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Hybrid Caputo-Type Fractional Parallel Schemes for Nonlinear Elliptic PDEs with Chaos- and Bifurcation-Based Acceleration
Barrier-Diffusion Controlled Adsorption at Anomalous Diffusion: Fractional Calculus Approach
Lagrange Adjoint System for Linear Fractional Systems with Distributed Delays

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 19.3 days after submission; acceptance to publication is undertaken in 2.8 days (median values for papers published in this journal in the second 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.
Journal Cluster of Mathematics and Its Applications: AppliedMath, Axioms, Computation, Fractal and Fractional, Geometry, International Journal of Topology, Logics, Mathematics and Symmetry.

Impact Factor: 3.3 (2024); 5-Year Impact Factor: 3.2 (2024)

Imprint Information Journal Flyer Open Access ISSN: 2504-3110

Latest Articles

20 pages, 2658 KB

Open AccessArticle

Variable-Coefficient Fractional High-Order Nonlinear Models: Establishment and Solutions

by Chunxia An, Jinling Zhang and Sheng Zhang

Fractal Fract. 2026, 10(6), 380; https://doi.org/10.3390/fractalfract10060380 (registering DOI) - 31 May 2026

This work extends the analytical operation of the Riemann–RHPHilbert approach (RHA) for fractional-order nonlinear integrable systems under the solvable meaning of inverse scattering transform (IST) to variable-coefficient fractional-order nonlinear models. Firstly, based on the matrix spectral problem proposed by Ablowitz, Kaup, Newell, and [...] Read more.

This work extends the analytical operation of the Riemann–RHPHilbert approach (RHA) for fractional-order nonlinear integrable systems under the solvable meaning of inverse scattering transform (IST) to variable-coefficient fractional-order nonlinear models. Firstly, based on the matrix spectral problem proposed by Ablowitz, Kaup, Newell, and Segur, this article derives an integer-order integrable system, which is abbreviated as the AKNS hierarchy. Secondly, by taking specific values of the operator in the derived AKNS hierarchy, a variable-coefficient fractional higher-order NLS hierarchy (vfhNLSH) is obtained, and its anomalous dispersion relation (ADR) is derived via formal solution. Significantly, the reductions of the vfhNLSH include three variable-coefficient fractional-order integrable models: the Hirota equation (vfHE), the Lakshmanan–Porsezian–Daniel equation (vfLPDE), and the fifth-order NLS equation (vffNLSE). Finally, we conduct a detailed study on the representative vfHE as an example rather than a special case and construct its explicit N-fold analytical solution based on the extension of the RHA. At the same time, numerical visualization simulations are conducted to demonstrate the waveform structure characteristics of the solutions under  <math> <mrow> <mi>N</mi><mo>=</mo><mn>1</mn></mrow> </math>  and  <math> <mrow> <mi>N</mi><mo>=</mo><mn>2</mn></mrow> </math>  conditions, including solitons, breathers, and their coupled nonlinear waves. The same process is fully applicable to the other two reduced models, with only some differences in the related results and the dynamic behavior of the solutions. It is shown that the temporal part of the Lax pair associated with the vfHE cannot yet be explicitly determined. Therefore, the fractional-order extension of the RHA presented in this article constitutes a formal or RHA-inspired construction, rather than a fully rigorous fractional-order RHA extension. Full article

(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition)

25 pages, 1691 KB

Open AccessArticle

A Parallel Krylov Subspace Iterative Scheme for Variable-Order Fractional Advection–Diffusion–Reaction Equation

by Fouad Mohammad Salama

Fractal Fract. 2026, 10(6), 378; https://doi.org/10.3390/fractalfract10060378 (registering DOI) - 31 May 2026

This paper is concerned with the numerical solution of the variable-order time fractional advection–diffusion–reaction equation (VO-TFADRE) in two space dimensions. We first propose a Crank–Nicolson (C-N) discretization scheme based on central difference operators and L1 formula for space and time variables, respectively. Then, [...] Read more.

This paper is concerned with the numerical solution of the variable-order time fractional advection–diffusion–reaction equation (VO-TFADRE) in two space dimensions. We first propose a Crank–Nicolson (C-N) discretization scheme based on central difference operators and L1 formula for space and time variables, respectively. Then, we apply the C-N scheme to construct a new algorithm, namely the explicit group (EG) method, for the model problem under consideration. The EG method utilizes the idea of small fixed-size groups of mesh points and comes with computational merits as compared with the C-N scheme. Stability and convergence analyses are given in this work. The resulting discretization leads to large sparse linear systems, which are solved using the Bi-CGSTAB iterative method. Numerical experiments demonstrate that both the C–N and EG schemes achieve accurate approximations, while the EG method significantly reduces computational time. To economize further on the computational cost, we propose a parallelized version of the EG method for solving the VO-TFADRE. Carried out numerical simulations reveal that the parallel algorithm is more efficient than the serial algorithm for solving the problem under consideration. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives, Second Edition)

34 pages, 4103 KB

Open AccessArticle

Statistical and Dynamical Analysis of Hidden Attractors in the Fractional Glukhovsky–Dolzhansky System

by Salem Mubarak Alzahrani, Ghaliah Alhamzi, Mona Bin-Asfour, Mansoor Alsulami, Khdija O. Taha, Najat Almutairi and Sayed Saber

Fractal Fract. 2026, 10(6), 377; https://doi.org/10.3390/fractalfract10060377 (registering DOI) - 30 May 2026

This study investigates the reliable numerical analysis of chaotic dynamics in the Glukhovsky–Dolzhansky system, which models convective fluid motion in a rotating ellipsoidal cavity. Hidden and self-excited attractors are localized using the numerical continuation method (NCM), Pyragas time-delayed feedback control, and Leonov’s analytical [...] Read more.

This study investigates the reliable numerical analysis of chaotic dynamics in the Glukhovsky–Dolzhansky system, which models convective fluid motion in a rotating ellipsoidal cavity. Hidden and self-excited attractors are localized using the numerical continuation method (NCM), Pyragas time-delayed feedback control, and Leonov’s analytical dimension formula following global stability loss. A critical assessment of Lyapunov exponents and Lyapunov dimensions in a finite-time setting shows that positive values over long but finite intervals may incorrectly indicate sustained chaos due to transient effects and shadowing breakdown. Furthermore, we demonstrate that the fractional order

γ

plays a bidirectional control role: it induces chaotic behavior at

ρ = 5

for

γ < 0.94

and suppresses chaos at

ρ = 15

for

γ < 0.93

. The multifractal spectrum and correlation dimension are used to quantify attractor complexity, where transient chaos exhibits a broader spectrum (

Δ α \approx 0.67

) compared to sustained chaos (

Δ α \approx 0.48

). Monte Carlo simulations, Sobol sensitivity analysis, Kaplan–Meier survival analysis, and bootstrap-based hypothesis testing confirm the robustness of the results. Overall, the findings provide a unified framework for analyzing hidden attractors, transient chaos, and fractional-order effects in nonlinear fluid dynamical systems. Full article

(This article belongs to the Special Issue Advances in Fractal and Fractional Dynamics)

27 pages, 1439 KB

Open AccessArticle

Order Modulation for Chaos Control and Hybrid Synchronization in a Variable-Order Fractional Arneodo System: Spectral Stability and Numerical Validation

by Thwiba A. Khalid, Nidal E. Taha, Manal Y. A. Juma, Mona Elmahi, Nuha Hassan Hagabdulla and Isra A. Ali

Fractal Fract. 2026, 10(6), 376; https://doi.org/10.3390/fractalfract10060376 (registering DOI) - 30 May 2026

We investigate chaos control and hybrid synchronization in a variable-order fractional Arneodo system by treating the differentiation order

α (t)

as a closed-loop control variable. A hybrid chaos indicator, combining a tracking error with a windowed estimate of the largest Lyapunov [...] Read more.

We investigate chaos control and hybrid synchronization in a variable-order fractional Arneodo system by treating the differentiation order

α (t)

as a closed-loop control variable. A hybrid chaos indicator, combining a tracking error with a windowed estimate of the largest Lyapunov exponent, drives both static and dynamic order modulation laws. The presence and uniqueness of solutions are demonstrated through two distinct methodologies: a piecewise constant-order decomposition with an explicit convergence rate and a direct contraction-mapping argument on the variable-order Volterra operator. Local stability is analyzed via Matignon’s spectral criterion under a quasi-static (frozen-time) approximation. The modulation laws are designed to steer

α (t)

below the critical order

α_{c} \approx 0.8632

, at which the nontrivial equilibria

E_{1, 2} = (\pm \sqrt{5.5}, 0, 0)

become locally asymptotically stable. A second-order predictor–corrector scheme attains its expected convergence rate. A controlled ablation study over 200 Monte Carlo runs demonstrates that the proposed laws reduce the terminal tracking error by 81% relative to the best fixed-order baseline, while requiring approximately eight orders of magnitude less control effort than classical active control. Hybrid synchronization (complete in

(u, v)

and anti-synchronization in w) is successfully achieved in the variable-order setting. Full article

(This article belongs to the Section General Mathematics, Analysis)

33 pages, 2675 KB

Open AccessArticle

Hybrid Modeling of Long-Memory Degradation Dynamics Using Fractional Difference Operators and Deep Reinforcement Learning

by Fengyun Xie, Zhenkai Pan, Shulei Wang, Huihang Chen, Haoran Sun and Zeyan Song

Fractal Fract. 2026, 10(6), 375; https://doi.org/10.3390/fractalfract10060375 (registering DOI) - 30 May 2026

Long-memory degradation processes in rotating machinery often exhibit nonlinear evolution, nonlocal temporal dependence, and hereditary characteristics, which are difficult to fully capture using conventional integer-order models or standard Markovian decision frameworks. To address this issue, this study proposes a hybrid fractional-dynamics and deep [...] Read more.

Long-memory degradation processes in rotating machinery often exhibit nonlinear evolution, nonlocal temporal dependence, and hereditary characteristics, which are difficult to fully capture using conventional integer-order models or standard Markovian decision frameworks. To address this issue, this study proposes a hybrid fractional-dynamics and deep reinforcement learning framework for predictive maintenance of memory-dependent degradation systems. First, the Grünwald–Letnikov fractional difference operator is introduced to construct a fractional-memory representation of degradation trajectories, enabling the model to explicitly encode long-range dependence and accumulated historical degradation effects. Then, a bidirectional gated recurrent unit network is employed to learn sequential degradation representations from the fractional-memory state space, while a deep Q-network is designed to optimize maintenance decisions under uncertain degradation evolution. Experimental results on the IEEE PHM 2012 bearing dataset show that the proposed FM-BiGRU-DQN with safety-guided execution achieved a mean maintenance lead time of 33.9 ± 12.6 steps, an in-band rate of 0.85 ± 0.06, a failure rate of 0.00 ± 0.00, and a deployment reliability of 1.00 ± 0.00 over 10 independent random seeds. Compared with NM-BiGRU-DQN, the in-band rate increased from 0.55 ± 0.10 to 0.85 ± 0.06, with a paired-test p-value of 0.013. Cross-dataset validation on the XJTU-SY bearing dataset further achieved an in-band rate of 0.80 and a failure rate of 0.00. These results indicate that embedding fractional-memory dynamics into deep reinforcement learning improves maintenance timing accuracy, policy robustness, and deployment reliability for complex memory-dependent degradation systems. Full article

(This article belongs to the Special Issue Hybrid Modeling Approaches: Deep Learning and Fractional Differential Equations in Complex Systems)

27 pages, 16012 KB

Open AccessArticle

Multifractal Characteristics and Controlling Factors of Tight Sandstone Reservoirs Across Lithofacies in the Benxi Formation, Ordos Basin, China

by Peipei Liu, Yuming Liu, Jiagen Hou, Lei Bao, Haowei Zhang and Qi Chen

Fractal Fract. 2026, 10(6), 374; https://doi.org/10.3390/fractalfract10060374 - 29 May 2026

The relationship between pore structure heterogeneity in tight sandstone reservoirs and their fractal characteristics is well documented. However, the impact of differential diagenesis across lithofacies on pore-throat structure and fractal properties remains unclear. In this study, we investigate the Carboniferous Benxi Formation in [...] Read more.

The relationship between pore structure heterogeneity in tight sandstone reservoirs and their fractal characteristics is well documented. However, the impact of differential diagenesis across lithofacies on pore-throat structure and fractal properties remains unclear. In this study, we investigate the Carboniferous Benxi Formation in the Ordos Basin using a suite of experiments to characterize pore-throat structure and multifractal behavior, and to assess the influence of diagenesis. The results reveal significant differences among lithofacies in mineral composition, pore types, pore throat structure, fractal dimensions, and petrophysical properties, primarily attributed to variations in sedimentary environments and diagenesis. Fractal characteristics were quantified by converting the T₂ spectra into pore-throat size distributions. Macropores exhibit the highest fractal dimensions, indicating the greatest structural complexity and heterogeneity, followed by mesopores, whereas micropores show the lowest heterogeneity (D₃ > D₂ > D₁). Quartz content mainly controls the fractal properties of macropores by enhancing structural stability, whereas clay minerals govern the fractal behavior of micropores and mesopores by increasing pore-throat complexity. High-energy depositional conditions promote sediment transportation and sorting, leading to quartzarenite lithofacies (QL) and sublitharenite lithofacies (SL) with lower fractal dimensions, more uniform pore structures, and better connectivity. In contrast, feldspathic litharenite lithofacies (FL) and litharenite lithofacies (LL) exhibit higher fractal dimensions due to stronger compaction, reduced primary porosity, and higher clay content, resulting in poorer reservoir quality. This study improves understanding of pore structure heterogeneity in tight sandstones and provides useful insights for predicting high-quality reservoirs in similar geological settings. Full article

(This article belongs to the Special Issue Analysis of Geological Pore Structure Based on Fractal Theory)

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17 pages, 2000 KB

Open AccessArticle

Improved Output Feedback Control for Underactuated Surface Vehicles via Fractional-Order Disturbance Observer

by Yusheng Jia and Jian Chen

Fractal Fract. 2026, 10(6), 373; https://doi.org/10.3390/fractalfract10060373 - 29 May 2026

This paper studies the output feedback control for underactuated surface vehicles with uncertainties and environmental disturbances. A novel finite-time state observer is proposed by using the three-order Levant differentiator to guarantee the Lyapunov finite-time stability of the estimation error and its first derivative. [...] Read more.

This paper studies the output feedback control for underactuated surface vehicles with uncertainties and environmental disturbances. A novel finite-time state observer is proposed by using the three-order Levant differentiator to guarantee the Lyapunov finite-time stability of the estimation error and its first derivative. Then, a fractional-order disturbance observer is established to estimate the uncertainties and the disturbances in the model. Combining the backstepping control with the command filter techniques, the output feedback controller is developed for the system. It is shown that the tracking error of the closed-loop system asymptotically converges to zero, and the tracking performance is improved. Numerical simulation illustrates the advantages of the proposed method. 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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25 pages, 5566 KB

Open AccessArticle

Proposal of a New Comprehensive Parameter to Characterize Directional Multi-Scale Morphological Complexity of Discontinuity in Rock Tunnel

by Wenguang Hao, Yuechao Pei, Anmin Wang, Chuanqiu Du, Yixin Shen, Junsong Huang and Qi Zhang

Fractal Fract. 2026, 10(6), 372; https://doi.org/10.3390/fractalfract10060372 - 29 May 2026

In rock tunnel engineering, the discontinuity roughness plays a crucial role in rock mass stability. Nevertheless, no suitable parameter is currently available to describe the multi-scale morphological complexity associated with varying shear directions. Improved 1D and 3D fractal dimensions are proposed and systematically [...] Read more.

In rock tunnel engineering, the discontinuity roughness plays a crucial role in rock mass stability. Nevertheless, no suitable parameter is currently available to describe the multi-scale morphological complexity associated with varying shear directions. Improved 1D and 3D fractal dimensions are proposed and systematically evaluated for their ability to characterize the anisotropy and morphology complexity. The results show that the trends of improved 1D and 3D fractal dimensions are consistent with the Grasselli parameter GP(θ) along the orthogonal direction of the major and minor axes, which effectively characterizes the anisotropy. Meanwhile, they effectively characterize the profile and discontinuity morphology complexity, respectively. On the basis of the evaluation results, a comprehensive parameter is proposed to quantify directional multi-scale morphological complexity, which characterizes the discontinuity anisotropy, local roughness, and global morphological complexity simultaneously. Furthermore, the proposed parameter is compared with GP(θ) based on the Huashansong tunnel, using the concordance correlation coefficient and macro-trend similarity as evaluation criteria. The results show a moderate correlation between the two parameters, with a concordance correlation coefficient of 0.716. The overall similarity score of 85.8 and the macroscopic trend similarity of 71.0 indicate strong consistency in morphological features. In addition, the RMSE and MAE of γ(θ) are 0.25 and 0.19, which is lower than that of the existing GP(θ) and PSD-based methods in the study of characterizing the dominant morphology complexity direction. Full article

(This article belongs to the Special Issue Applications of Fractal Dimensions in Rock Mechanics and Geomechanics)

17 pages, 534 KB

Open AccessArticle

Exact Dynamics and Optimization for a Discrete-Time SIR Model Using Time-Dependent Parameters

by Alina Alb Lupas, Pshtiwan Othman Mohammed, Balen Dlshad Yaseen, Majeed A. Yousif and Hamdy M. Youssef

Fractal Fract. 2026, 10(6), 371; https://doi.org/10.3390/fractalfract10060371 - 29 May 2026

This study extends the discrete version of the Susceptible–Infected–Recovered (SIR) model to the non-autonomous case in which the rates of infection and cure are time-dependent. This generalization addresses a critical open problem in the literature. We develop a non-standard finite difference approach that [...] Read more.

This study extends the discrete version of the Susceptible–Infected–Recovered (SIR) model to the non-autonomous case in which the rates of infection and cure are time-dependent. This generalization addresses a critical open problem in the literature. We develop a non-standard finite difference approach that maintains essential biological properties, such as population conservation and non-negativity, even under parameter variation. The main novelty here is the derivation of an exact semi-analytical solution to the non-autonomous nonlinear system. We also leverage the exact solution to develop an optimal control problem, which illustrates the potential for rigorous optimization of treatment strategies to minimize infection costs. The numerical simulations verify the theoretical results, showing not only the stability of the model when parameters change seasonally, as well as the efficiency of the optimal controls, but also the predictive power of the non-autonomous approach compared to classical autonomous models. Full article

(This article belongs to the Special Issue Fractal and Fractional Order Modeling of Real-World Phenomena)

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21 pages, 7734 KB

Open AccessArticle

Fractional Longitudinal Wave Dynamics in Magneto-Electro- Elastic Materials: A Neural Network-Based Approach

by Usman Younas, Aljethi Reem Abdullah, Fengping Yao and Jan Muhammad

Fractal Fract. 2026, 10(6), 370; https://doi.org/10.3390/fractalfract10060370 - 29 May 2026

Fractional derivatives introduce an effective mathematical structure to describe memory effects, long-range interactions, and anomalous transport processes that are not well represented by the traditional integer-order models. This paper presents the unidirectional fractional longitudinal wave equation as a governing equation where a model [...] Read more.

Fractional derivatives introduce an effective mathematical structure to describe memory effects, long-range interactions, and anomalous transport processes that are not well represented by the traditional integer-order models. This paper presents the unidirectional fractional longitudinal wave equation as a governing equation where a model is proposed to explain the steady wave propagation of solitary waves in a magneto-electro-elastic circular rod. Magneto-electro-elastic substances are a groundbreaking category of advanced functional materials with tremendous nanotechnology and biomedical engineering prospects because of their effective multi-field energy conversion and temperature responsiveness. In order to solve this complicated fractional nonlinear equation, we introduce a new computation-analysis approach: the Riccati subequation neural network method. This hybrid solution is a synergistic combination of an analytical solution structure and a neural network structure consisting of input, hidden, and output layers, with interconnection between neurons through weighted connections and activation functions. It is important to note that every neuron in the first hidden layer is coupled to the solutions of the Riccati equation, and this allows the systematic use of the new trial functions. With the suggested method, analytical solutions are obtained for the spacetime fractional partial differential equations of the unidirectional fractional longitudinal wave equation in the exact form of trigonometric, hyperbolic, and rational functions. This paper is the first attempt to combine the Riccati subequation method with a neural network model, which has given rise to new types of solitary wave solutions. The three-dimensional, two-dimensional, and contour plots are used to visualize the dynamic nature of these solutions and to display the rich nonlinear wave behavior. The effectiveness and the robustness of the implemented technique is not only proven through our findings but also provides more profound information about the nonlinear wave phenomena in the advanced multifunctional materials, which can inform future developments in energy harvesting and the design of biomedical devices. Full article

(This article belongs to the Special Issue Recent Computational Methods for Fractal and Fractional Nonlinear Partial Differential Equations)

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17 pages, 313 KB

Open AccessArticle

Novel Grüss-Type and Related Integral Inequalities via the Cotangent Fractional Integral with Respect to Another Function

by Ibtehal Alazman, Lakhlifa Sadek, Ahmad Shafee and Khalid Aldawsari

Fractal Fract. 2026, 10(6), 369; https://doi.org/10.3390/fractalfract10060369 - 28 May 2026

This article investigates new classes of integral inequalities within the framework of a recently introduced fractional operator: the cotangent fractional integral with respect to a rising function (RAF) ℵ. The cornerstone of this study is the derivation of a generalized Grüss-type inequality, [...] Read more.

This article investigates new classes of integral inequalities within the framework of a recently introduced fractional operator: the cotangent fractional integral with respect to a rising function (RAF) ℵ. The cornerstone of this study is the derivation of a generalized Grüss-type inequality, along with several related refinements, employing this novel operator. Our methodology strategically combines the properties of the cotangent fractional integral with classical analytical tools, including Young’s inequality and weighted arithmetic–geometric mean arguments. The results obtained are highly versatile, as they not only provide new bounds for this specific operator but also, through appropriate choices of the function ℵ and the parameter

r_{1}

, reduce elegantly to corresponding inequalities for well-known fractional integrals, such as those of Riemann–Liouville (RL), Hadamard, and Katugampola. Several corollaries are presented to illustrate these connections and recover existing results from the literature, thereby demonstrating the unifying nature of our approach. Full article

(This article belongs to the Special Issue Advances in Fractional Integral Inequalities: Theory and Applications, Second Edition)

41 pages, 3640 KB

Open AccessArticle

Orthogonal Self-Similarity Decomposition (OSSD): A Delay-Based Framework for Multiscale Time Series Analysis with Applications in Hydrological Forecasting

by Fatma Latifoğlu and Levent Latifoğlu

Fractal Fract. 2026, 10(6), 368; https://doi.org/10.3390/fractalfract10060368 - 28 May 2026

Decomposition of nonlinear, nonstationary multicomponent signals remains challenging for existing decomposition strategies, including frequency-based, data-driven, and subspace methods, which can suffer from mode mixing, leakage across components, and unreliable isolation of transients. Motivated by this gap, this study proposes Orthogonal Self-Similarity Decomposition (OSSD), [...] Read more.

Decomposition of nonlinear, nonstationary multicomponent signals remains challenging for existing decomposition strategies, including frequency-based, data-driven, and subspace methods, which can suffer from mode mixing, leakage across components, and unreliable isolation of transients. Motivated by this gap, this study proposes Orthogonal Self-Similarity Decomposition (OSSD), which exploits a self-similarity structure in delay-embedded orbit geometry so that temporal organization, rather than spectrum alone, guides component construction. OSSD-Basic introduces three algorithmic novelties within a single pipeline: (1) an adaptive proxy-correlation band merging on the delay axis, (2) a dominant-component cascade that prevents energy-dominant carriers from masking weaker components, and (3) a double MGS + LS reprojection that collapses the inter-mode orthogonality index to numerical zero, regardless of merging and pruning operations. Synthetic experiments with known ground truth show that OSSD-Basic provides a parsimonious four-mode representation with exact inter-mode orthogonality (OI = 9.4 × 10⁻¹⁸), the highest reconstruction SNR among the evaluated baselines (27.14 dB), and the highest ground-truth diagonal correlation sum (3.038) among the tested methods, while using two fewer modes than EMD, VMD, and SSA. Daily streamflow forecasting on a U.S. Geological Survey discharge record further shows that augmenting OSSD-derived inputs with fractal descriptors and fractional-order differencing features yields progressive accuracy gains over the AR-ANN baseline, with R² improving from 0.855 to 0.915 at one-step-ahead and from 0.388 to 0.699 at four-step-ahead forecasting in the single-input setting, within a single-station case study on USGS 01554000. Overall, OSSD-Basic offers an interpretable multiscale decomposition with guaranteed inter-mode orthogonality and a structured feature pathway for oscillatory–transient mixtures. Full article

(This article belongs to the Section Engineering)

17 pages, 1030 KB

Open AccessArticle

Analytical Solutions to Fractional Riccati Differential Economic Models with Memory Effects

by Faizah M. Alharbi, Mohamed A. Abdou, Eslam M. Youssef and Mai Taha

Fractal Fract. 2026, 10(6), 367; https://doi.org/10.3390/fractalfract10060367 - 28 May 2026

Fractional differential equations are highly beneficial in economics because they can be used to analyze nonlinear systems with memory effects. This research investigates a group of nonlinear fractional Riccati equations that show up in models of inventory and economic growth. The present work [...] Read more.

Fractional differential equations are highly beneficial in economics because they can be used to analyze nonlinear systems with memory effects. This research investigates a group of nonlinear fractional Riccati equations that show up in models of inventory and economic growth. The present work is a combined semi-analytical method for finding deterministic series solutions in the Caputo sense: the Adomian Decomposition–Sumudu Transform Method. Limited studies have examined its usage in memory-affected economic models. This method is effective with nonlinearities due to its ability to operate without the necessity of linearizing or discretizing them. The Mittag-Leffler function is employed to demonstrate that the series converges in a strict manner if it converges in a manner that is both absolute and uniform when the conditions are met. Finally, a Lyapunov stability study is conducted to ensure that the solution can accommodate modifications to the original data. Numerical models with different fractional orders show that the behavior of the system is controlled by the fractional parameter. When the fractional order is small, memory effects are increasing. As the order approaches closer to one, the solutions start to act like classical ones. These results indicate that the current methodology can be used for practical applications such as short-term currency exchange rates and volatility in financial markets. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications, 2nd Edition)

44 pages, 489 KB

Open AccessArticle

Solutions and Anti-Periodic Solutions for Non-Local Impulsive Differential Equations and Inclusions Containing the Weighted Generalized Atangana–Baleanu Fractional Derivative in Banach Spaces

by Zainab Alsheekhhussain, Ahmed Gamal Ibrahim, Mohammed Mossa Al-Sawalha and Marwa Ennaceur

Fractal Fract. 2026, 10(6), 366; https://doi.org/10.3390/fractalfract10060366 - 28 May 2026

This paper examines the sufficient conditions that guarantee the existence of solutions and anti-periodic solutions to five classes of fractional differential equations and inclusions involving the weighted generalized Atangana–Baleanu differential operator of order

δ \in (1, 2)

under non-local conditions [...] Read more.

This paper examines the sufficient conditions that guarantee the existence of solutions and anti-periodic solutions to five classes of fractional differential equations and inclusions involving the weighted generalized Atangana–Baleanu differential operator of order

δ \in (1, 2)

under non-local conditions and with instantaneous or non-instantaneous impulses in Banach spaces whose dimension is infinite. First, we deduce some novel properties of this differential operator, then derive the formula for the solutions and anti-periodic solutions, and investigate their existence for the problems presented. Our method relies on certain properties of the Atangana–Baleanu differential operator, which we will obtain, as well as the fixed-point theorems that can be applied to the functions and multi-valued functions. Our work generalizes recently published results. In the final section, we present some examples to illustrate how our theoretical results can be applied. Full article

(This article belongs to the Special Issue Fractal Functions: Theoretical Research and Application Analysis)

21 pages, 16434 KB

Open AccessArticle

A Visual and Quantitative Study of Fractal Mandelbrot Sets Using the IA-Iterative Algorithm for Complex Functions

by Asifa Tassaddiq, Muhammad Tanveer, Azza M. Alghamdi, Rabab Alharbi, Aiman Albarakati, Ruhaila Md Kasmani and Dalal Khalid Almutairi

Fractal Fract. 2026, 10(6), 365; https://doi.org/10.3390/fractalfract10060365 - 27 May 2026

This paper presents a visual and quantitative study of Mandelbrot sets generated by the IA-iterative algorithm for the transcendental complex map

T_{c} (z) = z^{p} + sinh (c^{q})

, where

p \geq 2

, [...] Read more.

This paper presents a visual and quantitative study of Mandelbrot sets generated by the IA-iterative algorithm for the transcendental complex map

T_{c} (z) = z^{p} + sinh (c^{q})

, where

p \geq 2

,

q \geq 1

, and

c \in C ∖ {0}

. A rigorous escape criterion is derived for the proposed map under the IA iteration scheme, providing the mathematical foundation for the generation and analysis of the associated fractal sets. Using this criterion, a broad family of Mandelbrot sets is constructed to examine the influence of the polynomial degree p and the transcendental parameter q on geometric complexity, symmetry, and branching behavior. To complement the visual analysis, a quantitative computational investigation is performed using the fractal dimension, non-escaping area index, average escape time, and average number of iterations. The numerical results reveal nonlinear relationships between these measures and the iteration parameters

(α, β, γ, λ)

, demonstrating the adaptability of the IA scheme for controlled fractal generation. A comparative analysis with CR iteration is also included to evaluate structural variation, computational behavior, and the broader relevance of the method for computer-based fractal modeling, visualization, and algorithmic design. Full article

(This article belongs to the Section Numerical and Computational Methods)

9 pages, 293 KB

Open AccessArticle

A Closed-Form Master Element for the Fractal P-Wave Equation: Galerkin Formulation and Euclidean Limit

by Hugo David Sánchez-Chávez, Alfredo Guadalupe Carreño-López, Leonardo Flores-Cano and Sergio Matias-Gutierres

Fractal Fract. 2026, 10(6), 364; https://doi.org/10.3390/fractalfract10060364 - 27 May 2026

This paper studies a two-dimensional P-wave model in a fractal-continuum setting, where the classical Laplacian is replaced by the Hausdorff Laplacian. Using a Galerkin finite-element formulation on the reference triangle, we derive exact element-level expressions for the master element. The derivation makes [...] Read more.

This paper studies a two-dimensional P-wave model in a fractal-continuum setting, where the classical Laplacian is replaced by the Hausdorff Laplacian. Using a Galerkin finite-element formulation on the reference triangle, we derive exact element-level expressions for the master element. The derivation makes explicit the role of the translated integration domain induced by the lower cutoffs

ℓ_{x}

and

ℓ_{y}

, which leads naturally to incomplete Beta/Gamma representations for the closed forms. In this way, the element integrations can be written analytically and the Euclidean limit can be checked in a consistent manner. A high-order quadrature verification confirms the shifted-domain closed-form expressions for the mass and stiffness matrices

M^{1}

and

K^{1}

while the corresponding Euclidean limit is recovered consistently. The resulting formulas provide reusable ingredients for implementations of related linear PDEs with the same operator structure, while avoiding element-level numerical quadrature. Full article

(This article belongs to the Section Mathematical Physics)

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24 pages, 1234 KB

Open AccessArticle

Fractional Memory-Driven Marine Predator Optimization for Nonconvex Economic Load Dispatch in Renewable-Integrated Power Systems

by Ibrahem E. Atawi, Abdul Wadood, Hani Albalawi, Omar H. Albalawi and Babar Sattar Khan

Fractal Fract. 2026, 10(6), 363; https://doi.org/10.3390/fractalfract10060363 - 27 May 2026

The increasing integration of renewable energy sources (RESs) and the nonlinear nature of economic load dispatch (ELD) have significantly intensified the complexity of modern power systems, demanding efficient optimization techniques. This study proposes a Fractional Marine Predators Algorithm (FMPA) to address the ELD [...] Read more.

The increasing integration of renewable energy sources (RESs) and the nonlinear nature of economic load dispatch (ELD) have significantly intensified the complexity of modern power systems, demanding efficient optimization techniques. This study proposes a Fractional Marine Predators Algorithm (FMPA) to address the ELD problem under uncertain and stochastic operating conditions. The proposed approach enhances the conventional MPA by incorporating fractional calculus (FC), which introduces memory-dependent search dynamics and improves the balance between exploration and exploitation. This modification enables the algorithm to utilize historical search information, resulting in smoother convergence behavior and reduced susceptibility to premature convergence. The performance of the proposed method is assessed using multiple benchmark systems, including 3-unit, 13-unit, and 40-unit configurations with integrated wind power. The optimization framework considers realistic system constraints such as generator limits, valve-point loading effects, and stochastic wind power modeled using probabilistic functions. Simulation results demonstrate that FMPA provides competitive and near-optimal generation costs with stable convergence behavior compared to several existing metaheuristic techniques. Statistical analysis over multiple independent runs confirms the robustness, stability, and reliability of the proposed method. The results indicate that FMPA provides a scalable and efficient solution for solving complex, nonlinear, and constrained ELD problems in modern power systems. Full article

(This article belongs to the Special Issue New Perspectives and Applications of Fractional Order Modeling in Energy Systems)

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19 pages, 4186 KB

Open AccessArticle

Identification of Complex Nonlinear Fractional-Order System Based on Fuzzy Multi-Model

by Qian Zhang, Chunlei Liu, Jingwen Chen, Hongwei Wang and Zheng Zhang

Fractal Fract. 2026, 10(6), 362; https://doi.org/10.3390/fractalfract10060362 - 27 May 2026

Fractional-order dynamic systems, due to their long memory and nonlocality, have significant advantages in describing the dynamic behavior of complex engineering systems. However, existing identification methods often struggle to balance modeling accuracy and model structural complexity under conditions of strong nonlinearity, strong coupling, [...] Read more.

Fractional-order dynamic systems, due to their long memory and nonlocality, have significant advantages in describing the dynamic behavior of complex engineering systems. However, existing identification methods often struggle to balance modeling accuracy and model structural complexity under conditions of strong nonlinearity, strong coupling, and multiple operating conditions. To address the challenge of modeling complex fractional-order systems with strong nonlinearity, strong coupling, and multiple operating conditions, this paper proposes a fuzzy multi-model modeling and identification method based on the decomposition-synthesis approach. First, a fractional-order fuzzy multi-model structure is constructed to characterize the dynamic characteristics of such complex systems. Second, an improved SKFCM hybrid clustering algorithm is proposed, combining K-means clustering and satisfactory fuzzy C-means clustering. This optimizes the cluster center selection strategy and overcomes the shortcomings of traditional satisfactory FCM algorithms, such as random initial membership and unreasonable cluster center selection, thus achieving a reasonable determination of the number of local models. Finally, the least-squares and Levenberg–Marquardt algorithms are integrated to interactively identify local model parameters, system fractional-order, and fuzzy scheduling function parameters, solving key difficulties such as unknown fractional-order s in antecedent variables, numerous parameter couplings, and difficulty in determining the antecedent space. Through academic examples and simulations of a robotic arm system, the proposed method effectively achieves high-precision modeling of complex fractional-order systems, demonstrating strong feasibility and superiority. Full article

(This article belongs to the Section Engineering)

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25 pages, 2213 KB

Open AccessArticle

Applying Iterated Function Systems in Waste Recycling Processes by Identifying Fractal-Like Patterns in Material Decomposition

by Ghaziyah Alsahli, Muhammad Nazam and Nura Alotaibi

Fractal Fract. 2026, 10(6), 361; https://doi.org/10.3390/fractalfract10060361 - 26 May 2026

This manuscript aims at the introduction of interpolative

FG

-contractions and their application to derive fixed-point results. As an application, one of the obtained results is used to analyze the fractional-order Aizawa model. Moreover, we introduce a Hutchinson–Barnsley operator defined in terms of [...] Read more.

This manuscript aims at the introduction of interpolative

FG

-contractions and their application to derive fixed-point results. As an application, one of the obtained results is used to analyze the fractional-order Aizawa model. Moreover, we introduce a Hutchinson–Barnsley operator defined in terms of interpolative

FG

-contractions and a related iterated function system to prove the existence of a unique fractal. The theoretical findings are supported with illustrative examples and graphical demonstrations. This manuscript also sheds light on a theoretical framework for analyzing material decomposition and recycling processes. Full article

(This article belongs to the Section General Mathematics, Analysis)

32 pages, 416 KB

Open AccessArticle

Averaging Effects and Their Applications to Fractional Elliptic and Parabolic Equations

by Wenxiong Chen and Yahong Guo

Fractal Fract. 2026, 10(6), 360; https://doi.org/10.3390/fractalfract10060360 - 26 May 2026

The averaging effect is a distinctive property possessed by fractional operators. In recent years, it has emerged as a powerful tool in the study of qualitative properties of solutions to fractional elliptic and parabolic equations. In this article, we systematically summarize and prove [...] Read more.

The averaging effect is a distinctive property possessed by fractional operators. In recent years, it has emerged as a powerful tool in the study of qualitative properties of solutions to fractional elliptic and parabolic equations. In this article, we systematically summarize and prove various forms of the averaging effects for both fractional elliptic and parabolic equations, from the simplest one to the one under very relaxed conditions, including versions for antisymmetric functions. We then present examples to illustrate how to apply these effects to obtain radial symmetry and monotonicity for solutions in a unit ball and in a half space. In addition, we derive averaging effects for fractional Monge–Ampère operators and for fractional p-Laplacians, which will be potentially applied to obtain qualitative properties for solutions to equations involving these operators. Compared with the traditional approaches, methods based on the averaging effect require substantially weaker regularity assumptions and can even accommodate unbounded solutions. Full article

(This article belongs to the Special Issue Fractional Elliptic and Parabolic Equations: Analysis and Related Topics)

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