Fractal and Fractional

25 pages, 2056 KB

Open AccessArticle

Analysis of Stability and Quasi-Synchronization in Fractional-Order Neural Networks with Mixed Delays, Uncertainties, and External Disturbances

by Tian-Zeng Li, Xiao-Wen Tan, Yu Wang and Qian-Kun Wang

Fractal Fract. 2026, 10(1), 73; https://doi.org/10.3390/fractalfract10010073 - 22 Jan 2026

Viewed by 93

Abstract

This study addresses the stability and quasi-synchronization of fractional-order neural networks that incorporate mixed delays, system uncertainties, and external disturbances. Accordingly, a more realistic neural network model is constructed. For fractional-order neural networks incorporating mixed delays and uncertainties (FONNMDU), this study establishes a [...] Read more.

This study addresses the stability and quasi-synchronization of fractional-order neural networks that incorporate mixed delays, system uncertainties, and external disturbances. Accordingly, a more realistic neural network model is constructed. For fractional-order neural networks incorporating mixed delays and uncertainties (FONNMDU), this study establishes a criterion for uniform asymptotic stability and proves the existence and uniqueness of the equilibrium solution. Furthermore, it investigates the global uniform stability and stability regions of fractional-order neural networks with mixed delays, uncertainties, and external disturbances (FONNMDUED). Then, to address the quasi-synchronization problem, a controller is designed and some novel sufficient conditions for achieving quasi-synchronization are established. The results show that tuning the control parameters can adjust the error bound. These findings not only enrich the theoretical foundation of fractional-order neural networks but also offer practical insights for applications in complex systems. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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19 pages, 2755 KB

Open AccessArticle

Fractional Modelling of Hereditary Vibrations in Coupled Circular Plate System with Creep Layers

by Julijana Simonović

Fractal Fract. 2026, 10(1), 72; https://doi.org/10.3390/fractalfract10010072 - 21 Jan 2026

Viewed by 97

Abstract

This paper presents an analytical model for the hereditary vibrations of a coupled circular plate system interconnected by viscoelastic creep layers. The system is represented as a discrete-continuous chain of thin, isotropic plates with time-dependent material properties. Based on the theory of hereditary [...] Read more.

This paper presents an analytical model for the hereditary vibrations of a coupled circular plate system interconnected by viscoelastic creep layers. The system is represented as a discrete-continuous chain of thin, isotropic plates with time-dependent material properties. Based on the theory of hereditary viscoelasticity and D’Alembert’s principle, a system of partial integro-differential equations is derived and reduced to ordinary integro-differential equations using Bernoulli’s method and Laplace transforms. Analytical expressions for natural frequencies, mode shapes, and time-dependent response functions are obtained. The results reveal the emergence of multi-frequency vibration regimes, with modal families remaining temporally uncoupled. This enables the identification of resonance conditions and dynamic absorption phenomena. The fractional parameter serves as a tunable damping factor: lower values result in prolonged oscillations, while higher values cause rapid decay. Increasing the kinetic stiffness of the coupling layers raises vibration frequencies and enhances sensitivity to hereditary effects. This interplay provides deeper insight into dynamic behavior control. The model is applicable to multilayered structures in aerospace, civil engineering, and microsystems, where long-term loading and time-dependent material behavior are critical. The proposed framework offers a powerful tool for designing systems with tailored dynamic responses and improved stability. Full article

(This article belongs to the Special Issue From Fractals to Fractional Calculus: Nonlinear Dynamics and Hybrid Complex Systems)

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27 pages, 642 KB

Open AccessArticle

Advanced Hermite-Hadamard-Mercer Type Inequalities with Refined Error Estimates and Applications

by Arslan Munir, Hüseyin Budak, Artion Kashuri and Loredana Ciurdariu

Fractal Fract. 2026, 10(1), 71; https://doi.org/10.3390/fractalfract10010071 - 20 Jan 2026

Viewed by 174

Abstract

The purpose of this research is to develop a set of Hermite–Hadamard–Mercer-type inequalities that involve different types of fractional integral operators such as classical Riemann–Liouville fractional integral operators. Furthermore, some fractional integral inequalities are obtained for three-times differentiable convex functions with respect to [...] Read more.

The purpose of this research is to develop a set of Hermite–Hadamard–Mercer-type inequalities that involve different types of fractional integral operators such as classical Riemann–Liouville fractional integral operators. Furthermore, some fractional integral inequalities are obtained for three-times differentiable convex functions with respect to the right-hand side of the Hermite–Hadamard–Mercer-type inequality. Moreover, several new results regarding Young’s inequality, bounded function and L-Lipschitzian function are deduced. The paper presents additional remarks and comments on the results to make sense of them. To illustrate the key findings, graphical representations are provided, and applications involving special means, midpoint formula, q-digamma function and modified Bessel function are presented to demonstrate the practical utility of the derived inequalities. Full article

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

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18 pages, 5751 KB

Open AccessArticle

Prediction of Dielectric Constant of Polyurethane Grouting Materials Based on Fractal Characteristics

by Meili Meng, Xiao Zhao, Shuangliang Song and Maolin Yang

Fractal Fract. 2026, 10(1), 70; https://doi.org/10.3390/fractalfract10010070 - 20 Jan 2026

Viewed by 178

Abstract

The microstructure of polyurethane (PU) grouting material is the key determinant of its macroscopic dielectric properties. In this study, based on its microscopic fractal characteristics and combined with effective medium theory and the Menger sponge structure, an n-stage fractal dielectric model was constructed. [...] Read more.

The microstructure of polyurethane (PU) grouting material is the key determinant of its macroscopic dielectric properties. In this study, based on its microscopic fractal characteristics and combined with effective medium theory and the Menger sponge structure, an n-stage fractal dielectric model was constructed. This model correlates the material’s dielectric response with its fractal dimension and porosity. The fractal dimensions of PU specimens with densities ranging from 0.29734 g/cm³ to 0.41817 g/cm³ were calculated using the box-counting method. Within this density range, the fractal dimension of the PU specimens showed no significant variation, with a calculated value of approximately 2.7355. By approximating the microscopic unit as an n-stage fractal cube based on the Menger sponge structure and incorporating series-parallel dielectric models, an analytical expression for the dielectric constant was derived. A comparison with experimental data shows that the model’s predictions are in good agreement with the measured values, with a mean relative error (MRE) of only 4%. Full article

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17 pages, 1002 KB

Open AccessArticle

Finite-Time Synchronization of Uncertain Fractional-Order Quaternion-Valued Neural Networks with Discontinuous Activation Function

by Zhongwen Wu, Kui Ding and Xiaoan Wang

Fractal Fract. 2026, 10(1), 69; https://doi.org/10.3390/fractalfract10010069 - 20 Jan 2026

Viewed by 92

Abstract

This study explores finite-time synchronization (FTS) in fractional-order quaternion-valued neural networks (FQVNNs) characterized by discontinuous activation functions and uncertainties in parameters. Initially, leveraging the properties of the Mittag-Leffler function along with fractional-order (F-O) delayed differential inequalities, a novel finite-time stability theorem for F-O [...] Read more.

This study explores finite-time synchronization (FTS) in fractional-order quaternion-valued neural networks (FQVNNs) characterized by discontinuous activation functions and uncertainties in parameters. Initially, leveraging the properties of the Mittag-Leffler function along with fractional-order (F-O) delayed differential inequalities, a novel finite-time stability theorem for F-O systems is established, building upon previous research findings. Next, based on norm definitions, two state feedback controllers employing quaternion 1-norm and quaternion 2-norm are devised to ensure FTS for the system under consideration. Following this, by utilizing differential inclusion theory, examining the quaternion sign function, employing advanced inequality methods, applying principles of F-O differential equations, and using the Lyapunov functional approach, new criteria for achieving FTS in FQVNNs are formulated. Additionally, precise estimates for the settling time are presented. In conclusion, two carefully designed numerical examples are included to corroborate the theoretical results derived. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Chaotic and Complex Systems)

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25 pages, 14211 KB

Open AccessArticle

Constructing New Fractal-like Particle Aggregates Using Seeded DLA and Concentration Gradient Diffusion Approaches

by Sancho Salcedo-Sanz, Pablo Álvarez-Couso, Luis Castelo-Sardina and Jorge Pérez-Aracil

Fractal Fract. 2026, 10(1), 68; https://doi.org/10.3390/fractalfract10010068 - 19 Jan 2026

Viewed by 137

Abstract

Hybridization of existing fractal aggregate construction methods has been used to obtain new fractal-like structures, with different properties and fractal dimensions to aggregates obtained using the hybridized methods alone. In this paper we propose the hybridization of the Diffusion-Limited Aggregation (DLA) approach with [...] Read more.

Hybridization of existing fractal aggregate construction methods has been used to obtain new fractal-like structures, with different properties and fractal dimensions to aggregates obtained using the hybridized methods alone. In this paper we propose the hybridization of the Diffusion-Limited Aggregation (DLA) approach with other methods for constructing fractal-like aggregates, such as Iterated Function Systems (IFSs), Lindenmayer systems (L-Systems), Strange Attractors (SAs) or Percolation-based fractal construction approaches. The proposed approach is a variation of the seeded DLA algorithm used previously in the literature, which consists of considering existing fractal aggregates as condensation nuclei before the DLA simulation. In this case, we revisit the seeded DLA scheme and test different existing fractals as nuclei, such as Strange Attractors or different IFS fractals. We also introduce a simple algorithm for simulating the diffusion of particle aggregate structures, based on concentration gradient diffusion. We show how different fractal aggregates diffuse using this model, and how the diffused versions of the fractal aggregates can then be used themselves as condensation nuclei for the seeded DLA algorithm, obtaining new fractal aggregates. We characterize the new fractal-like aggregates constructed by means of their fractal dimensions, calculated by using the box-counting approach. The obtained fractal-like aggregates have potential applications in computer graphics and multi-media art, due to their esthetic and visually attractive structures based on particles. Applications of the aggregates in statistical and material physics, as well as the modeling of new aggregate types using condensation nuclei and their applications in the development of algorithms, mathematical operators or antenna design, are also reported. Full article

(This article belongs to the Special Issue Multifractal Analysis and Complex Systems)

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14 pages, 298 KB

Open AccessArticle

On Global Solutions to a ψ-Caputo Fractional Inequality

by Mohammed D. Kassim

Fractal Fract. 2026, 10(1), 67; https://doi.org/10.3390/fractalfract10010067 - 19 Jan 2026

Viewed by 182

Abstract

This paper investigates the absence of globally defined, non-trivial solutions for some fractional differential inequalities of the

ψ

-Caputo type, with polynomial sources involving fractional derivatives. We rigorously establish these results within an appropriate function space using properties of

ψ

-fractional integrals and [...] Read more.

This paper investigates the absence of globally defined, non-trivial solutions for some fractional differential inequalities of the

ψ

-Caputo type, with polynomial sources involving fractional derivatives. We rigorously establish these results within an appropriate function space using properties of

ψ

-fractional integrals and derivatives and the test function technique. To demonstrate the applicability and enhance understanding, we provide three illustrative examples. Our findings broaden the scope of previous literature, encompassing existing results as special cases. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)

28 pages, 5991 KB

Open AccessArticle

Particle Transport in Self-Affine Rough Rock Fractures: A CFD–DEM Analysis of Multiscale Flow–Particle Interactions

by Junce Xu, Kangsheng Xue, Hai Pu and Xingji He

Fractal Fract. 2026, 10(1), 66; https://doi.org/10.3390/fractalfract10010066 - 19 Jan 2026

Viewed by 187

Abstract

Understanding particle transport in rough-walled fractures is essential for predicting flow behavior, clogging, and permeability evolution in natural and engineered subsurface systems. This study develops a fully coupled CFD–DEM framework to investigate how self-affine fractal roughness, represented by the Joint Roughness Coefficient (JRC), [...] Read more.

Understanding particle transport in rough-walled fractures is essential for predicting flow behavior, clogging, and permeability evolution in natural and engineered subsurface systems. This study develops a fully coupled CFD–DEM framework to investigate how self-affine fractal roughness, represented by the Joint Roughness Coefficient (JRC), governs fluid–particle interactions across multiple scales. Nine fracture geometries with controlled roughness were generated using a fractal-based surface model, enabling systematic isolation of roughness effects. The results show that increasing JRC introduces a hierarchy of geometric perturbations that reorganize the flow field, amplify shear and velocity-gradient fluctuations, and enhance particle–wall interactions. Particle migration exhibits a nonlinear response to roughness due to the competing influences of disturbance amplification and the formation of preferential high-velocity pathways. Furthermore, roughness-controlled scaling relations are identified for mean particle velocity, residence time, and energy dissipation, revealing JRC as a fundamental parameter linking geometric complexity to transport efficiency. Based on these findings, a unified mechanistic framework is established that conceptualizes fractal roughness as a multiscale geometric forcing mechanism governing hydrodynamic heterogeneity, particle dynamics, and dissipative processes. This framework provides new physical insight into transport behavior in rough fractures and offers a scientific basis for improved prediction of clogging, proppant placement, and transmissivity evolution in subsurface engineering applications. Full article

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16 pages, 664 KB

Open AccessArticle

The Fractal Timoshenko Beam Equation

by Helvio Mollinedo, Ernesto Pineda León, David De-León, Andriy Kryvko, Israel Miguel-Andrés, Didier Samayoa and Lucero Damián-Adame

Fractal Fract. 2026, 10(1), 65; https://doi.org/10.3390/fractalfract10010065 - 18 Jan 2026

Viewed by 172

Abstract

A fractal approach for the Timoshenko beam theory by applying differential vector calculus in a three-dimensional continuum with a fractal metric is developed. First, a summary of the tools needed, mathematical relationships, and background of fractal continuum mechanics is presented. Then, the static [...] Read more.

A fractal approach for the Timoshenko beam theory by applying differential vector calculus in a three-dimensional continuum with a fractal metric is developed. First, a summary of the tools needed, mathematical relationships, and background of fractal continuum mechanics is presented. Then, the static and dynamical parts of the Timoshenko beam equation are extended to fractal manifolds. Afterwards, an intrafractal beam constructed as a Cartesian product is suggested and the fractal dimensionalities of the Balankin beam are scrutinized. This allows comparing both intrafractal beams when they have the same Hausdorff dimension but different connectivity. Finally, the effects of fractal attributes on the mechanical properties of the deformable fractal medium are highlighted. Some applications of the developed tools are briefly outlined. Full article

(This article belongs to the Special Issue Fractional and Fractal Methods with Their Mechanics Applications)

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24 pages, 1474 KB

Open AccessArticle

A Fractional Hybrid Strategy for Reliable and Cost-Optimal Economic Dispatch in Wind-Integrated Power Systems

by Abdul Wadood, Babar Sattar Khan, Bakht Muhammad Khan, Herie Park and Byung O. Kang

Fractal Fract. 2026, 10(1), 64; https://doi.org/10.3390/fractalfract10010064 - 16 Jan 2026

Viewed by 216

Abstract

Economic dispatch in wind-integrated power systems is a critical challenge, yet many recent metaheuristics suffer from premature convergence, heavy parameter tuning, and limited ability to escape local optima in non-smooth valve-point landscapes. This study proposes a new hybrid optimization framework, the Fractional Grasshopper [...] Read more.

Economic dispatch in wind-integrated power systems is a critical challenge, yet many recent metaheuristics suffer from premature convergence, heavy parameter tuning, and limited ability to escape local optima in non-smooth valve-point landscapes. This study proposes a new hybrid optimization framework, the Fractional Grasshopper Optimization algorithm (FGOA), which integrates fractional-order calculus into the standard Grasshopper Optimization algorithm (GOA) to enhance its search efficiency. The FGOA method is applied to the economic load dispatch (ELD) problem, a nonlinear and nonconvex task that aims to minimize fuel and wind-generation costs while satisfying practical constraints such as valve-point loading effects (VPLEs), generator operating limits, and the stochastic behavior of renewable energy sources. Owing to the increasing role of wind energy, stochastic wind power is modeled through the incomplete gamma function (IGF). To further improve computational accuracy, FGOA is hybridized with Sequential Quadratic Programming (SQP), where FGOA provides global exploration and SQP performs local refinement. The proposed FGOA-SQP approach is validated on systems with 3, 13, and 40 generating units, including mixed thermal and wind sources. Comparative evaluations against recent metaheuristic algorithms demonstrate that FGOA-SQP achieves more accurate and reliable dispatch outcomes. Specifically, the proposed approach achieves fuel cost reductions ranging from 0.047% to 0.71% for the 3-unit system, 0.31% to 27.25% for the 13-unit system, and 0.69% to 12.55% for the 40-unit system when compared with state-of-the-art methods. Statistical results, particularly minimum fitness values, further confirm the superior performance of the FGOA-SQP framework in addressing the ELD problem under wind power uncertainty. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 3rd Edition)

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8 pages, 224 KB

Open AccessEditorial

Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods, and Applications

by Wenying Feng

Fractal Fract. 2026, 10(1), 63; https://doi.org/10.3390/fractalfract10010063 - 16 Jan 2026

Viewed by 269

Abstract

This special issue is devoted to the advancement of fractional-order differential equations and their wide-ranging applications [...] Full article

(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)

18 pages, 1014 KB

Open AccessArticle

New Fractional Hermite–Hadamard-Type Inequalities for Caputo Derivative and MET-(p, s)-Convex Functions with Applications

by Muhammad Sajid Zahoor, Amjad Hussain and Yuanheng Wang

Fractal Fract. 2026, 10(1), 62; https://doi.org/10.3390/fractalfract10010062 - 15 Jan 2026

Viewed by 192

Abstract

This article investigates fractional Hermite–Hadamard integral inequalities through the framework of Caputo fractional derivatives and MET-

(p, s)

-convex functions. In particular, we introduce new modifications to two classical fractional extensions of Hermite–Hadamard-type inequalities, formulated for both MET- [...] Read more.

This article investigates fractional Hermite–Hadamard integral inequalities through the framework of Caputo fractional derivatives and MET-

(p, s)

-convex functions. In particular, we introduce new modifications to two classical fractional extensions of Hermite–Hadamard-type inequalities, formulated for both MET-

(p, s)

-convex functions and logarithmic

(p, s)

-convex functions. Moreover, we obtain enhancements of inequalities like the Hermite–Hadamard, midpoint, and Fejér types for two extended convex functions by employing the Caputo fractional derivative. The research presents a numerical example with graphical representations to confirm the correctness of the obtained results. Full article

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

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23 pages, 3846 KB

Open AccessArticle

A Fractal-Enhanced Mohr–Coulomb (FEMC) Model for Strength Prediction in Rough Rock Discontinuities

by Dina Kon, Sage Ngoie, Jisen Shu, Yadah Mbuyu and Dave Mbako

Fractal Fract. 2026, 10(1), 61; https://doi.org/10.3390/fractalfract10010061 - 15 Jan 2026

Viewed by 313

Abstract

Accurate prediction of the shear strength of rock discontinuities requires accounting for surface roughness, which is a factor neglected in the classical Mohr–Coulomb criterion. This study proposes a fractal-enhanced Mohr–Coulomb model that incorporates the surface fractal dimension Ds as a geometric state variable [...] Read more.

Accurate prediction of the shear strength of rock discontinuities requires accounting for surface roughness, which is a factor neglected in the classical Mohr–Coulomb criterion. This study proposes a fractal-enhanced Mohr–Coulomb model that incorporates the surface fractal dimension Ds as a geometric state variable governing both the cohesion and internal friction angle. The fractal dimension is treated as an objective, scale-invariant descriptor, computable via established methods, such as box-counting and power spectral density analysis, which are known to yield consistent results when applied to joint topography. The model predicts a nonlinear increase in shear strength with Ds, producing a dynamically adjustable failure envelope that can exceed the classical Mohr–Coulomb estimates by 25–40% for rough joints, which is consistent with trends observed in experimental shear tests. By linking strength parameters directly to measurable surface geometry, the framework provides a physically interpretable bridge between micro-scale roughness and macro-scale mechanical response. Although the current formulation assumes monotonic, dry, and quasi-static conditions, the explicit dependence on Ds offers a foundation for future extensions that incorporate anisotropy, damage evolution, and hydro-mechanical coupling. Full article

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

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19 pages, 4153 KB

Open AccessArticle

Pore Structure and Heterogeneity in Deep Coal Reservoirs: Macrolithotype Controls and Implications for CBM Development

by Bo Hu, Xiongxiong Yang, Kui Chen, Shuheng Tang, Xiaohui Li, Songhang Zhang, Jingchen Ding and Ming Zhao

Fractal Fract. 2026, 10(1), 60; https://doi.org/10.3390/fractalfract10010060 - 15 Jan 2026

Viewed by 199

Abstract

The heterogeneity of pore structure in deep coal reservoirs is a critical factor controlling the storage and transport capacity of coalbed methane (CBM). However, the fundamental control exerted by macrolithotypes remains inadequately quantified. This study systematically investigates the No. 8 coal seam of [...] Read more.

The heterogeneity of pore structure in deep coal reservoirs is a critical factor controlling the storage and transport capacity of coalbed methane (CBM). However, the fundamental control exerted by macrolithotypes remains inadequately quantified. This study systematically investigates the No. 8 coal seam of the Taiyuan Formation in the Daniudi gas field, Ordos Basin, using an integrated multi-technique approach including high-pressure mercury intrusion (HPMI), low-temperature N₂ adsorption (LTGA-N₂), and low-pressure CO₂ adsorption (LPGA-CO₂). Results reveal a consistent bimodal pore structure across all samples, dominated by well-developed micropores and macropores, whereas mesopores are relatively underdeveloped. More importantly, a clear macrolithotype control is established: as coal brightness decreases from bright to dull coal, the proportions of micropores and macropores decline significantly, leading to a substantial reduction in total pore volume and specific surface area. Fractal analysis further indicates that dull and semi-dull coals exhibit larger fractal dimensions, reflecting more complex pore structures and stronger heterogeneity compared to bright and semi-bright coals. This heterogeneity shows a positive correlation with ash and mineral contents, but a negative correlation with vitrinite and fixed carbon contents, suggesting that coal composition plays a primary governing role. These findings underscore that bright and semi-bright coals, with their superior micropore storage capacity and well-connected macropore networks, represent the most favorable targets for deep CBM exploration. This work establishes macrolithotype as a practical key indicator for reservoir quality assessment and production strategy optimization in deep CBM plays. Full article

(This article belongs to the Section Engineering)

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15 pages, 373 KB

Open AccessArticle

Nonlinear F-Contractions in Relational Metric Space and Applications to Fractional Differential Equations

by Doaa Filali, Amal F. Alharbi, Faizan Ahmad Khan, Fahad M. Alamrani, Esmail Alshaban and Adel Alatawi

Fractal Fract. 2026, 10(1), 59; https://doi.org/10.3390/fractalfract10010059 - 14 Jan 2026

Viewed by 106

Abstract

During the last decade, F-contraction has been a widely investigated problem in the fixed point theory. There are various outcomes regarding the extensions and generalizations of F-contraction in different perspectives, along with the findings concerning the application of those ideas, mostly in the [...] Read more.

During the last decade, F-contraction has been a widely investigated problem in the fixed point theory. There are various outcomes regarding the extensions and generalizations of F-contraction in different perspectives, along with the findings concerning the application of those ideas, mostly in the area of differential and difference equations, fractional calculus, etc. The present article concludes some existence and uniqueness outcomes on fixed points for

(φ, F)

–contractions in the context of a metric space endowed with a local class of transitive binary relations. Some illustrative examples are furnished to justify that our contraction conditions are more general than many others in this area. The findings presented herein are used to obtain a unique solution to certain fractional boundary value problems. Full article

(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations, Second Edition)

29 pages, 8386 KB

Open AccessArticle

Multifractal Characteristics of the Pore Structure and Resistance to Chloride Ion Penetration of Cement Mortar Modified with a Waterborne Nanosilicate-Based Densifier

by Xin Wang, Rongxin Guo, Haiting Xia, Dian Guan and Zhuo Liu

Fractal Fract. 2026, 10(1), 58; https://doi.org/10.3390/fractalfract10010058 - 14 Jan 2026

Viewed by 239

Abstract

Cementitious composites are heterogeneous porous materials whose pore structure plays a critical role in resistance to chloride-ion penetration. A waterborne nano-silicate-based densifier (CF-S5) was used to examine its influence on the pore structure and resistance to the chloride ion penetration of mortar. We [...] Read more.

Cementitious composites are heterogeneous porous materials whose pore structure plays a critical role in resistance to chloride-ion penetration. A waterborne nano-silicate-based densifier (CF-S5) was used to examine its influence on the pore structure and resistance to the chloride ion penetration of mortar. We investigated the resistance to the chloride ion penetration of mortar with added CF-S5 admixture through the Rapid Chloride Permeability Test (RCPT). We investigated the pore structure characteristics of mortar by mercury intrusion porosimetry (MIP) coupled with fractal theory and investigated the degree of hydration of the cement paste by thermogravimetric analysis (TG). Ultimately, the degree of correlation between multifractal parameters and the chloride ion migration coefficient of mortar was examined using gray relational analysis (GRA). Results indicate that the CF-S5 admixture reduces mortar porosity and the content of harmful pores while increasing pore tortuosity, thus improving the resistance to the chloride ion penetration of mortar. Multifractal analysis indicated that the CF-S5 admixture decreased the connectivity and increased the complexity of the mortar pore structure. The CF-S5 admixture did not reduce the hydration degree of cement paste at 28 d. Additionally, the multifractal parameters show a high gray relational degree with the chloride migration coefficient; therefore, they may serve as potential indicators to reflect the resistance to the chloride ion penetration of mortar. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Materials Science)

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19 pages, 329 KB

Open AccessArticle

Ulam-Type Stability Results for Fractional Integro-Delay Differential and Integral Equations via the ψ-Hilfer Operator

by Cemil Tunç and Osman Tunç

Fractal Fract. 2026, 10(1), 57; https://doi.org/10.3390/fractalfract10010057 - 14 Jan 2026

Viewed by 229

Abstract

In this article, we investigate a nonlinear

ψ

-Hilfer fractional order Volterra integro-delay differential equation (

ψ

-Hilfer FRVIDDE) and a nonlinear

ψ

-Hilfer fractional Volterra delay integral equation (

ψ

-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish [...] Read more.

In this article, we investigate a nonlinear

ψ

-Hilfer fractional order Volterra integro-delay differential equation (

ψ

-Hilfer FRVIDDE) and a nonlinear

ψ

-Hilfer fractional Volterra delay integral equation (

ψ

-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish sufficient conditions for the existence of a unique solution and the Ulam–Hyers stability (U-H stability) of both the

ψ

-Hilfer FRVIDDE and

ψ

-the Hilfer FRVDIE through two new main results. The proof technique relies on the Banach contraction mapping principle, properties of the Hilfer operator, and some additional analytical tools. The considered

ψ

-Hilfer FRVIDDE and

ψ

-Hilfer FRVDIE are new fractional mathematical models in the relevant literature. They extend and improve some available related fractional mathematical models from cases without delay to models incorporating multiple variable time delays, and they also provide new contributions to the qualitative theory of fractional delay differential and fractional delay integral equations. We also give two new examples to verify the applicability of main results of the article. Finally, the article presents substantial and novel results with new examples, contributing to the relevant literature. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

11 pages, 800 KB

Open AccessArticle

Convergence of a Structure-Preserving Scheme for the Space-Fractional Ginzburg–Landau–Schrödinger Equation

by Hongyu Qin, Haoyue Jiang and Xiaoli Chen

Fractal Fract. 2026, 10(1), 56; https://doi.org/10.3390/fractalfract10010056 - 14 Jan 2026

Viewed by 155

Abstract

We present a linearly implicit and structure-preserving scheme to solve the space-fractional Ginzburg–Landau–Schrödinger equation. The fully discrete scheme is obtained by combining the modified leap-frog method in the temporal direction and the finite difference methods in the spatial direction. It is shown that [...] Read more.

We present a linearly implicit and structure-preserving scheme to solve the space-fractional Ginzburg–Landau–Schrödinger equation. The fully discrete scheme is obtained by combining the modified leap-frog method in the temporal direction and the finite difference methods in the spatial direction. It is shown that the scheme can be unconditionally energy-stable. In particular, the equation becomes the space-fractional Schrödinger equation. Then, the scheme can keep both the discrete mass and energy conserved. Moreover, convergence of the scheme is obtained. Numerical experiments are performed to confirm the theoretical results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 3rd Edition)

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41 pages, 6499 KB

Open AccessArticle

Cascaded Optimized Fractional Controller for Green Hydrogen-Based Microgrids with Mitigating False Data Injection Attacks

by Nadia A. Nagem, Mokhtar Aly, Emad A. Mohamed, Aisha F. Fareed, Dokhyl M. Alqahtani and Wessam A. Hafez

Fractal Fract. 2026, 10(1), 55; https://doi.org/10.3390/fractalfract10010055 - 13 Jan 2026

Viewed by 251

Abstract

Green hydrogen production and the use of fuel cells (FCs) in microgrid (MG) systems have become viable and feasible solutions due to their continuous cost reduction and advancements in technology. Furthermore, green hydrogen electrolyzers and FC can mitigate fluctuations in renewable energy generation [...] Read more.

Green hydrogen production and the use of fuel cells (FCs) in microgrid (MG) systems have become viable and feasible solutions due to their continuous cost reduction and advancements in technology. Furthermore, green hydrogen electrolyzers and FC can mitigate fluctuations in renewable energy generation and various demand-related disturbances. Proper incorporation of electrolyzers and FCs can enhance load frequency control (LFC) in MG systems. However, they are subjected to multiple false data injection attacks (FDIAs), which can deteriorate MG stability and availability. Moreover, most existing LFC control schemes—such as conventional PID-based methods, single-degree-of-freedom fractional-order controllers, and various optimization-based structures—lack robustness against coordinated and multi-point FDIAs, leading to significant degradation in frequency regulation performance. This paper presents a new, modified, multi-degree-of-freedom, cascaded fractional-order controller for green hydrogen-based MG systems with high fluctuating renewable and demand sources. The proposed LFC is a cascaded control structure that combines a 1+TID controller with a filtered fractional-order PID controller (FOPIDF), namely the cascaded 1+TID/FOPIDF LFC control. Furthermore, another tilt-integrator derivative electric vehicle (EV) battery frequency regulation controller is proposed to benefit from EVs installed in MG systems. The proposed cascaded 1+TID/FOPIDF LFC control and EV TID LFC methods are designed using the powerful capability of the exponential distribution optimizer (EDO), which determines the optimal set of design parameters, leading to guaranteed optimal performance. The effectiveness of the newly proposed cascaded 1+TID/FOPIDF LFC control and design approach employing multi-generational-based two-area MG systems is studied by taking into account a variety of projected scenarios of FDIAs and renewable/load fluctuation scenarios. In addition, performance comparisons with some featured controllers are provided in the paper. For example, in the case of fluctuation in RESs, the measured indices are as follows: ISE (1.079, 0.5306, 0.3515, 0.0104); IAE (15.011, 10.691, 9.527, 1.363); ITSE (100.613, 64.412, 53.649, 1.323); and ITAE (2120, 1765, 1683, 241.32) for TID, FOPID, FOTID, and proposed, respectively, which confirm superior frequency deviation mitigation using the proposed optimized cascaded 1+TID/FOPIDF and EV TID LFC control method. Full article

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28 pages, 652 KB

Open AccessArticle

A Generalized Fractional Legendre-Type Differential Equation Involving the Atangana–Baleanu–Caputo Derivative

by Muath Awadalla and Dalal Alhwikem

Fractal Fract. 2026, 10(1), 54; https://doi.org/10.3390/fractalfract10010054 - 13 Jan 2026

Viewed by 121

Abstract

This paper introduces a fractional generalization of the classical Legendre differential equation based on the Atangana–Baleanu–Caputo (ABC) derivative. A novel fractional Legendre-type operator is rigorously defined within a functional framework of continuously differentiable functions with absolutely continuous derivatives. The associated initial value problem [...] Read more.

This paper introduces a fractional generalization of the classical Legendre differential equation based on the Atangana–Baleanu–Caputo (ABC) derivative. A novel fractional Legendre-type operator is rigorously defined within a functional framework of continuously differentiable functions with absolutely continuous derivatives. The associated initial value problem is reformulated as an equivalent Volterra integral equation, and existence and uniqueness of classical solutions are established via the Banach fixed-point theorem, supported by a proved Lipschitz estimate for the ABC derivative. A constructive solution representation is obtained through a Volterra–Neumann series, explicitly revealing the role of Mittag–Leffler functions. We prove that the fractional solutions converge uniformly to the classical Legendre polynomials as the fractional order approaches unity, with a quantitative convergence rate of order

O (1 - α)

under mild regularity assumptions on the Volterra kernel. A fully reproducible quadrature-based numerical scheme is developed, with explicit kernel formulas and implementation algorithms provided in appendices. Numerical experiments for the quadratic Legendre mode confirm the theoretical convergence and illustrate the smooth interpolation between fractional and classical regimes. An application to time-fractional diffusion in spherical coordinates demonstrates that the operator arises naturally in physical models, providing a mathematically consistent tool for extending classical angular analysis to fractional settings with memory. Full article

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14 pages, 383 KB

Open AccessArticle

From Mathematics to Art: A Petri Net Representation of the Fibonacci Sequence and Its Fractal Geometry

by David Mailland and Iwona Grobelna

Fractal Fract. 2026, 10(1), 53; https://doi.org/10.3390/fractalfract10010053 - 13 Jan 2026

Viewed by 304

Abstract

Mathematics, as Bertrand Russell noted, possesses both truth and beauty. In this work, we revisit the classical Fibonacci recurrence thanks to a minimal Petri net. Starting from a minimal layered construction that mirrors the second-order additive rule [...] Read more.

Mathematics, as Bertrand Russell noted, possesses both truth and beauty. In this work, we revisit the classical Fibonacci recurrence thanks to a minimal Petri net. Starting from a minimal layered construction that mirrors the second-order additive rule

F_{n} = F_{n - 1} + F_{n - 2}

, we show that the marking dynamics of the associated net generate a combinatorial triangle whose parity structure reveals a self-similar, Sierpiński-like pattern. To the best of our knowledge, this oblique fractal geometry has never been formally documented. We provide a formal definition of the underlying Petri net, analyse its computational properties, and explore the emergence of higher-order harmonics when token markings are considered modulo primes. The study highlights how a classical recurrence gives rise to previously unnoticed geometric regularities at the intersection of mathematics and art. Beyond its mathematical interest, the construction illustrates how minimal Petri net dynamics can be used as formal specification patterns for distributed, event-driven systems. Full article

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44 pages, 642 KB

Open AccessArticle

A Fractional q-Rung Orthopair Fuzzy Tensor Framework for Dynamic Group Decision-Making: Application to Smart City Renewable Energy Planning

by Muhammad Bilal, Chaoqian Li, A. K. Alzahrani and A. K. Aljahdali

Fractal Fract. 2026, 10(1), 52; https://doi.org/10.3390/fractalfract10010052 - 13 Jan 2026

Viewed by 136

Abstract

In complex decision-making scenarios, such as smart city renewable energy project selection, decision-makers must contend with multi-dimensional uncertainty, conflicting expert opinions, and evolving temporal dynamics. This study introduces a novel Fractional q-Rung Orthopair Fuzzy Tensor (Fq-ROFT)-based group decision-making methodology that integrates the flexibility [...] Read more.

In complex decision-making scenarios, such as smart city renewable energy project selection, decision-makers must contend with multi-dimensional uncertainty, conflicting expert opinions, and evolving temporal dynamics. This study introduces a novel Fractional q-Rung Orthopair Fuzzy Tensor (Fq-ROFT)-based group decision-making methodology that integrates the flexibility of q-rung orthopair fuzzy sets with tensorial representation and fractional-order dynamics. The proposed framework allows for the modeling of positive and negative membership degrees in a multi-dimensional, time-dependent structure while capturing memory effects inherent in expert evaluations. A detailed case study involving six renewable energy alternatives and six criteria demonstrates the method’s ability to aggregate expert opinions, compute fractional dynamic scores, and provide robust, reliable rankings. Comparative analysis with existing approaches, including classical q-ROFSs, intuitionistic fuzzy sets, and weighted sum methods, highlights the superior discriminative power, consistency, and dynamic sensitivity of the Fq-ROFT approach. Sensitivity analysis confirms the robustness of the top-ranked alternatives under variations in expert weights and fractional orders and membership perturbations. The study concludes by discussing the advantages, limitations, and future research directions of the proposed methodology, establishing Fq-ROFT as a powerful tool for dynamic, high-dimensional, and uncertain group decision-making applications. Full article

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18 pages, 3053 KB

Open AccessArticle

Dynamics and Chaos Analysis of the Fractional-Order Lü System Using a Hybrid Approach

by Mohamed Elbadri, Naseam Al-kuleab, Rania Saadeh, Mohamed Hafez and Mohamed A. Abdoon

Fractal Fract. 2026, 10(1), 51; https://doi.org/10.3390/fractalfract10010051 - 13 Jan 2026

Viewed by 183

Abstract

In this study, an analysis of fractional-order Lü systems is performed through a framework approach consisting of analytical solution strategies in combination with numerical methods. On the analytical methodology front, the recently developed form of the new generalized differential transform method (NGDTM) is [...] Read more.

In this study, an analysis of fractional-order Lü systems is performed through a framework approach consisting of analytical solution strategies in combination with numerical methods. On the analytical methodology front, the recently developed form of the new generalized differential transform method (NGDTM) is adopted for its efficiency in providing an approximate solution with high capability in tracking the behavior of these systems. On the other hand, the Grünwald–Letnikov via Riemann–Liouville scheme (GLNS) is adopted within this study as one of its tools in confirming whether chaos exists within these systems. The performance and accuracy of the proposed method are also rigorously tested, and comparisons are made numerically with the Adams–Bashforth–Moulton method, which is used here as a standard method for validation purposes. It is clear from the results that the combination of analytical and numerical methods can greatly enhance both the speed of computation and the accuracy of results. Additionally, the proposed method or approach is found to be quite robust and accurate and can thus be employed for analyzing various fractional dynamical systems that display chaotic attractors. The proposed method can also be expanded upon in the future for analyzing complex models in science and engineering. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Chaotic and Complex Systems)

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23 pages, 838 KB

Open AccessArticle

Stability for Caputo–Hadamard Fractional Uncertain Differential Equation

by Shida Peng, Zhi Li, Jun Zhang, Yuncong Zhu and Liping Xu

Fractal Fract. 2026, 10(1), 50; https://doi.org/10.3390/fractalfract10010050 - 12 Jan 2026

Viewed by 165

Abstract

This paper focuses on the Caputo-Hadamard fractional uncertain differential equations (CH-FUDEs) governed by Liu processes, which combine the Caputo–Hadamard fractional derivative with uncertain differential equations to describe dynamic systems involving memory characteristics and uncertain information. Within the framework of uncertain theory, this Liu [...] Read more.

This paper focuses on the Caputo-Hadamard fractional uncertain differential equations (CH-FUDEs) governed by Liu processes, which combine the Caputo–Hadamard fractional derivative with uncertain differential equations to describe dynamic systems involving memory characteristics and uncertain information. Within the framework of uncertain theory, this Liu process serves as the counterpart to Brownian motion. We establish some new Bihari type fractional inequalities that are easy to apply in practice and can be considered as a more general tool in some situations. As applications of those inequalities, we establish the well-posedness of a proposed class of equations under specified non-Lipschitz conditions. Building upon this result, we establish the notions of stability in distribution and stability in measure solutions to CH-FUDEs, deriving sufficient conditions to ensure these stability properties. Finally, the theoretical findings are verified through two numerical examples. Full article

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

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25 pages, 522 KB

Open AccessArticle

Fractional Integral Estimates of Boole Type: Majorization and Convex Function Approach with Applications

by Saad Ihsan Butt, Mohammed Alammar and Youngsoo Seol

Fractal Fract. 2026, 10(1), 49; https://doi.org/10.3390/fractalfract10010049 - 12 Jan 2026

Viewed by 159

Abstract

The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex [...] Read more.

The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex functions with classical inequalities like the Power mean and Hölder inequalities, as well as the Niezgoda–Jensen–Mercer (NJM) inequality for majorized tuples. Additionally, the study presents real-world examples involving special functions and examines pertinent quadrature rules. This work’s primary contribution is the extension and generalization of a number of results that are already known in the current body of mathematical literature. Full article

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

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20 pages, 937 KB

Open AccessArticle

Quasi-Consensus of Fractional-Order Multi-Agent Systems with Mixed Delays and External Disturbance via Impulsive Pinning Control

by Tao Chen, Zhiwen Fu, Caimao Su, Ning Chen and Wanli Lin

Fractal Fract. 2026, 10(1), 48; https://doi.org/10.3390/fractalfract10010048 - 12 Jan 2026

Viewed by 225

Abstract

In this brief, the quasi-consensus issue is examined for fractional-order multi-agent systems (FROMASs) subject to both mixed delays and external disturbances, employing an impulsive pinning control (IPC) strategy. Unlike mainstream pinning strategies with fixed nodes or static rules, a dynamic pinning mechanism based [...] Read more.

In this brief, the quasi-consensus issue is examined for fractional-order multi-agent systems (FROMASs) subject to both mixed delays and external disturbances, employing an impulsive pinning control (IPC) strategy. Unlike mainstream pinning strategies with fixed nodes or static rules, a dynamic pinning mechanism based on consensus error distances is proposed, which adaptively adjusts the set of pinned nodes at each impulsive instant. By integrating the Razumikhin method, Lyapunov stability theory, and the comparison system approach, sufficient conditions for achieving quasi-consensus of FROMASs are established. Moreover, the convergence bound of consensus errors is quantitatively estimated and shown to be explicitly determined by the intensity of external disturbances. This paper organically integrates the dynamic node pinning mechanism, FRO impulsive control, and Razumikhin stability analysis. It effectively handles mixed delays and external disturbances while significantly reducing control costs. Finally, numerical simulations show that the synchronization errors are strictly bounded within the threshold

M = 13.6372

, which effectively validates both the proposed control scheme and the theoretical analysis. Full article

(This article belongs to the Topic Modeling, Stability, and Control of Dynamic Systems and Their Applications)

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15 pages, 2130 KB

Open AccessArticle

Characterization and Analysis of Hybrid Fractal Antennas for Multiband Communication and Radar Applications

by Abdelbasset Azzouz, Rachid Bouhmidi, Mehr E. Munir, Moustafa M. Nasralla and Mohammed Chetioui

Fractal Fract. 2026, 10(1), 47; https://doi.org/10.3390/fractalfract10010047 - 12 Jan 2026

Viewed by 278

Abstract

This work introduces the development and performance analysis of a hybrid fractal antenna combining a Koch snowflake outer geometry with a center slot patterned as a Sierpinski rectangular carpet. The antenna is fabricated on an FR4 board (

ε_{r} = 4.7

, [...] Read more.

This work introduces the development and performance analysis of a hybrid fractal antenna combining a Koch snowflake outer geometry with a center slot patterned as a Sierpinski rectangular carpet. The antenna is fabricated on an FR4 board (

ε_{r} = 4.7

,

\tan δ = 0.0197

) with dimensions

40 \times 60 \times 0.8

mm³. Electromagnetic simulations are performed using Ansys HFSS v15, revealing seven distinct resonances at 2.11, 3.06, 5.78, 6.94, 8.48, 9.23, and 9.56 GHz. The corresponding impedance bandwidths are 90, 37, 67, 100, 90, 130, and 220 MHz, with return losses of −14, −12, −16, −10, −30, −16, and −17 dB, and VSWR values ranging from 1.06 to 1.80. The gains at these resonances are 3.92, 8.24, 6.90, 11.66, 19.38, 16.76, and 12.06 dBi. Frequency allocation analysis indicates compatibility with UMTS/LTE (2.11 GHz), S-band 5G and radar (3.06 GHz), ISM/UNII-3 Wi-Fi and ITS (5.78 GHz), C-band satellite uplink (6.94 GHz), and X-band radar/satellite downlink (8.48–9.56 GHz). The proposed geometry demonstrates wide multi-band coverage, making it a strong candidate for integration into multi-standard communication and radar platforms requiring compact, broadband, and high-directivity performance. Full article

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26 pages, 3417 KB

Open AccessArticle

Optimal Fractional Order PID Controller Design for Hydraulic Turbines Using a Multi-Objective Imperialist Competitive Algorithm

by Mohamed Nejlaoui, Abdullah Alghafis and Nasser Ayidh Alqahtani

Fractal Fract. 2026, 10(1), 46; https://doi.org/10.3390/fractalfract10010046 - 11 Jan 2026

Cited by 1 | Viewed by 235

Abstract

This paper introduces a novel approach for designing a Fractional Order Proportional-Integral-Derivative (FOPID) controller for the Hydraulic Turbine Regulating System (HTRS), aiming to overcome the challenge of tuning its five complex parameters (

K_{p}, K_{i}, K_{d}

, λ [...] Read more.

This paper introduces a novel approach for designing a Fractional Order Proportional-Integral-Derivative (FOPID) controller for the Hydraulic Turbine Regulating System (HTRS), aiming to overcome the challenge of tuning its five complex parameters (

K_{p}, K_{i}, K_{d}

, λ and μ). The design is formulated as a multi-objective optimization problem, minimized using the Multi-Objective Imperialist Competitive Algorithm (MOICA). The goal is to minimize two key transient performance metrics: the Integral of Squared Error (ISE) and the Integral of the Time Multiplied Squared Error (ITSE). MOICA efficiently generates a Pareto-front of non-dominated solutions, providing control system designers with diverse trade-off options. The resulting optimal FOPID controller demonstrated superior robustness when evaluated against simulated variations in key HTRS parameters (mg, eg and Tw). Comparative simulations against an optimally tuned integer-order PID and established literature methods (FOPID-GA, FOPID-MOPSO and FOPID-MOHHO) confirm the enhanced dynamic response and stable operation of the MOICA-based FOPID. The MOICA-tuned FOPID demonstrated superior performance for Setpoint Tracking, achieving up to a 26% faster settling speed (ITSE) and an 8% higher accuracy (ISE). Furthermore, for Disturbance Rejection, it showed enhanced robustness, leading to up to a 23% quicker recovery speed (ITSE) and an 18.9% greater error suppression (ISE). Full article

(This article belongs to the Section Engineering)

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6 pages, 185 KB

Open AccessEditorial

Advances in Fractional Modeling and Computation

by Yuri Dimitrov, Slavi Georgiev, Venelin Todorov and Jordan Hristov

Fractal Fract. 2026, 10(1), 45; https://doi.org/10.3390/fractalfract10010045 - 11 Jan 2026

Viewed by 173

Abstract

Fractional calculus, the study of derivatives and integrals of arbitrary order [...] Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

25 pages, 554 KB

Open AccessArticle

Dynamic Analysis and Optimal Prevention Strategies for Monkeypox Spread Modeled via the Mittag–Leffler Kernel

by Mine Yurtoğlu, Dilara Yapışkan, Ebenezer Bonyah, Beyza Billur İskender Eroğlu, Derya Avcı and Delfim F. M. Torres

Fractal Fract. 2026, 10(1), 44; https://doi.org/10.3390/fractalfract10010044 - 10 Jan 2026

Viewed by 206

Abstract

Monkeypox is a viral disease belonging to the smallpox family. Although it has milder symptoms than smallpox in humans, it has become a global threat in recent years, especially in African countries. Initially, incidental immunity against monkeypox was provided by smallpox vaccines. However, [...] Read more.

Monkeypox is a viral disease belonging to the smallpox family. Although it has milder symptoms than smallpox in humans, it has become a global threat in recent years, especially in African countries. Initially, incidental immunity against monkeypox was provided by smallpox vaccines. However, the eradication of smallpox over time and thus the lack of vaccination has led to the widespread and clinical importance of monkeypox. Although mathematical epidemiology research on the disease is complementary to clinical studies, it has attracted attention in the last few years. The present study aims to discuss the indispensable effects of three control strategies such as vaccination, treatment, and quarantine to prevent the monkeypox epidemic modeled via the Atangana–Baleanu operator. The main purpose is to determine optimal control measures planned to reduce the rates of exposed and infected individuals at the minimum costs. For the controlled model, the existence-uniqueness of the solutions, stability, and sensitivity analysis, and numerical optimal solutions are exhibited. The optimal system is numerically solved using the Adams-type predictor–corrector method. In the numerical simulations, the efficacy of the vaccination, treatment, and quarantine controls is evaluated in separate analyzes as single-, double-, and triple-control strategies. The results demonstrate that the most effective strategy for achieving the aimed outcome is the simultaneous application of vaccination, treatment, and quarantine controls. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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