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 19.9 days after submission; acceptance to publication is undertaken in 2.7 days (median values for papers published in this journal in the first half of 2025).
- 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:
3.3 (2024);
5-Year Impact Factor:
3.2 (2024)
Latest Articles
Multifractal-Aware Convolutional Attention Synergistic Network for Carbon Market Price Forecasting
Fractal Fract. 2025, 9(7), 449; https://doi.org/10.3390/fractalfract9070449 - 7 Jul 2025
Abstract
Accurate carbon market price prediction is crucial for promoting a low-carbon economy and sustainable engineering. Traditional models often face challenges in effectively capturing the multifractality inherent in carbon market prices. Inspired by the self-similarity and scale invariance inherent in fractal structures, this study
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Accurate carbon market price prediction is crucial for promoting a low-carbon economy and sustainable engineering. Traditional models often face challenges in effectively capturing the multifractality inherent in carbon market prices. Inspired by the self-similarity and scale invariance inherent in fractal structures, this study proposes a novel multifractal-aware model, MF-Transformer-DEC, for carbon market price prediction. The multi-scale convolution (MSC) module employs multi-layer dilated convolutions constrained by shared convolution kernel weights to construct a scale-invariant convolutional network. By projecting and reconstructing time series data within a multi-scale fractal space, MSC enhances the model’s ability to adapt to complex nonlinear fluctuations while significantly suppressing noise interference. The fractal attention (FA) module calculates similarity matrices within a multi-scale feature space through multi-head attention, adaptively integrating multifractal market dynamics and implicit associations. The dynamic error correction (DEC) module models error commonality through variational autoencoder (VAE), and uncertainty-guided dynamic weighting achieves robust error correction. The proposed model achieved an average R2 of 0.9777 and 0.9942 for 7-step ahead predictions on the Shanghai and Guangdong carbon price datasets, respectively. This study pioneers the interdisciplinary integration of fractal theory and artificial intelligence methods for complex engineering analysis, enhancing the accuracy of carbon market price prediction. The proposed technical pathway of “multi-scale deconstruction and similarity mining” offers a valuable reference for AI-driven fractal modeling.
Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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Open AccessArticle
Fractal Feature of Manufactured Sand Ultra-High-Performance Concrete (UHPC) Based on MIP
by
Xinlin Wang, Tinghong Pan, Yang Yang, Rongqing Qi, Dian Guan, Kaihe Dong, Run-Sheng Lin and Rongxin Guo
Fractal Fract. 2025, 9(7), 448; https://doi.org/10.3390/fractalfract9070448 - 5 Jul 2025
Abstract
To alleviate environmental pressures, manufactured sand (MS) are increasingly being used in the production of ultra-high-performance concrete (UHPC) due to their consistent supply and environmental benefits. However, manufactured sand properties are critically influenced by processing and production techniques, resulting in substantial variations in
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To alleviate environmental pressures, manufactured sand (MS) are increasingly being used in the production of ultra-high-performance concrete (UHPC) due to their consistent supply and environmental benefits. However, manufactured sand properties are critically influenced by processing and production techniques, resulting in substantial variations in fundamental characteristics that directly impact UHPC matrix pore structure and ultimately compromise performance. Traditional testing methods inadequately characterize UHPC’s pore structure, necessitating multifractal theory implementation to enhance pore structural interpretation capabilities. In this study, UHPC specimens were fabricated with five types of MS exhibiting distinct properties and at varying water to binder (w/b) ratios. The flowability, mechanical strength, and pore structure of the specimens were systematically evaluated. Additionally, multifractal analysis was conducted on each specimen group using mercury intrusion porosimetry (MIP) data to characterize pore complexity. SM-type sands have a more uniform distribution of pores of different scales, better pore structure and matrix homogeneity due to their finer particles, moderate stone powder content, and better cleanliness. Both excessively high and low stone powder content, as well as low cleanliness, will lead to pore aggregation and poor closure, degrading the pore structure.
Full article
(This article belongs to the Special Issue Fractal and Fractional in Construction Materials)
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Nonfragile State Estimator Design for Memristor-Based Fractional-Order Neural Networks with Randomly Occurring Hybrid Time Delays and Stochastic Cyber-Attacks
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Qifeng Niu, Xiaoguang Shao, Yanjuan Lu, Yibo Zhao and Jie Zhang
Fractal Fract. 2025, 9(7), 447; https://doi.org/10.3390/fractalfract9070447 - 4 Jul 2025
Abstract
This paper addresses the design of nonfragile state estimators for memristor-based fractional-order neural networks that are subject to stochastic cyber-attacks and hybrid time delays. To mitigate the issue of limited bandwidth during signal transmission, quantitative processing is introduced to reduce network burden and
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This paper addresses the design of nonfragile state estimators for memristor-based fractional-order neural networks that are subject to stochastic cyber-attacks and hybrid time delays. To mitigate the issue of limited bandwidth during signal transmission, quantitative processing is introduced to reduce network burden and prevent signal blocking. In real network environments, the outputs may be compromised by cyber-attacks, which can disrupt data transmission systems. To better reflect the actual conditions of fractional-order neural networks, a Bernoulli variable is utilized to describe the statistical properties. Additionally, novel conditions are presented to ensure the stochastic asymptotic stability of the augmented error system through a new fractional-order free-matrix-based integral inequality. Finally, the effectiveness of the proposed estimation methods is demonstrated through two numerical simulations.
Full article
(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application, 2nd Edition)
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Quantum Dynamics in a Comb Geometry: Green Function Solutions with Nonlocal and Fractional Potentials
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Enrique C. Gabrick, Ervin K. Lenzi, Antonio S. M. de Castro, José Trobia and Antonio M. Batista
Fractal Fract. 2025, 9(7), 446; https://doi.org/10.3390/fractalfract9070446 - 4 Jul 2025
Abstract
We investigate a generalized quantum Schrödinger equation in a comb-like structure that imposes geometric constraints on spatial variables. The model is extended by the introduction of nonlocal and fractional potentials to capture memory effects in both space and time. We consider four distinct
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We investigate a generalized quantum Schrödinger equation in a comb-like structure that imposes geometric constraints on spatial variables. The model is extended by the introduction of nonlocal and fractional potentials to capture memory effects in both space and time. We consider four distinct scenarios: (i) a time-dependent nonlocal potential, (ii) a spatially nonlocal potential, (iii) a combined space–time nonlocal interaction with memory kernels, and (iv) a fractional spatial derivative, which is related to distributions asymptotically governed by power laws and to a position-dependent effective mass. For each scenario, we propose solutions based on the Green’s function for arbitrary initial conditions and analyze the resulting quantum dynamics. Our results reveal distinct spreading regimes, depending on the type of non-locality and the fractional operator applied to the spatial variable. These findings contribute to the broader generalization of comb models and open new questions for exploring quantum dynamics in backbone-like structures.
Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications, 2nd Edition)
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Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear Damping
by
Shahid Hussain, Xinlong Feng, Arafat Hussain and Ahmed Bakhet
Fractal Fract. 2025, 9(7), 445; https://doi.org/10.3390/fractalfract9070445 - 4 Jul 2025
Abstract
We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative ( ) with mixed finite element methods (P1b–P1 and
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We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative ( ) with mixed finite element methods (P1b–P1 and – ) for spatial discretization of velocity and pressure. This approach addresses the key challenges of fractional models, including nonlocality and memory effects, while maintaining stability in the presence of the nonlinear damping term , for . We prove unconditional stability for both semi-discrete and fully discrete schemes and derive optimal error estimates for the velocity and pressure components. Numerical experiments validate the theoretical results. Convergence tests using exact solutions, along with benchmark problems such as backward-facing channel flow and lid-driven cavity flow, confirm the accuracy and reliability of the method. The computed velocity contours and streamlines show close agreement with analytical expectations. This scheme is particularly effective for capturing anomalous diffusion in Newtonian and turbulent flows, and it offers a strong foundation for future extensions to viscoelastic and biological fluid models.
Full article
(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)
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A Multidisciplinary Approach to Triangular Shapes: Philosophy, Art, Mathematical Properties, and Application Purposes for High-Frequency Signal Processing Using Sierpiński Geometry
by
Romolo Marcelli
Fractal Fract. 2025, 9(7), 444; https://doi.org/10.3390/fractalfract9070444 - 4 Jul 2025
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Triangular shapes have been studied from different perspectives over a wide temporal frame since ancient times. Initially, fundamental theorems were formulated to demonstrate their geometrical properties. Philosophy and art leveraged the peculiar aspects of triangles as building blocks for more complex geometrical shapes.
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Triangular shapes have been studied from different perspectives over a wide temporal frame since ancient times. Initially, fundamental theorems were formulated to demonstrate their geometrical properties. Philosophy and art leveraged the peculiar aspects of triangles as building blocks for more complex geometrical shapes. This paper will review triangles by adopting a multidisciplinary approach, recalling ancient science and Plato’s arguments in relation to their connection with philosophy. It will then consider the artistic utilization of triangles, particularly in compositions created during the medieval era, as exemplified by the Cosmati Italian family’s masterpieces. Various scientific environments have explored triangular 2D and 3D shapes for different purposes, which will be briefly reviewed here. After that, Sierpiński geometry and its properties will be introduced, focusing on the equilateral shape and its internal complexity generated by subdividing the entire triangle into smaller sub-triangles. Finally, examples of triangular planar shapes that fulfill the Sierpiński geometry will be presented as an application in signal processing for high-frequency signals in the microwave and millimeter-wave range.
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Open AccessArticle
Development of Fractional Newton-Type Inequalities Through Extended Integral Operators
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Abd-Allah Hyder, Areej A. Almoneef, Mohamed A. Barakat, Hüseyin Budak and Özge Aktaş
Fractal Fract. 2025, 9(7), 443; https://doi.org/10.3390/fractalfract9070443 - 4 Jul 2025
Abstract
This paper introduces a new class of Newton-type inequalities (NTIs) within the framework of extended fractional integral operators. This study begins by establishing a fundamental identity for generalized fractional Riemann–Liouville (FR-L) operators, which forms the basis for deriving various inequalities under different assumptions
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This paper introduces a new class of Newton-type inequalities (NTIs) within the framework of extended fractional integral operators. This study begins by establishing a fundamental identity for generalized fractional Riemann–Liouville (FR-L) operators, which forms the basis for deriving various inequalities under different assumptions on the integrand. In particular, fractional counterparts of the classical and Simpson rules are obtained when the modulus of the first derivative is convex. The analysis is further extended to include functions that satisfy a Lipschitz condition or have bounded first derivatives. Moreover, an additional NTI is presented for functions of bounded variation, expressed in terms of their total variation. In all scenarios, the proposed results reduce to classical inequalities when the fractional parameters are specified accordingly, thus offering a unified perspective on numerical integration through fractional operators.
Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)
Open AccessArticle
Efficient Solution Criteria for a Coupled Fractional Laplacian System on Some Infinite Domains
by
Abdelkader Moumen, Sabri T. M. Thabet, Hussien Albala, Khaled Aldwoah, Hicham Saber, Eltigani I. Hassan and Alawia Adam
Fractal Fract. 2025, 9(7), 442; https://doi.org/10.3390/fractalfract9070442 - 3 Jul 2025
Abstract
This article concerns a novel coupled implicit differential system under –Riemann–Liouville ( ) fractional derivatives with -Laplacian operator and multi-point strip boundary conditions on unbounded domains. An applicable Banach space is introduced to define solutions on unbounded domains
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This article concerns a novel coupled implicit differential system under –Riemann–Liouville ( ) fractional derivatives with -Laplacian operator and multi-point strip boundary conditions on unbounded domains. An applicable Banach space is introduced to define solutions on unbounded domains . The explicit iterative solution’s existence and uniqueness ( ) are established by employing the Banach fixed point strategy. The different types of Ulam–Hyers–Rassias ( ) stabilities are investigated. Ultimately, we provide a numerical application of a coupled - fractional turbulent flow model to illustrate and test the effectiveness of our outcomes.
Full article
(This article belongs to the Special Issue Recent Advances in Nonlocal Problems Involving the Fractional Laplacian Operators)
Open AccessArticle
A Unified Framework for Fractional and Non-Fractional Operators in Some Function Spaces
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Mieczysław Cichoń, Wafa Shammakh and Hussein A. H. Salem
Fractal Fract. 2025, 9(7), 441; https://doi.org/10.3390/fractalfract9070441 - 3 Jul 2025
Abstract
This paper examines a family of operators that combine the features of fractional-order and classical operators. Our goal is to obtain results on their invertibility in function spaces, based on their inherent improving properties. The class of proportional operators we study is extensive
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This paper examines a family of operators that combine the features of fractional-order and classical operators. Our goal is to obtain results on their invertibility in function spaces, based on their inherent improving properties. The class of proportional operators we study is extensive and includes both fractional-order and classical operators. This leads to interesting function spaces in which we obtain the right- and left-handed properties of invertibility. Thus, we extend and unify results concerning fractional-order and proportional operators. To confirm the relevance of our results, we have supplemented the paper with a series of results on the equivalence of differential and integral forms for various problems, including terminal value problems.
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(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
Open AccessArticle
Synchronization of Short-Memory Fractional Directed Higher-Order Networks
by
Xiaoqin Wang, Weiyuan Ma and Jiayu Zou
Fractal Fract. 2025, 9(7), 440; https://doi.org/10.3390/fractalfract9070440 - 3 Jul 2025
Abstract
This paper addresses the synchronization problem in complex networks characterized by short-memory fractional dynamics and directed higher-order interactions. Sufficient conditions for global synchronization are rigorously derived using Lyapunov-based analysis and an effective pinning control strategy. To further enhance the adaptability and robustness of
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This paper addresses the synchronization problem in complex networks characterized by short-memory fractional dynamics and directed higher-order interactions. Sufficient conditions for global synchronization are rigorously derived using Lyapunov-based analysis and an effective pinning control strategy. To further enhance the adaptability and robustness of the network, an adaptive control law is constructed, accommodating uncertainties and time-varying coupling strengths. An improved predictor–corrector numerical algorithm is also proposed to efficiently solve the underlying short-memory systems. A numerical simulation is conducted to demonstrate the validity of the proposed theoretical results. This work deepens the theoretical understanding of synchronization in higher-order fractional networks and provides practical guidance for the design and control of complex systems with short-memory and higher-order effects.
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(This article belongs to the Special Issue Modeling and Dynamic Analysis of Fractional-Order Systems)
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Pore-Throat Structure, Fractal Characteristics, and Main Controlling Factors in Extremely Low-Permeability Sandstone Reservoirs: The Case of Chang 3 Section in Huachi Area, Ordos Basin
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Huanmeng Zhang, Chenyang Wang, Jinkuo Sui, Yujuan Lv, Ling Guo and Zhiyu Wu
Fractal Fract. 2025, 9(7), 439; https://doi.org/10.3390/fractalfract9070439 - 3 Jul 2025
Abstract
The pore-throat structure of the extremely low-permeability sandstone reservoir in the Huachi area of the Ordos Basin is complex and highly heterogeneous. Currently, there are issues such as unclear understanding of the micro-pore-throat structural characteristics, primary controlling factors of reservoir quality, and classification
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The pore-throat structure of the extremely low-permeability sandstone reservoir in the Huachi area of the Ordos Basin is complex and highly heterogeneous. Currently, there are issues such as unclear understanding of the micro-pore-throat structural characteristics, primary controlling factors of reservoir quality, and classification boundaries of the reservoir in the study area, which seriously restricts the exploration and development effectiveness of the reservoir in this region. It is necessary to use a combination of various analytical techniques to comprehensively characterize the pore-throat structure and establish reservoir classification evaluation standards in order to better understand the reservoir. This study employs a suite of analytical and testing techniques, including cast thin sections (CTS), scanning electron microscopy (SEM), cathodoluminescence (CL), X-ray diffraction (XRD), as well as high-pressure mercury injection (HPMI) and constant-rate mercury injection (CRMI), and applies fractal theory for analysis. The research findings indicate that the extremely low-permeability sandstone reservoir of the Chang 3 section primarily consists of arkose and a minor amount of lithic arkose. The types of pore-throat are diverse, with intergranular pores, feldspar dissolution pores, and clay interstitial pores and microcracks being the most prevalent. The throat types are predominantly sheet-type, followed by pore shrinkage-type and tubular throats. The pore-throat network of low-permeability sandstone is primarily composed of nanopores (pore-throat radius r < 0.01 μm), micropores (0.01 < r < 0.1 μm), mesopores (0.1 < r < 1.0 μm), and macropores (r > 1.0 μm). The complexity of the reservoir pore-throat structure was quantitatively characterized by fractal theory. Nanopores do not exhibit ideal fractal characteristics. By splicing high-pressure mercury injection and constant-rate mercury injection at a pore-throat radius of 0.12 μm, a more detailed characterization of the full pore-throat size distribution can be achieved. The average fractal dimensions for micropores (Dh2), mesopores (Dc3), and macropores (Dc4) are 2.43, 2.75, and 2.95, respectively. This indicates that the larger the pore-throat size, the rougher the surface, and the more complex the structure. The degree of development and surface roughness of large pores significantly influence the heterogeneity and permeability of the reservoir in the study area. Dh2, Dc3, and Dc4 are primarily controlled by a combination of pore-throat structural parameters, sedimentary processes, and diagenetic processes. Underwater diversion channels and dissolution are key factors in the formation of effective storage space. Based on sedimentary processes, reservoir space types, pore-throat structural parameters, and the characteristics of mercury injection curves, the study area is divided into three categories. This classification provides a theoretical basis for predicting sweet spots in oil and gas exploration within the study area.
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(This article belongs to the Special Issue Pore Structure and Fractal Characteristics in Unconventional Oil and Gas Reservoirs)
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Analytical Solitary Wave Solutions of Fractional Tzitzéica Equation Using Expansion Approach: Theoretical Insights and Applications
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Wael W. Mohammed, Mst. Munny Khatun, Mohamed S. Algolam, Rabeb Sidaoui and M. Ali Akbar
Fractal Fract. 2025, 9(7), 438; https://doi.org/10.3390/fractalfract9070438 - 3 Jul 2025
Abstract
In this study, we investigate the fractional Tzitzéica equation, a nonlinear evolution equation known for modeling complex phenomena in various scientific domains such as solid-state physics, crystal dislocation, electromagnetic waves, chemical kinetics, quantum field theory, and nonlinear optics. Using the (G′/
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In this study, we investigate the fractional Tzitzéica equation, a nonlinear evolution equation known for modeling complex phenomena in various scientific domains such as solid-state physics, crystal dislocation, electromagnetic waves, chemical kinetics, quantum field theory, and nonlinear optics. Using the (G′/G, 1/G)-expansion approach, we derive different categories of exact solutions, like hyperbolic, trigonometric, and rational functions. The beta fractional derivative is used here to generalize the classical idea of the derivative, which preserves important principles. The derived solutions with broader nonlinear wave structures are periodic waves, breathers, peakons, W-shaped solitons, and singular solitons, which enhance our understanding of nonlinear wave dynamics. In relation to these results, the findings are described by showing the solitons’ physical behaviors, their stabilities, and dispersions under fractional parameters in the form of contour plots and 2D and 3D graphs. Comparisons with earlier studies underscore the originality and consistency of the (G′/G, 1/G)-expansion approach in addressing fractional-order evolution equations. It contributes new solutions to analytical problems of fractional nonlinear integrable systems and helps understand the systems’ dynamic behavior in a wider scope of applications.
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(This article belongs to the Special Issue Recent Computational Methods for Fractal and Fractional Nonlinear Partial Differential Equations)
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Existence and Uniqueness Analysis for (k, ψ)-Hilfer and (k, ψ)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary Conditions
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Furkan Erkan, Nuket Aykut Hamal, Sotiris K. Ntouyas and Jessada Tariboon
Fractal Fract. 2025, 9(7), 437; https://doi.org/10.3390/fractalfract9070437 - 2 Jul 2025
Abstract
This paper investigates the existence and uniqueness of solutions to a class of sequential fractional differential equations and inclusions involving the -Hilfer and -Caputo derivatives under non-separated boundary conditions. By reformulating the problems
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This paper investigates the existence and uniqueness of solutions to a class of sequential fractional differential equations and inclusions involving the -Hilfer and -Caputo derivatives under non-separated boundary conditions. By reformulating the problems into equivalent fixed-point systems, several classical fixed-point theorems, including those of Banach, Krasnosel’ski ’s, Schaefer, and the Leray–Schauder alternative, are employed to derive rigorous results. The study is further extended to the multi-valued setting, where existence results are established for both convex- and nonconvex-valued multi-functions using appropriate fixed-point techniques. Numerical examples are provided to illustrate the applicability and effectiveness of the theoretical findings.
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(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)
Open AccessArticle
Soliton Dynamics of the Nonlinear Kodama Equation with M-Truncated Derivative via Two Innovative Schemes: The Generalized Arnous Method and the Kudryashov Method
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Khizar Farooq, Ali. H. Tedjani, Zhao Li and Ejaz Hussain
Fractal Fract. 2025, 9(7), 436; https://doi.org/10.3390/fractalfract9070436 - 2 Jul 2025
Abstract
The primary aim of this research article is to investigate the soliton dynamics of the M-truncated derivative nonlinear Kodama equation, which is useful for optical solitons on nonlinear media, shallow water waves over complex media, nonlocal internal waves, and fractional viscoelastic wave propagation.
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The primary aim of this research article is to investigate the soliton dynamics of the M-truncated derivative nonlinear Kodama equation, which is useful for optical solitons on nonlinear media, shallow water waves over complex media, nonlocal internal waves, and fractional viscoelastic wave propagation. We utilized two recently developed analytical techniques, the generalized Arnous method and the generalized Kudryashov method. First, the nonlinear Kodama equation is transformed into a nonlinear ordinary differential equation using the homogeneous balance principle and a traveling wave transformation. Next, various types of soliton solutions are constructed through the application of these effective methods. Finally, to visualize the behavior of the obtained solutions, three-dimensional, two-dimensional, and contour plots are generated using Maple (2023) mathematical software.
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(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition)
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Discrete-Time Fractional-Order Sliding Mode Attitude Control of Multi-Spacecraft Systems Based on the Fully Actuated System Approach
by
Yiqi Chen and Shuyi Shao
Fractal Fract. 2025, 9(7), 435; https://doi.org/10.3390/fractalfract9070435 - 1 Jul 2025
Abstract
In practical applications, most systems operate based on digital signals obtained through sampling. Applying fractional-order control to spacecraft attitude control is meaningful for achieving better performance, especially in the coordination of the multi-spacecraft attitude system. In this paper, a discrete-time fractional-order sliding mode
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In practical applications, most systems operate based on digital signals obtained through sampling. Applying fractional-order control to spacecraft attitude control is meaningful for achieving better performance, especially in the coordination of the multi-spacecraft attitude system. In this paper, a discrete-time fractional-order sliding mode attitude control problem is studied for multi-spacecraft systems based on the fully actuated system approach. Firstly, a discrete-time disturbance observer based on the fractional-order theory is constructed to estimate the disturbance. Secondly, a discrete-time fractional-order sliding mode controller is designed by combining the transformed fully actuated discrete-time system and the disturbance observer. Subsequently, every spacecraft can track the desired attitude under the designed controller. Finally, the simulation results show that the developed control method achieves faster convergence, smaller overshoot, and higher control accuracy.
Full article
(This article belongs to the Special Issue Fractional Dynamics and Control in Multi-Agent Systems and Networks)
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Double Local Fractional Yang–Laplace Transform for Local Fractional PDEs on Fractal Domains
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Djelloul Ziane, Mountassir Hamdi Cherif, Carlo Cattani and Abdelhamid Mohammed Djaouti
Fractal Fract. 2025, 9(7), 434; https://doi.org/10.3390/fractalfract9070434 - 1 Jul 2025
Abstract
This study introduces a novel analytical technique known as the double local fractional Yang–Laplace transform method ( ) and rigorously investigates its foundational properties, including linearity, differentiation, and convolution. The proposed method is formulated via double local fractional
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This study introduces a novel analytical technique known as the double local fractional Yang–Laplace transform method ( ) and rigorously investigates its foundational properties, including linearity, differentiation, and convolution. The proposed method is formulated via double local fractional integrals, enabling a robust mechanism for addressing local fractional partial differential equations defined on fractal domains, particularly Cantor sets. Through a series of illustrative examples, we demonstrate the applicability and efficacy of the transform in solving complex local fractional partial differential equation models. Special emphasis is placed on the local fractional Laplace equation, the linear local fractional Klein–Gordon equation, and other models, wherein the method reveals significant computational and analytical advantages. The results substantiate the method’s potential as a powerful tool for broader classes of problems governed by local fractional dynamics on fractal geometries.
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Open AccessArticle
Fractional Calculus for Neutrosophic-Valued Functions and Its Application in an Inventory Lot-Sizing Problem
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Rakibul Haque, Mostafijur Rahaman, Adel Fahad Alrasheedi, Dimplekumar Chalishajar and Sankar Prasad Mondal
Fractal Fract. 2025, 9(7), 433; https://doi.org/10.3390/fractalfract9070433 - 30 Jun 2025
Abstract
Past experiences and memory significantly contribute to self-learning and improved decision-making. These can assist decision-makers in refining their strategies for better outcomes. Fractional calculus is a tool that captures a system’s memory or past experience through its repeating patterns. In the realm of
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Past experiences and memory significantly contribute to self-learning and improved decision-making. These can assist decision-makers in refining their strategies for better outcomes. Fractional calculus is a tool that captures a system’s memory or past experience through its repeating patterns. In the realm of uncertainty, neutrosophic set theory demonstrates greater suitability, as it independently assesses membership, non-membership, and indeterminacy. In this article, we aim to extend the theory further by introducing fractional calculus for neutrosophic-valued functions. The proposed method is applied to an economic lot-sizing problem. Numerical simulations of the lot-sizing model suggest that strong memory employment with a memory index of 0.1 can lead to an increase in average profit in memory-independent phenomena with a memory index of 1 by approximately 44% to 49%. Additionally, the neutrosophic environment yields superior profitability results compared to both precise and imprecise settings. The synergy of fractional-order dynamics and neutrosophic uncertainty modeling paves the way for enhanced decision-making in complex, ambiguous environments.
Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition)
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Multi-Focus Image Fusion Based on Dual-Channel Rybak Neural Network and Consistency Verification in NSCT Domain
by
Ming Lv, Sensen Song, Zhenhong Jia, Liangliang Li and Hongbing Ma
Fractal Fract. 2025, 9(7), 432; https://doi.org/10.3390/fractalfract9070432 - 30 Jun 2025
Abstract
In multi-focus image fusion, accurately detecting and extracting focused regions remains a key challenge. Some existing methods suffer from misjudgment of focus areas, resulting in incorrect focus information or the unintended retention of blurred regions in the fused image. To address these issues,
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In multi-focus image fusion, accurately detecting and extracting focused regions remains a key challenge. Some existing methods suffer from misjudgment of focus areas, resulting in incorrect focus information or the unintended retention of blurred regions in the fused image. To address these issues, this paper proposes a novel multi-focus image fusion method that leverages a dual-channel Rybak neural network combined with consistency verification in the nonsubsampled contourlet transform (NSCT) domain. Specifically, the high-frequency sub-bands produced by NSCT decomposition are processed using the dual-channel Rybak neural network and a consistency verification strategy, allowing for more accurate extraction and integration of salient details. Meanwhile, the low-frequency sub-bands are fused using a simple averaging approach to preserve the overall structure and brightness information. The effectiveness of the proposed method has been thoroughly evaluated through comprehensive qualitative and quantitative experiments conducted on three widely used public datasets: Lytro, MFFW, and MFI-WHU. Experimental results show that our method consistently outperforms several state-of-the-art image fusion techniques, including both traditional algorithms and deep learning-based approaches, in terms of visual quality and objective performance metrics ( , , , , , , , , , and ). These results clearly demonstrate the robustness and superiority of the proposed fusion framework in handling multi-focus image fusion tasks.
Full article
(This article belongs to the Special Issue Fractional Order Complex Systems: Advanced Control, Intelligent Estimation and Reinforcement Learning Image Processing Algorithms, Second Edition)
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Heterogeneity of Deep Tight Sandstone Reservoirs Using Fractal and Multifractal Analysis Based on Well Logs and Its Correlation with Gas Production
by
Peiqiang Zhao, Qiran Lv, Yi Xin and Ning Wu
Fractal Fract. 2025, 9(7), 431; https://doi.org/10.3390/fractalfract9070431 - 30 Jun 2025
Abstract
Deep tight sandstone reservoirs are characterized by low porosity and permeability, complex pore structure, and strong heterogeneity. Conducting research on the heterogeneity characteristics of reservoirs could lay a foundation for evaluating their effectiveness and accurately identifying advantageous reservoirs, which is of great significance
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Deep tight sandstone reservoirs are characterized by low porosity and permeability, complex pore structure, and strong heterogeneity. Conducting research on the heterogeneity characteristics of reservoirs could lay a foundation for evaluating their effectiveness and accurately identifying advantageous reservoirs, which is of great significance for searching for “sweet spot” oil and gas reservoirs in tight reservoirs. In this study, the deep tight sandstone reservoir in the Dibei area, northern Kuqa depression, Tarim Basin, China, is taken as the research object. Firstly, statistical methods are used to calculate the coefficient of variation (CV) and coefficient of heterogeneity (TK) of core permeability, and the heterogeneity within the reservoir is evaluated by analyzing the variations in the reservoir permeability. Then, based on fractal theory, the fractal and multifractal parameters of the GR and acoustic logs are calculated using the box dimension, correlation dimension, and the wavelet leader methods. The results show that the heterogeneity revealed by the box dimension, correlation dimension, and multifractal singular spectrum calculated based on well logs is consistent and in good agreement with the parameters calculated based on core permeability. The heterogeneity of gas layers is comparatively weaker, while that of dry layers is stronger. In addition, the fractal parameters of GR and the acoustic logs of three wells with the oil test in the study area were analyzed, and the relationship between reservoir heterogeneity and production was determined: As reservoir heterogeneity decreases, production increases. Therefore, reservoir heterogeneity can be used as an indicator of production; specifically, reservoirs with weak heterogeneity have high production.
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(This article belongs to the Special Issue Pore Structure and Fractal Characteristics in Unconventional Oil and Gas Reservoirs)
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Open AccessArticle
Pore Structure and Multifractal Characteristics of the Upper Lianggaoshan Formation in the Northeastern Sichuan Basin, China
by
Jingjing Guo, Guotao Luo, Haitao Wang and Liehui Zhang
Fractal Fract. 2025, 9(7), 430; https://doi.org/10.3390/fractalfract9070430 - 30 Jun 2025
Abstract
The Upper Lianggaoshan (LGS) Formation in the northeastern Sichuan Basin, composed of shale with interbedded siltstone, is a promising target layer for shale oil. Accurate evaluation of pore structures is essential for effective exploration of shale oil. This study investigated pore structures of
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The Upper Lianggaoshan (LGS) Formation in the northeastern Sichuan Basin, composed of shale with interbedded siltstone, is a promising target layer for shale oil. Accurate evaluation of pore structures is essential for effective exploration of shale oil. This study investigated pore structures of siltstone and shale samples from the Upper LGS Formation using low-pressure CO2 adsorption (LTCA), low-temperature N2 adsorption (LTNA), high-pressure mercury intrusion (HPMI), and nuclear magnetic resonance (NMR) methods. The single-exponent and multifractal dimensions of samples were determined, and the relationships between fractal dimensions and pore structures were explored. Results show that the pore size distribution (PSD) of siltstone and shale samples exhibits multi-peak characteristics, with mesopores (2–50 nm) being dominant in the total pore volumes. The multi-scaled pores in shale and siltstone samples exhibit fractal characteristics. The average values of single-fractal dimensions (D1, D2) obtained by LTNA data are 2.39 and 2.62 for shale samples, and 2.24 and 2.59 for siltstone samples, respectively. Compared to siltstones, the pore structures of shale samples exhibit greater complexity, indicated by larger fractal dimensions. The samples from subsections Liang 2 and Liang 3 exhibit greater heterogeneity compared to subsection Liang 1. The single-fractal dimensions of micropores and mesopores show positive correlations with specific surface area (SSA) and pore volume (PV), while the fractal dimension of macropores shows a negative correlation with average pore diameter and median radius. The average values of single-fractal dimension D3 obtained from HPMI data are 2.9644 and 2.9471 for shale and siltstone samples, respectively, indicating more complex structures of macropores in shale samples compared to siltstone samples. The average value of and singularity strength range Δα obtained by a multifractal model for core samples from subsection Liang 1 are 1.868 and 2.155, respectively, which are the smallest among all of the three subsections, indicating that the heterogeneity of pore structures of subsection Liang 1 is the weakest. This research provides valuable guidance for shale oil development in the northeastern Sichuan Basin, China.
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(This article belongs to the Special Issue Analysis of Geological Pore Structure Based on Fractal Theory)
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