
Full article
(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)
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Augmenting the Stability of Automatic Voltage Regulators through Sophisticated Fractional-Order Controllers
by
Emad A. Mohamed, Mokhtar Aly, Waleed Alhosaini and Emad M. Ahmed
Fractal Fract. 2024, 8(5), 300; https://doi.org/10.3390/fractalfract8050300 - 20 May 2024
Abstract
The transition from traditional to renewable energy sources is a critical issue in current energy-generation systems, which aims to address climate change and the increased demand for energy. This shift, however, imposes additional burdens on control systems to maintain power system stability and
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The transition from traditional to renewable energy sources is a critical issue in current energy-generation systems, which aims to address climate change and the increased demand for energy. This shift, however, imposes additional burdens on control systems to maintain power system stability and quality within predefined limits. Addressing these challenges, this paper proposes an innovative Modified Hybrid Fractional-Order (MHFO) automatic voltage regulator (AVR) equipped with a fractional-order tilt integral and proportional derivative with a filter plus a second-order derivative with a filter FOTI-PDND2N2 controller. This advanced controller combines the benefits of a (FOTI) controller, known for enhancing dynamic performance and steady-state response, with a (PDND2N2) controller to improve system robustness and adaptability. The proposed MHFO controller stands out with its nine tunable parameters, providing more extensive control options than the conventional three-parameter PID controller and the five-parameter FOPID controller. Furthermore, a recent optimization approach using a growth optimizer (GO) has been formulated and applied to optimally adjust the MHFO controller’s parameters simultaneously. The performance of the proposed AVR based on the MHFO-GO controller is scrutinized by contrasting it with various established and developed optimization algorithms. The comparative study shows that the AVR based on the MHFO-GO controller surpasses other AVR controllers from the stability, robustness, and dynamic response speed points of view.
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(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications, 2nd Edition)
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Unveiling the Complexity of HIV Transmission: Integrating Multi-Level Infections via Fractal-Fractional Analysis
by
Yasir Nadeem Anjam, Rubayyi Turki Alqahtani, Nadiyah Hussain Alharthi and Saira Tabassum
Fractal Fract. 2024, 8(5), 299; https://doi.org/10.3390/fractalfract8050299 - 20 May 2024
Abstract
This article presents a non-linear deterministic mathematical model that captures the evolving dynamics of HIV disease spread, considering three levels of infection in a population. The model integrates fractal-fractional order derivatives using the Caputo operator and undergoes qualitative analysis to establish the existence
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This article presents a non-linear deterministic mathematical model that captures the evolving dynamics of HIV disease spread, considering three levels of infection in a population. The model integrates fractal-fractional order derivatives using the Caputo operator and undergoes qualitative analysis to establish the existence and uniqueness of solutions via fixed-point theory. Ulam-Hyer stability is confirmed through nonlinear functional analysis, accounting for small perturbations. Numerical solutions are obtained using the fractional Adam-Bashforth iterative scheme and corroborated through MATLAB simulations. The results, plotted across various fractional orders and fractal dimensions, are compared with integer orders, revealing trends towards HIV disease-free equilibrium points for infective and recovered populations. Meanwhile, susceptible individuals decrease towards this equilibrium state, indicating stability in HIV exposure. The study emphasizes the critical role of controlling transmission rates to mitigate fatalities, curb HIV transmission, and enhance recovery rates. This proposed strategy offers a competitive advantage, enhancing comprehension of the model’s intricate dynamics.
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(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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Exploring the Exact Solution of the Space-Fractional Stochastic Regularized Long Wave Equation: A Bifurcation Approach
by
Bashayr Almutairi, Muneerah Al Nuwairan and Anwar Aldhafeeri
Fractal Fract. 2024, 8(5), 298; https://doi.org/10.3390/fractalfract8050298 - 18 May 2024
Abstract
This study explores the effects of using space-fractional derivatives and adding multiplicative noise, modeled by a Wiener process, on the solutions of the space-fractional stochastic regularized long wave equation. New fractional stochastic solutions are constructed, and the consistency of the obtained solutions is
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This study explores the effects of using space-fractional derivatives and adding multiplicative noise, modeled by a Wiener process, on the solutions of the space-fractional stochastic regularized long wave equation. New fractional stochastic solutions are constructed, and the consistency of the obtained solutions is examined using the transition between phase plane orbits. Their bifurcation and dependence on initial conditions are investigated. Some of these solutions are shown graphically, illustrating both the individual and combined influences of fractional order and noise on selected solutions. These effects appear as alterations in the amplitude and width of the solutions, and as variations in their smoothness.
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(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
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Synchronization of Fractional Delayed Memristive Neural Networks with Jump Mismatches via Event-Based Hybrid Impulsive Controller
by
Huiyu Wang, Shutang Liu, Xiang Wu, Jie Sun and Wei Qiao
Fractal Fract. 2024, 8(5), 297; https://doi.org/10.3390/fractalfract8050297 - 18 May 2024
Abstract
This study investigates the asymptotic synchronization in fractional memristive neural networks of the Riemann–Liouville type, considering mixed time delays and jump mismatches. Addressing the challenges associated with discrepancies in the circuit switching speed and the accuracy of the memristor, this paper introduces an
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This study investigates the asymptotic synchronization in fractional memristive neural networks of the Riemann–Liouville type, considering mixed time delays and jump mismatches. Addressing the challenges associated with discrepancies in the circuit switching speed and the accuracy of the memristor, this paper introduces an enhanced model that effectively navigates these complexities. We propose two novel event-based hybrid impulsive controllers, each characterized by unique triggering conditions. Utilizing advanced techniques in inequality and hybrid impulsive control, we establish the conditions necessary for achieving synchronization through innovative Lyapunov functions. Importantly, the developed controllers are theoretically optimized to minimize control costs, an essential consideration for their practical deployment. Finally, the effectiveness of our proposed approach is demonstrated through two illustrative simulation examples.
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(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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Noether’s Theorem of Herglotz Type for Fractional Lagrange System with Nonholonomic Constraints
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Yuanyuan Deng and Yi Zhang
Fractal Fract. 2024, 8(5), 296; https://doi.org/10.3390/fractalfract8050296 - 18 May 2024
Abstract
This research aims to investigate the Noether symmetry and conserved quantity for the fractional Lagrange system with nonholonomic constraints, which are based on the Herglotz principle. Firstly, the fractional-order Herglotz principle is given, and the Herglotz-type fractional-order differential equations of motion for the
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This research aims to investigate the Noether symmetry and conserved quantity for the fractional Lagrange system with nonholonomic constraints, which are based on the Herglotz principle. Firstly, the fractional-order Herglotz principle is given, and the Herglotz-type fractional-order differential equations of motion for the fractional Lagrange system with nonholonomic constraints are derived. Secondly, by introducing infinitesimal generating functions of space and time, the Noether symmetry of the Herglotz type is defined, along with its criteria, and the conserved quantity of the Herglotz type is given. Finally, to demonstrate how to use this method, two examples are provided.
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