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 23.7 days after submission; acceptance to publication is undertaken in 2.7 days (median values for papers published in this journal in the second half of 2024).
- 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:
3.6 (2023);
5-Year Impact Factor:
3.5 (2023)
Latest Articles
Design of a Novel Fractional Whale Optimization-Enhanced Support Vector Regression (FWOA-SVR) Model for Accurate Solar Energy Forecasting
Fractal Fract. 2025, 9(1), 35; https://doi.org/10.3390/fractalfract9010035 (registering DOI) - 11 Jan 2025
Abstract
This study presents a novel Fractional Whale Optimization Algorithm-Enhanced Support Vector Regression (FWOA-SVR) framework for solar energy forecasting, addressing the limitations of traditional SVR in modeling complex relationships within data. The proposed framework incorporates fractional calculus in the Whale Optimization Algorithm (WOA) to
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This study presents a novel Fractional Whale Optimization Algorithm-Enhanced Support Vector Regression (FWOA-SVR) framework for solar energy forecasting, addressing the limitations of traditional SVR in modeling complex relationships within data. The proposed framework incorporates fractional calculus in the Whale Optimization Algorithm (WOA) to improve the balance between exploration and exploitation during hyperparameter tuning. The FWOA-SVR model is comprehensively evaluated against traditional SVR, Long Short-Term Memory (LSTM), and Backpropagation Neural Network (BPNN) models using training, validation, and testing datasets. Experimental results show that FWOA-SVR achieves superior performance with the lowest MSE values (0.036311, 0.03942, and 0.03825), RMSE values (0.19213, 0.19856, and 0.19577), and the highest R2 values (0.96392, 0.96104, and 0.96192) for training, validation, and testing, respectively. These results highlight the significant improvements of FWOA-SVR in prediction accuracy and efficiency, surpassing benchmark models in capturing complex patterns within the data. The findings highlight the effectiveness of integrating fractional optimization techniques into machine learning frameworks for advancing solar energy forecasting solutions.
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(This article belongs to the Special Issue Fractional-Order Learning Systems: Theory, Algorithms, and Emerging Applications)
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Fractional Electrodamage in A549 Human Lung Cancer Cells
by
Hilario Martines-Arano, Jose Alberto Arano-Martinez, Manuel Alejandro Mosso-Pani, Alejandra Valdivia-Flores, Martin Trejo-Valdez, Blanca Estela García-Pérez and Carlos Torres-Torres
Fractal Fract. 2025, 9(1), 34; https://doi.org/10.3390/fractalfract9010034 - 10 Jan 2025
Abstract
Fractional electrodamage in A549 human lung cancer cells was analyzed by introducing a non-integer order parameter to model the influence of electrical stimulation on cellular behavior. Numerical simulations were conducted to evaluate the conversion of electrical energy to heat within A549 cancer cells,
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Fractional electrodamage in A549 human lung cancer cells was analyzed by introducing a non-integer order parameter to model the influence of electrical stimulation on cellular behavior. Numerical simulations were conducted to evaluate the conversion of electrical energy to heat within A549 cancer cells, emphasizing the electrocapacitive effects and electrical conductivity in modulating dielectric properties. Using the Riemann–Liouville fractional calculus framework, experimental results were accurately fitted, demonstrating the non-integer nature of electrodamage processes. The study identified a strong dependency of electrical behavior on frequency, revealing a critical role of fractional dynamics in the dielectric breakdown and susceptibility of A549 cells to voltage changes. These findings advance our understanding of cellular responses to electrical fields and provide insights into applications in cancer diagnostics, monitoring, and potential therapeutic treatments.
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(This article belongs to the Special Issue Fractal and Fractional Analysis in Biomedical Sciences and Engineering)
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A Time–Frequency Composite Recurrence Plots-Based Series Arc Fault Detection Method for Photovoltaic Systems with Different Operating Conditions
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Zhendong Yin, Hongxia Ouyang, Junchi Lu, Li Wang and Shanshui Yang
Fractal Fract. 2025, 9(1), 33; https://doi.org/10.3390/fractalfract9010033 - 8 Jan 2025
Abstract
Series arc faults (SAFs) pose a significant threat to the safety of photovoltaic (PV) systems. However, the complex operating conditions of PV systems make accurate SAF detection challenging. To tackle this issue, this article proposes a SAF detection method based on time–frequency composite
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Series arc faults (SAFs) pose a significant threat to the safety of photovoltaic (PV) systems. However, the complex operating conditions of PV systems make accurate SAF detection challenging. To tackle this issue, this article proposes a SAF detection method based on time–frequency composite recurrence plots (TFCRPs). Initially, variational mode decomposition (VMD) is employed to decompose the current into distinct modes. Subsequently, the proposed TFCRP transforms these modes into two-dimensional matrices, enabling the measurement of composite similarity between different phase states. Lastly, extra tree (ET) is utilized to fuse the fractional recurrence entropy (FRE) and the singular values extracted from the matrices, thereby achieving SAF detection. Experimental results indicate that the proposed method achieves a detection accuracy of 98.75% and can accurately detect SAFs under various operating conditions. Comparisons with different methods further highlight the advancement of the proposed method. Furthermore, the detection time of the proposed method (209 ms) meets the requirements of standard UL1699B.
Full article
(This article belongs to the Special Issue Implementations and Applications of Algorithms Based on Fractional Calculus to Engineering Problems)
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A Multiscale Fractal Approach for Determining Cushioning Curves of Low-Density Polymer Foams
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Mariela C. Bravo-Sánchez, Luis M. Palacios-Pineda, José L. Gómez-Color, Oscar Martínez-Romero, Imperio A. Perales-Martínez, Daniel Olvera-Trejo, Jorge A. Estrada-Díaz and Alex Elías-Zúñiga
Fractal Fract. 2025, 9(1), 32; https://doi.org/10.3390/fractalfract9010032 - 8 Jan 2025
Abstract
This study investigates the impact response of polymer foams commonly used in protective packaging, considering the fractal nature of their material microstructure. The research begins with static material characterization and impact tests on two low-density polyethylene foams. To capture the multiscale nature of
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This study investigates the impact response of polymer foams commonly used in protective packaging, considering the fractal nature of their material microstructure. The research begins with static material characterization and impact tests on two low-density polyethylene foams. To capture the multiscale nature of the dynamic response behavior of two low-density foams to sustain impact loads, fractional differential equations of motion are used to qualitatively and quantitatively describe the dynamic response behavior, assuming restoring forces for each foam characterized, respectively, by a polynomial of heptic degree and by a trigonometric tangential function. A two-scale transform is employed to solve the mathematical model and predict the material’s behavior under impact loads, accounting for the fractal structure of the material’s molecular configuration. To assess the accuracy of the mathematical model, we performed impact tests considering eight dropping heights and two plate weights. We found good predictions from the mathematical models compared to experimental data when the fractal derivatives were between 1.86 and 1.9, depending on the cushioning material used. The accuracy of the theoretical predictions achieved using fractal calculus elucidates how to predict multiscale phenomena associated with foam heterogeneity across space, density, and average pore size, which influence the foam chain’s molecular motion during impact loading conditions.
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(This article belongs to the Special Issue Fractal Theory and Models in Nonlinear Dynamics and Their Applications)
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Shallow-Water Wave Dynamics: Butterfly Waves, X-Waves, Multiple-Lump Waves, Rogue Waves, Stripe Soliton Interactions, Generalized Breathers, and Kuznetsov–Ma Breathers
by
Sarfaraz Ahmed, Ujala Rehman, Jianbo Fei, Muhammad Irslan Khalid and Xiangsheng Chen
Fractal Fract. 2025, 9(1), 31; https://doi.org/10.3390/fractalfract9010031 - 8 Jan 2025
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A nonlinear -dimensional nonlinear Geng equation that can be utilized to explain the dynamics of shallow-water waves in fluids is given special attention. Various wave solutions are produced with the aid of the Hirota bilinear and Cole–Hopf transformation
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A nonlinear -dimensional nonlinear Geng equation that can be utilized to explain the dynamics of shallow-water waves in fluids is given special attention. Various wave solutions are produced with the aid of the Hirota bilinear and Cole–Hopf transformation techniques. By selecting the appropriate polynomial function and implementing the distinct transformations in bilinear form, bright lump waves, dark lump waves, and rogue waves (RWs) are generated. A positive quadratic transformation and cosine function are combined in Hirota bilinear form to evaluate the RW solutions. Typically, RWs have crests that are noticeably higher than those of surrounding waves. These waves are also known as killer, freak, or monster waves. The lump periodic solutions (LPSs) are obtained using a combination of the cosine and positive quadratic functions. The lump-one stripe solutions are computed by using a mix of positive quadratic and exponential transformations to the governing equation. The lump two-stripe solutions are obtained by using a mix of positive quadratic and exponential transformations to the governing equation. The interactional solutions of lump, kink, and periodic wave solutions are obtained. Additionally, mixed solutions with butterfly waves, X-waves and lump waves are computed. The Ma breather (MB), Kuznetsov–Ma breather (KMB), and generalized breathers GBs are generated. Furthermore, solitary wave solution is obtained and a relation for energy of the wave via ansatz function technique.
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The Unbalanced Control Research of Fractional-Order Cascaded H-Bridge Multilevel STATCOM
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Junhua Xu, Guopeng He, Songqin Tang, Zheng Gong, Chunwei Wang and Yue Lan
Fractal Fract. 2025, 9(1), 30; https://doi.org/10.3390/fractalfract9010030 - 7 Jan 2025
Abstract
Recent research on fractional-order cascaded H-bridge multilevel static compensator (FCHM-STATCOM) indicates that it has better performance than the traditional cascaded H-bridge multilevel static compensator (CHM-STATCOM). The existing FCHM-STATCOM control system lacks some special control links for dealing with unbalanced operative situations, which is
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Recent research on fractional-order cascaded H-bridge multilevel static compensator (FCHM-STATCOM) indicates that it has better performance than the traditional cascaded H-bridge multilevel static compensator (CHM-STATCOM). The existing FCHM-STATCOM control system lacks some special control links for dealing with unbalanced operative situations, which is demanded by power systems. This paper improves the existing FCHM-STATCOM control system to satisfy the demand for power systems, creating the potential for it to be applied in the electrical industry. The improvement of the FCHM-STATCOM control system is based on traditional CHM-STATCOM control systems, introducing special control loops for unbalanced operative situations. The improved FCHM-STATCOM control system constructed in this paper consists of an outer control loop that can apply special strategies for different control objectives, two inner control loops, respectively, for positive- and negative-sequence currents, and an inter-phase balancing control loop for balancing its arm operation. At the end of this paper, the results of digital simulations verify the FCHM-STATCOM complete control system’s capacity to regulate its negative-sequence currents and balance its arm operation to deal with unbalanced operative situations. Moreover, based on different strategies, the control system shows effectiveness in eliminating power oscillations and current balancing.
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(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing, 2nd Edition)
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Fish-Tail Structured Fractal Monopole Printed Antenna with Dual Broadband Characteristics for Sub–6GHz 5G and X–Band Radar Applications
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Guntamukkala Yaminisasi, Pokkunuri Pardhasaradhi, Nagandla Prasad, Boddapati Taraka Phani Madhav, Abeer D. Algarni, Sudipta Das and Mohammed El Ghzaoui
Fractal Fract. 2025, 9(1), 29; https://doi.org/10.3390/fractalfract9010029 - 7 Jan 2025
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This article presents a printed antenna, designed with a fractal-shaped patch with fish-tail structured outer edges, a tapered feedline, and a rectangular notch-based defected partial ground structure (DPGS). The presented design has been printed on a FR-4 substrate, which has a dielectric constant
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This article presents a printed antenna, designed with a fractal-shaped patch with fish-tail structured outer edges, a tapered feedline, and a rectangular notch-based defected partial ground structure (DPGS). The presented design has been printed on a FR-4 substrate, which has a dielectric constant of 4.4 and a loss tangent of 0.035. The overall dimension of the proposed antenna is 24 × 40 × 1.6 mm3. The proposed fractal antenna achieved dual broad-band functionality by maintaining the compact size of the radiator. The designed fractal radiator can operate at three distinct resonant frequencies (3.22, 7.64, and 9.41 GHz), covering two distinct frequency bands, extending from 2.5 to 4.2 GHz and 7 to 9.8 GHz. A thorough parametric analysis has been carried out using CST Studio suite 2019 licensed version to achieve better performance in terms of S11 (dB), radiation efficiency, and gain over the operating frequency range. The operating bands fall within the S, C, and X bands to support sub-6GHz 5G and Radar applications at the microwave frequency range.
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Computational Study of a Fractional-Order HIV Epidemic Model with Latent Phase and Treatment
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Sana Abdulkream Alharbi and Nada A. Almuallem
Fractal Fract. 2025, 9(1), 28; https://doi.org/10.3390/fractalfract9010028 - 7 Jan 2025
Abstract
In this work, we propose and investigate a model of the dynamical behavior of HIV/AIDS transmission by considering a new compartment of the population with HIV: the latent asymptomatic class. The infection reproduction number that stabilizes the global dynamics of the model is
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In this work, we propose and investigate a model of the dynamical behavior of HIV/AIDS transmission by considering a new compartment of the population with HIV: the latent asymptomatic class. The infection reproduction number that stabilizes the global dynamics of the model is evaluated. We analyze the model’s global asymptotic stability using the Lyapunov function and LaSalle’s invariance principle. To identify the primary factors affecting the dynamics of HIV/AIDS, a sensitivity analysis of the model parameters is conducted. We also examine a fractional-order HIV model using the Caputo fractional differential operator. Through qualitative analysis and applications, we determine the existence and uniqueness of the model’s solutions. We derive some results from the fixed-point theorem and Ulam–Hyers stability. Ultimately, the obtained numerical simulation results are in agreement with the analytical outcomes obtained from the model analysis. Our findings illustrate the efficacy of the fractional model in depicting the dynamics of the HIV/AIDS epidemic and offering critical insights for the formulation of effective control strategies. The results show that early intervention and treatment in the latent phase of infection can decrease the spread of the disease and its progression to AIDS, as well as increase the success of treatment strategies.
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(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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Open AccessReview
Multifractal Applications in Hydro-Climatology: A Comprehensive Review of Modern Methods
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Shamseena Vahab and Adarsh Sankaran
Fractal Fract. 2025, 9(1), 27; https://doi.org/10.3390/fractalfract9010027 - 6 Jan 2025
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Complexity evaluation of hydro-climatic datasets is a challenging but essential pre-requisite for accurate modeling and subsequent planning. Changes in climate and anthropogenic interventions amplify the complexity of hydro-climatic time-series. Understanding persistence and fractal features may help us to develop new and robust modeling
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Complexity evaluation of hydro-climatic datasets is a challenging but essential pre-requisite for accurate modeling and subsequent planning. Changes in climate and anthropogenic interventions amplify the complexity of hydro-climatic time-series. Understanding persistence and fractal features may help us to develop new and robust modeling frameworks which can work well under non-stationary and non-linear environments. Classical fractal hydrology, rooted in statistical physics, has been developed since the 1980s and the modern alternatives based on de-trending, complex network, and time–frequency principles have been developed since 2002. More specifically, this review presents the procedures of Multifractal Detrended Fluctuation Analysis (MFDFA) and Arbitrary Order Hilbert Spectral Analysis (AOHSA), along with their applications in the field of hydro-climatology. Moreover, this study proposes a complex network-based fractal analysis (CNFA) framework for the multifractal analysis of daily streamflows as an alternative. The case study proves the efficacy of CNMFA and shows that it has the flexibility to be applied in visibility and inverted visibility schemes, which is effective in complex datasets comprising both high- and low-amplitude fluctuations. The comprehensive review showed that more than 75% of the literature focuses on characteristic analysis of the time-series using MFDFA rather than modeling. Among the variables, about 70% of studies focused on analyzing fine-resolution streamflow and rainfall datasets. This study recommends the use of CNMF in hydro-climatology and advocates the necessity of knowledge integration from multiple fields to enhance the multifractal modeling applications. This study further asserts that transforming the characterization into operational hydrology is highly warranted.
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Discrete Element Study of Particle Size Distribution Shape Governing Critical State Behavior of Granular Material
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Mingdong Jiang, Daniel Barreto, Zhi Ding and Kaifang Yang
Fractal Fract. 2025, 9(1), 26; https://doi.org/10.3390/fractalfract9010026 - 6 Jan 2025
Abstract
Granular soil is a porous medium composed of particles with different sizes and self-similar structures, exhibiting fractal characteristics. It is well established that variations in these fractal properties, such as particle size distribution (PSD), significantly influence the mechanical behavior of the soil. In
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Granular soil is a porous medium composed of particles with different sizes and self-similar structures, exhibiting fractal characteristics. It is well established that variations in these fractal properties, such as particle size distribution (PSD), significantly influence the mechanical behavior of the soil. In this paper, a three-dimensional (3D) Discrete Element Method (DEM) is applied to study the mechanical and critical-state behavior of the idealized granular assemblages, in which various PSD shape parameters are considered, including the coefficient of uniformity (Cu), the coefficient of curvature (Cc), and the coefficient of size span (Cs). In addition, the same PSDs but with different mean particle sizes (D50) are also employed in the numerical simulations to examine the particle size effect on the mechanical behavior of the granular media. Numerical triaxial tests are carried out by imposing axial compression under constant mean effective pressure conditions. A unique critical-state stress ratio in p′-q space is observed, indicating that the critical friction angle is independent of the shape of the PSD. However, in the e-p′ plane, the critical state line (CSL) shifts downward and rotates counterclockwise, as the grading becomes more widely distributed, i.e., the increasing coefficient of span (Cs). Additionally, a decrease in the coefficient of curvature (Cc) would also move the CSL downward but with negligible rotation. However, it is found that the variations in the mean particle size (D50) and coefficient of uniformity (Cu) do not affect the position of the CSL in the e-p′ plane. The numerical findings may shed some light on the development of constitutive models of sand that undergo variations in the grading due to crushing and erosion, and address fractal problems related to micro-mechanics in soils.
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(This article belongs to the Special Issue Fractal and Fractional Models in Soil Mechanics)
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New Approaches to Fractal–Fractional Bullen’s Inequalities Through Generalized Convexity
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Wedad Saleh, Hamid Boulares, Abdelkader Moumen, Hussien Albala and Badreddine Meftah
Fractal Fract. 2025, 9(1), 25; https://doi.org/10.3390/fractalfract9010025 - 3 Jan 2025
Abstract
This paper introduces a new identity involving fractal–fractional integrals, which allow us to derive several new Bullen-type inequalities via generalized convexity. This study provides a significant advancement in the area of fractal–fractional inequalities, presenting a range of results not only for fractional integrals
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This paper introduces a new identity involving fractal–fractional integrals, which allow us to derive several new Bullen-type inequalities via generalized convexity. This study provides a significant advancement in the area of fractal–fractional inequalities, presenting a range of results not only for fractional integrals and fractal calculus, but also offering a refinement of the well-known Bullen-type inequality. We further explore the connections between generalized convexity and fractal–fractional integrals, showing how the concept of generalized convexity enables the establishment of error bounds for fractal–fractional integrals involving lower-order derivatives, with an emphasis on their applications in various fields. The findings expand the current understanding of fractal–fractional inequalities and offer new insights into the use of local fractional derivatives for analyzing functions with fractional-order properties.
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(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)
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Modeling Anomalous Transport of Cosmic Rays in the Heliosphere Using a Fractional Fokker–Planck Equation
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José Luis Díaz Palencia
Fractal Fract. 2025, 9(1), 24; https://doi.org/10.3390/fractalfract9010024 - 2 Jan 2025
Abstract
Cosmic rays exhibit anomalous diffusion behaviors in the heliospheric environment that cannot be adequately described by classical diffusion models. In this paper, we develop a theoretical framework employing a fractional Fokker–Planck equation to model the anomalous transport of cosmic rays. This approach accounts
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Cosmic rays exhibit anomalous diffusion behaviors in the heliospheric environment that cannot be adequately described by classical diffusion models. In this paper, we develop a theoretical framework employing a fractional Fokker–Planck equation to model the anomalous transport of cosmic rays. This approach accounts for the observed non-Gaussian distributions, long-range correlations and memory effects in cosmic ray fluxes. We derive analytical solutions using the Adomian Decomposition Method and express them in terms of Mittag-Leffler functions and Lévy stable distributions. The model parameters, including the fractional orders and and the entropic index q, are estimated by a short comparison between theoretical predictions and observational data from cosmic ray experiments. Our findings suggest that the integration of fractional calculus and non-extensive statistics can be employed for describing the cosmic ray propagation and the anomalous diffusion observed in the heliosphere.
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(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)
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Analytical Solutions of Time-Fractional Navier–Stokes Equations Employing Homotopy Perturbation–Laplace Transform Method
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Awatif Muflih Alqahtani, Hamza Mihoubi, Yacine Arioua and Brahim Bouderah
Fractal Fract. 2025, 9(1), 23; https://doi.org/10.3390/fractalfract9010023 - 31 Dec 2024
Abstract
The aim of this article is to introduce analytical and approximate techniques to obtain the solution of time-fractional Navier–Stokes equations. This proposed technique consists is coupling the homotopy perturbation method (HPM) and Laplace transform (LT). The time-fractional derivative used is the Caputo–Hadamard fractional
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The aim of this article is to introduce analytical and approximate techniques to obtain the solution of time-fractional Navier–Stokes equations. This proposed technique consists is coupling the homotopy perturbation method (HPM) and Laplace transform (LT). The time-fractional derivative used is the Caputo–Hadamard fractional derivative (CHFD). The effectiveness of this method is demonstrated and validated through two test problems. The results show that the proposed method is robust, efficient, and easy to implement for both linear and nonlinear problems in science and engineering. Additionally, its computational efficiency requires less computation compared to other schemes.
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(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)
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Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects
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José Luis Echenausía-Monroy, Luis Alberto Quezada-Tellez, Hector Eduardo Gilardi-Velázquez, Omar Fernando Ruíz-Martínez, María del Carmen Heras-Sánchez, Jose E. Lozano-Rizk, José Ricardo Cuesta-García, Luis Alejandro Márquez-Martínez, Raúl Rivera-Rodríguez, Jonatan Pena Ramirez and Joaquín Álvarez
Fractal Fract. 2025, 9(1), 22; https://doi.org/10.3390/fractalfract9010022 - 31 Dec 2024
Abstract
Fractional calculus (or arbitrary order calculus) refers to the integration and derivative operators of an order different than one and was developed in 1695. They have been widely used to study dynamical systems, especially chaotic systems, as the use of arbitrary-order operators broke
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Fractional calculus (or arbitrary order calculus) refers to the integration and derivative operators of an order different than one and was developed in 1695. They have been widely used to study dynamical systems, especially chaotic systems, as the use of arbitrary-order operators broke the milestone of restricting autonomous continuous systems of order three to obtain chaotic behavior and triggered the study of fractional chaotic systems. In this paper, we study the chaotic behavior in fractional systems in more detail and characterize the geometric variations that the dynamics of the system undergo when using arbitrary-order operators by asking the following question: is the Lyapunov exponent sufficient to describe the dynamical variations in a chaotic system of fractional order? By quantifying the convex envelope generated by the 2D projection of the system into all its phase portraits, the changes in the area of the system, as well as the volume of the attractor, are characterized. The results are compared with standard metrics for the study of chaotic systems, such as the Kaplan–Yorke dimension and the fractal dimension, and we also evaluate the frequency fluctuations in the dynamical response. It is found that our methodology can better describe the changes occurring in the systems, while the traditional dimensions are limited to confirming chaotic behaviors; meanwhile, the frequency spectrum hardly changes. The results deepen the study of fractional-order chaotic systems, contribute to understanding the implications and effects observed in the dynamics of the systems, and provide a reference framework for decision-making when using arbitrary-order operators to model dynamical systems.
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(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems, 2nd Edition)
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Automatic Voltage Regulator Betterment Based on a New Fuzzy FOPI+FOPD Tuned by TLBO
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Mokhtar Shouran and Mohammed Alenezi
Fractal Fract. 2025, 9(1), 21; https://doi.org/10.3390/fractalfract9010021 - 31 Dec 2024
Abstract
This paper presents a novel Fuzzy Logic Controller (FLC) framework aimed at enhancing the performance and stability of Automatic Voltage Regulators (AVRs) in power systems. The proposed system combines fuzzy control theory with the Fractional Order Proportional Integral Derivative (FOPID) technique and employs
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This paper presents a novel Fuzzy Logic Controller (FLC) framework aimed at enhancing the performance and stability of Automatic Voltage Regulators (AVRs) in power systems. The proposed system combines fuzzy control theory with the Fractional Order Proportional Integral Derivative (FOPID) technique and employs cascading control theory to significantly improve reliability and robustness. The unique control architecture, termed Fuzzy Fractional Order Proportional Integral (PI) plus Fractional Order Proportional Derivative (PD) plus Integral (Fuzzy FOPI+FOPD+I), integrates advanced control methodologies to achieve superior performance. To optimize the controller parameters, the Teaching–Learning-Based Optimization (TLBO) algorithm is utilized in conjunction with the Integral Time Absolute Error (ITAE) objective function, ensuring precise tuning for optimal control behavior. The methodology is validated through comparative analyses with controllers reported in prior studies, highlighting substantial improvements in performance metrics. Key findings demonstrate significant reductions in peak overshoot, peak undershoot, and settling time, emphasizing the proposed controller’s effectiveness. Additionally, the robustness of the controller is extensively evaluated under challenging scenarios, including parameter uncertainties and load disturbances. Results confirm its ability to maintain stability and performance across a wide range of conditions, outperforming existing methods. This study presents a notable contribution by introducing an innovative control structure that addresses critical challenges in AVR systems, paving the way for more resilient and efficient power system operations.
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(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)
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A Study on the Existence, Uniqueness, and Stability of Fractional Neutral Volterra-Fredholm Integro-Differential Equations with State-Dependent Delay
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Prabakaran Raghavendran, Tharmalingam Gunasekar, Junaid Ahmad and Walid Emam
Fractal Fract. 2025, 9(1), 20; https://doi.org/10.3390/fractalfract9010020 - 31 Dec 2024
Abstract
This paper presents an analysis of the existence, uniqueness, and stability of solutions to fractional neutral Volterra-Fredholm integro-differential equations, incorporating Caputo fractional derivatives and semigroup operators with state-dependent delays. By employing Krasnoselskii’s fixed point theorem, conditions under which solutions exist are established. To
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This paper presents an analysis of the existence, uniqueness, and stability of solutions to fractional neutral Volterra-Fredholm integro-differential equations, incorporating Caputo fractional derivatives and semigroup operators with state-dependent delays. By employing Krasnoselskii’s fixed point theorem, conditions under which solutions exist are established. To ensure uniqueness, the Banach Contraction Principle is applied, and the contraction condition is verified. Stability is analyzed using Ulam’s stability concept, emphasizing the resilience of solutions to perturbations and providing insights into their long-term behavior. An example is included, accompanied by graphical analysis that visualizes the solutions and their dynamic properties.
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(This article belongs to the Section General Mathematics, Analysis)
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Dubovsky’s Class of Mathematical Models for Describing Economic Cycles with Heredity Effects
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Danil Makarov, Roman Parovik and Zafar Rakhmonov
Fractal Fract. 2025, 9(1), 19; https://doi.org/10.3390/fractalfract9010019 - 31 Dec 2024
Abstract
The article is devoted to the study of economic cycles and crises, which are studied within the framework of the theory of N.D. Kondratiev long waves (K-waves). The object of the study is the fractional mathematical models of S. V. Dubovsky, consisting of
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The article is devoted to the study of economic cycles and crises, which are studied within the framework of the theory of N.D. Kondratiev long waves (K-waves). The object of the study is the fractional mathematical models of S. V. Dubovsky, consisting of two nonlinear differential equations of fractional order and describing the dynamics of the efficiency of new technologies and the efficiency of capital productivity, taking into account constant and variable heredity. Fractional mathematical models also take into account the dependence of the rate of accumulation on capital productivity and the influx of external investment and new technological solutions. The effects of heredity lead to a delayed effect of the response of the system in question to the impact. The property of heredity in mathematical models is taken into account using fractional derivatives of constant and variable orders in the sense of Gerasimov–Caputo. The fractional mathematical models of S. V. Dubovsky are further studied numerically using the Adams–Bashforth–Moulton algorithm. Using a numerical algorithm, oscillograms and phase trajectories were constructed for various values and model parameters. It is shown that the fractional mathematical models of S. V. Dubovsky may have limit cycles, which are not always stable.
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(This article belongs to the Section Mathematical Physics)
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Open AccessArticle
Dynamic Behavior and Fixed-Time Synchronization Control of Incommensurate Fractional-Order Chaotic System
by
Xianchen Wang, Zhen Wang and Shihong Dang
Fractal Fract. 2025, 9(1), 18; https://doi.org/10.3390/fractalfract9010018 - 30 Dec 2024
Abstract
In this paper, an incommensurate fractional-order chaotic system is established based on Chua’s system. Combining fractional-order calculus theory and the Adomian algorithm, the dynamic phenomena of the incommensurate system caused by different fractional orders are studied. Meanwhile, the incommensurate system parameters and initial
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In this paper, an incommensurate fractional-order chaotic system is established based on Chua’s system. Combining fractional-order calculus theory and the Adomian algorithm, the dynamic phenomena of the incommensurate system caused by different fractional orders are studied. Meanwhile, the incommensurate system parameters and initial values are used as variables to study the dynamic characteristics of the incommensurate system. It is found that there are rich coexistence bifurcation diagrams and coexistence Lyapunov exponent spectra which are further verified with the phase diagrams. Moreover, a special dynamic phenomenon, such as chaotic degenerate dynamic behavior, is found in the incommensurate system. Secondly, for the feasibility of practical application, the equivalent analog circuit of incommensurate system is realized according to fractional-order time–frequency frequency domain algorithm. Finally, in order to overcome the limitation that the convergence time of the finite-time synchronization control scheme depends on the initial value, a fixed-time synchronization control scheme is proposed in the selection of synchronization control scheme. The rationality of this scheme is proved by theoretical analysis and numerical simulation.
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(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)
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Open AccessArticle
An Adsorption Model Considering Fictitious Stress
by
Xiaohua Tan, Xinjian Ma, Xiaoping Li and Yilong Li
Fractal Fract. 2025, 9(1), 17; https://doi.org/10.3390/fractalfract9010017 - 30 Dec 2024
Abstract
The adsorption of coalbed methane alters the pore structure of reservoirs, subsequently affecting the coal seam’s gas adsorption capacity. However, traditional gas adsorption models often neglect this crucial aspect. In this article, we introduce a fractal capillary bundle model that accounts for the
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The adsorption of coalbed methane alters the pore structure of reservoirs, subsequently affecting the coal seam’s gas adsorption capacity. However, traditional gas adsorption models often neglect this crucial aspect. In this article, we introduce a fractal capillary bundle model that accounts for the expansion of coal seam adsorption. We utilize curvature fractal dimension and capillary fractal dimension to characterize the complexity of the coal seam’s pore structure. By incorporating the concept of fictitious stress, we have described the relationship between gas adsorption, matrix porosity, and permeability changes. We have developed a model that describes the changes in matrix porosity and permeability during the gas adsorption process. After fitting this model to experimental data, it demonstrated high accuracy in predictions. Furthermore, our investigation into how factors such as curvature fractal dimension, capillary fractal dimension, and fictitious stress influence gas adsorption capacity reveals several key findings. Firstly, the specific surface area within the pore structure of coal seams is the primary factor controlling gas adsorption capacity. Secondly, the virtual stress generated during the gas adsorption process alters the coal seam’s maximum gas adsorption capacity, a factor that cannot be overlooked. Lastly, we found that gas adsorption primarily affects the gas migration process, while under high-pressure conditions, gas desorption does not cause significant changes in the matrix porosity and permeability.
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(This article belongs to the Section Engineering)
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Open AccessArticle
The Use of Artificial Intelligence in Data Analysis with Error Recognitions in Liver Transplantation in HIV-AIDS Patients Using Modified ABC Fractional Order Operators
by
Hasib Khan, Jehad Alzabut, D. K. Almutairi and Wafa Khalaf Alqurashi
Fractal Fract. 2025, 9(1), 16; https://doi.org/10.3390/fractalfract9010016 - 30 Dec 2024
Abstract
In this article, we focused on the fractional order modeling, simulations and neural networking to observe the correlation between severity of infection in HIV-AIDS patients and the role of treatments and control. The model is structured with eight classes and a modified Atangana–Baleanu
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In this article, we focused on the fractional order modeling, simulations and neural networking to observe the correlation between severity of infection in HIV-AIDS patients and the role of treatments and control. The model is structured with eight classes and a modified Atangana–Baleanu derivative in Caputo’s sense. The model has several interlinking parameters which show the rates of transmission between classes. We assumed natural death and death on the disease severity in patients. The model was analyzed mathematically as well as computationally. In the mathematical aspects, was plotted for different cases which play a vital role in the infection spread in the population. The model was passed through qualitative analysis for the existence of solutions and stability results. A computational scheme is developed for the model and is applied for the numerical results to analyze the intricate dynamics of the infection. It has been observed that there is a good resemblance in the results for the correlation between the hospitalization, vaccination and recovery rate of the patients. These are reaffirmed with the neural networking tools for the regression, probability, clustering, mean square error and fitting data.
Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)
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