Fractal and Fractional

41 pages, 8076 KB

Open AccessArticle

THMD Coupling Modelling and Crack Propagation Analysis of Coal Rock Under In Situ Liquid Nitrogen Fracturing

by Qiang Li, Yunbo Li, Dangyu Song, Rongqi Wang, Jienan Pan, Zhenzhi Wang and Chengtao Wang

Fractal Fract. 2026, 10(4), 274; https://doi.org/10.3390/fractalfract10040274 - 21 Apr 2026

Viewed by 322

Liquid nitrogen (LN₂) fracturing is a highly promising stimulation technology for unconventional reservoirs. Understanding its in situ fracture network formation mechanism is essential for engineering practice. This study investigates coal rock fracturing driven by the synergistic effect of thermal stress and [...] Read more.

Liquid nitrogen (LN₂) fracturing is a highly promising stimulation technology for unconventional reservoirs. Understanding its in situ fracture network formation mechanism is essential for engineering practice. This study investigates coal rock fracturing driven by the synergistic effect of thermal stress and fluid pressure during LN₂ injection. A coupled thermal–hydraulic–mechanical–damage (THMD) numerical model is developed, incorporating in situ stress conditions and LN₂ phase change behavior. Through true triaxial LN₂ fracturing simulations validated against physical experiments, the multi-field dynamic coupling behavior is systematically analyzed, revealing the synergistic mechanism of fracture propagation and permeability enhancement under cryogenic conditions. The results show the following: (1) The proposed model effectively reproduces the true triaxial LN₂ fracturing process, with simulation results in good agreement with physical experiments. (2) LN₂ fracturing exhibits distinct stage-wise characteristics: cryogenic temperatures induce thermal stress that triggers micro-crack initiation; the self-enhancing effects of damage and permeability significantly promote fracture propagation; fluid pressure then becomes the dominant driving force. (3) Coal rock damage follows a four-stage evolution—wellbore crack initiation, stable propagation, unstable propagation, and through-going failure—ultimately forming a complex spatial fracture network. (4) The horizontal stress ratio is a key factor controlling fracture morphology: a single dominant fracture forms under a high stress difference, whereas a multi-directional complex network develops under equal confining pressure. Fractal analysis reveals significant anisotropy and a non-monotonic stress response in the fracture complexity, reflecting structural evolution from multi-directional propagation to main channel connection. This study provides theoretical support for understanding LN₂ fracturing mechanisms and optimizing field treatment parameters. Full article

(This article belongs to the Special Issue Recent Advances in Fractal Analysis for Hydrocarbon Dynamics and Flow Modeling)

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42 pages, 4491 KB

Open AccessArticle

Fractional Diffusion on Graphs: Superposition of Laplacian Semigroups Incorporating Memory

by Nikita Deniskin and Ernesto Estrada

Fractal Fract. 2026, 10(4), 273; https://doi.org/10.3390/fractalfract10040273 - 21 Apr 2026

Viewed by 315

Abstract

Subdiffusion on graphs is often modeled by time-fractional diffusion equations; yet, its structural and dynamical consequences remain unclear. We show that subdiffusive transport on graphs is a memory-driven process generated by a random time change that compresses operational time, produces long-tailed waiting times, [...] Read more.

Subdiffusion on graphs is often modeled by time-fractional diffusion equations; yet, its structural and dynamical consequences remain unclear. We show that subdiffusive transport on graphs is a memory-driven process generated by a random time change that compresses operational time, produces long-tailed waiting times, and breaks Markovianity while preserving linearity and mass conservation. While the subordination representation and complete monotonicity properties of the Mittag-Leffler function are classical, we develop a graph-based synthesis in which Mittag-Leffler dynamics admit an exact convex, mass-preserving representation as a superposition of Laplacian semigroups evaluated at rescaled times. This perspective reveals fractional diffusion as ordinary diffusion acting across multiple intrinsic time scales and enables new structural and dynamical interpretations of graphs. This framework uncovers heterogeneous, vertex-dependent memory effects and induces transport biases absent in classical diffusion, including algebraic relaxation, degree-dependent waiting times, and early-time asymmetries between sources and neighbors. These features define a subdiffusive geometry on graphs, enabling the recovery of global shortest paths, in contrast to the graph exploration of diffusive geometry, while simultaneously favoring high-degree regions. Finally, we show that time-fractional diffusion can be interpreted as a singular limit of multi-rate diffusion, in an appropriate asymptotic sense. Full article

(This article belongs to the Special Issue Fractal Analysis and Data-Driven Complex Systems)

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19 pages, 397 KB

Open AccessArticle

On a Class of Nonlocal Integro-Delay Problems with Generalized Tempered Fractional Operators

by Marwa Ennaceur, Mohammed S. Abdo, Osman Osman, Amel Touati, Amer Alsulami, Neama Haron and Khaled Aldwoah

Fractal Fract. 2026, 10(4), 272; https://doi.org/10.3390/fractalfract10040272 - 21 Apr 2026

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Abstract

This paper proposes and studies a new class of nonlinear nonlocal problem driven by a tempered Caputo-type fractional derivative with respect to an arbitrary smooth kernel. The novelty lies in treating a single nonlocal integro-delay setting that simultaneously couples an arbitrary kernel, exponential [...] Read more.

This paper proposes and studies a new class of nonlinear nonlocal problem driven by a tempered Caputo-type fractional derivative with respect to an arbitrary smooth kernel. The novelty lies in treating a single nonlocal integro-delay setting that simultaneously couples an arbitrary kernel, exponential tempering, a delayed state, a lower-order distributed fractional memory term, and multipoint nonlocal initial data, rather than introducing a new fractional operator. The resulting problem can be viewed as a rigorous well-posedness prototype for hereditary systems with delayed feedback, tempered memory, and nonlocal initialization. First, an equivalent Volterra integral equation is derived. Then, the existence and uniqueness of solutions are obtained by the Banach contraction principle in a suitable Banach space of continuous functions. Next, a Picard successive approximation procedure is introduced and shown to converge uniformly to the unique solution, together with an explicit a priori error estimate. Moreover, a continuous dependence result is proved with respect to perturbations in the initial constants, the multipoint coefficients, and the nonlinear term. Finally, the main results are illustrated with two examples enhanced by graphs of explicit Picard approximations and convergence tables. Full article

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

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23 pages, 4334 KB

Open AccessArticle

Pore Structure and Fractal Characteristics of Low-Maturity Shales in the Upper-Fourth Shahejie Formation, Minfeng Sag

by Chijun Huang, Shaohua Li, Changsheng Lu, Zhihui Peng, Long Jiang, Yu Li and Siyu Yu

Fractal Fract. 2026, 10(4), 271; https://doi.org/10.3390/fractalfract10040271 - 21 Apr 2026

Viewed by 349

Abstract

An integrated analysis incorporating total organic carbon (TOC) content measurement, X-ray diffraction (XRD), scanning electron microscopy (SEM), and gas adsorption experiments was performed on core samples from Well FY1-4 of the upper-fourth Shahejie Formation (Es₄) in the Minfeng Sag. To address [...] Read more.

An integrated analysis incorporating total organic carbon (TOC) content measurement, X-ray diffraction (XRD), scanning electron microscopy (SEM), and gas adsorption experiments was performed on core samples from Well FY1-4 of the upper-fourth Shahejie Formation (Es₄) in the Minfeng Sag. To address the lack of systematic research on the pore and fractal characteristics of organic-rich low-maturity shales in the Minfeng Sag (against the preponderance of studies on high-maturity shales), this study characterized the lithofacies, reservoir space and pore fractal features of the target low-maturity shale interval and clarified the sedimentary controls on lithofacies and key factors regulating pore fractal heterogeneity. The results reveal that the shale in the Es₄ of the study area exhibits low thermal maturity, with six distinct lithofacies identified. Organic-rich laminated calcareous shale lithofacies (RL-1) and organic-rich laminated calcareous/argillaceous mixed shale lithofacies (RL-2) represent the most favorable lithofacies, which are dominated by large mesopores and macropores. Their reservoir spaces were primarily composed of intergranular pores, intragranular pores, and organic pores, whereas the other lithofacies are dominated by small mesopores. The pore surface fractal dimension (D) was calculated using the Frenkel–Halsey–Hill (FHH) model based on low-temperature N₂ adsorption (LTNA) data. The meso-macropore system shows higher heterogeneity than the micropore system (D₂ > D₁). Both D₁ and D₂ exhibit a weak negative correlation with TOC and carbonate content and a positive correlation with clay content. In the initial depositional stage of the Es₄, the arid climate, weak terrigenous input, shallow lake depth, and high salinity resulted in the strongly reducing saline depositional environment with relatively low organic matter enrichment. As the climate became progressively humid in the middle and late stages, hydrodynamic conditions intensified, leading to a lithofacies transition from mixed shales to argillaceous calcareous shales. Increased TOC and carbonate contents reduce the pore fractal dimension of shale. Smaller fractal dimensions directly indicate a simple pore structure and regular pore surface in the shale oil reservoir of the Minfeng Sag, where reservoir space is dominated by large pores such as intercrystalline pores and dissolved pores. Such pore characteristics are more favorable for the enrichment of shale oil. Full article

(This article belongs to the Special Issue Fractal Analysis in Unconventional Reservoirs: Theory and Applications)

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45 pages, 7742 KB

Open AccessArticle

Fractional-Order Typhoid Fever Dynamics and Parameter Identification via Physics-Informed Neural Networks

by Mallika Arjunan Mani, Kavitha Velusamy, Sowmiya Ramasamy and Seenith Sivasundaram

Fractal Fract. 2026, 10(4), 270; https://doi.org/10.3390/fractalfract10040270 - 21 Apr 2026

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Abstract

This paper presents a unified analytical and computational framework for the study of typhoid fever transmission dynamics governed by a Caputo fractional-order compartmental model of order

κ \in (0, 1]

. The population is stratified into five epidemiological classes, namely [...] Read more.

This paper presents a unified analytical and computational framework for the study of typhoid fever transmission dynamics governed by a Caputo fractional-order compartmental model of order

κ \in (0, 1]

. The population is stratified into five epidemiological classes, namely susceptible

(S)

, asymptomatic

(A)

, symptomatic

(I)

, hospitalised

(H)

, and recovered

(R)

, and the governing system explicitly incorporates asymptomatic transmission, treatment dynamics, and temporary immunity with waning. The use of the Caputo fractional derivative is motivated by the well-documented existence of chronic asymptomatic Salmonella Typhi carriers, whose heavy-tailed sojourn times in the carrier state are naturally encoded by the Mittag–Leffler waiting-time distribution arising from the fractional operator. A complete qualitative analysis of the fractional system is carried out: the basic reproduction number

R_{0}

is derived via the next-generation matrix method; local and global asymptotic stability of both the disease-free equilibrium

E_{0}

(when

R_{0} \leq 1

) and the endemic equilibrium

E^{*}

(when

R_{0} > 1

) are established using fractional Lyapunov theory and the LaSalle invariance principle; and the normalised sensitivity indices of

R_{0}

are computed to identify transmission-amplifying and transmission-suppressing parameters. Existence, uniqueness, and Ulam–Hyers stability of solutions are established via Banach and Leray–Schauder fixed-point arguments. To complement the analytical results, a fractional physics-informed neural network (

PINN

) framework is developed to simultaneously reconstruct compartmental trajectories and identify unknown biological parameters from sparse synthetic observations.

PINN

embeds the L1-Caputo discretisation directly into the training residuals and employs a four-stage Adam–L-BFGS optimisation strategy to recover five trainable parameters

Θ

=

{ϕ, μ, σ, ψ, β}

across three fractional orders

κ \in {1.0, 0.95, 0.9}

. The estimated parameters show strong agreement with the true values at the classical limit

κ = 1.0

(

MAPE = 2.27 %

), with the natural mortality rate

μ

recovered with

APE \leq 0.51 %

and the transmission rate

β

with

APE \leq 3.63 %

across all fractional orders, confirming the structural identifiability of the model. Pairwise correlation analysis of the learned parameters establishes the absence of equifinality, validating that

β

can be reliably included in the trainable set. Noise robustness experiments under Gaussian perturbations of

1 %

,

3 %

, and

5 %

demonstrate graceful degradation (

MAPE

:

0.82 % \to 3.10 % \to 7.31 %

), confirming the reliability of the proposed framework under realistic observational conditions. Full article

(This article belongs to the Special Issue Fractional Dynamics Systems: Modeling, Forecasting, and Control)

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44 pages, 7084 KB

Open AccessArticle

Fractional-Order Anteater Foraging Optimization Algorithm for Compact Layout Design of Electro-Hydrostatic Actuator Controllers

by Shuai Cao, Wei Xu, Weibo Li, Kangzheng Huang and Xiaoqing Deng

Fractal Fract. 2026, 10(4), 269; https://doi.org/10.3390/fractalfract10040269 - 20 Apr 2026

Viewed by 434

Abstract

The development of More Electric Aircraft (MEA) necessitates that Electro-Hydrostatic Actuator (EHA) controllers achieve exceptional power density within rigorously constrained volumes. However, the compact layout design of these controllers constitutes a challenging NP-hard problem, characterized by strong multi-physics coupling—such as electromagnetic, thermal, and [...] Read more.

The development of More Electric Aircraft (MEA) necessitates that Electro-Hydrostatic Actuator (EHA) controllers achieve exceptional power density within rigorously constrained volumes. However, the compact layout design of these controllers constitutes a challenging NP-hard problem, characterized by strong multi-physics coupling—such as electromagnetic, thermal, and structural fields—and complex nonlinear constraints. Traditional meta-heuristic algorithms frequently suffer from premature convergence and struggle to balance global exploration with local exploitation. To address these challenges, the core contribution of this paper is the proposal of a novel Fractional-Order Anteater Foraging Optimization Algorithm (AFO), which is successfully applied to an established EHA controller layout optimization model. At the algorithmic level, by incorporating the Grünwald–Letnikov fractional derivative, the algorithm exploits the inherent memory property of fractional calculus to dynamically adjust the search step size and direction based on historical evolutionary information, thereby preventing stagnation in local optima. At the engineering application level, a high-fidelity mathematical model of the EHA controller is established, comprising 11 design variables and 10 critical physical constraints, including parasitic inductance minimization, thermal radiation efficiency, and electromagnetic interference (EMI) isolation. Extensive validation against the CEC2005 and CEC2022 benchmark functions demonstrates the superior convergence accuracy and stability of the AFO algorithm. In a specific EHA case study, the proposed method reduced the controller volume by 33.9% while strictly satisfying all multi-physics constraints, compared to traditional methods. Furthermore, a physical prototype was fabricated based on the optimized layout, and experimental tests confirmed its stable operation and excellent thermal performance. The results validate the efficacy of incorporating fractional calculus into bio-inspired algorithms to solve complex, high-dimensional engineering optimization problems. Full article

(This article belongs to the Section Engineering)

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28 pages, 702 KB

Open AccessArticle

A Hybrid Neural Network Approach to Controllability in Caputo Fractional Neutral Integro-Differential Systems for Cryptocurrency Forecasting

by Prabakaran Raghavendran and Yamini Parthiban

Fractal Fract. 2026, 10(4), 268; https://doi.org/10.3390/fractalfract10040268 - 18 Apr 2026

Viewed by 356

Abstract

This research paper demonstrates how to manage Caputo fractional neutral integro-differential equations which include both integral and nonlinear elements through a unified framework that models dynamic systems with memory-based dynamics. The research establishes sufficient conditions for controllability through fixed point theory in a [...] Read more.

This research paper demonstrates how to manage Caputo fractional neutral integro-differential equations which include both integral and nonlinear elements through a unified framework that models dynamic systems with memory-based dynamics. The research establishes sufficient conditions for controllability through fixed point theory in a Banach space framework which requires particular assumptions while the study focuses on the

K_{1} < 1

condition which leads to the existence of a controllable solution. The proposed criteria are demonstrated through a numerical example which tests the theoretical results. The real-world case study uses artificial neural network (ANN) technology to predict Litecoin prices through the application of the fractional controllability model which analyzes historical financial data. The hybrid framework enables precise forecasting of nonlinear time series because it combines fractional calculus mathematical principles with ANN learning abilities. The proposed method demonstrates its predictive efficiency. The method shows robust performance through experimental results using cross-validation and performance metrics. The proposed model demonstrates competitive performance while providing additional advantages such as incorporation of memory effects and theoretical controllability. The research establishes a novel connection between fractional dynamical systems and machine learning which serves as an essential tool for studying complicated systems in theoretical research and practical applications. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section 2026)

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25 pages, 7570 KB

Open AccessArticle

Relationship of Multifractal and Entropic Properties of Global Seismic Noise with Major Earthquakes, 1997–2025

by Alexey Lyubushin and Eugeny Rodionov

Fractal Fract. 2026, 10(4), 267; https://doi.org/10.3390/fractalfract10040267 - 17 Apr 2026

Viewed by 361

Abstract

A method for analyzing long-term (1997–2025) continuous records of low-frequency global seismic noise measured at a network of 229 broadband seismic stations distributed across the Earth’s surface is proposed in this study. The method is based on the use of nonlinear multifractal and [...] Read more.

A method for analyzing long-term (1997–2025) continuous records of low-frequency global seismic noise measured at a network of 229 broadband seismic stations distributed across the Earth’s surface is proposed in this study. The method is based on the use of nonlinear multifractal and entropy statistics, evaluated daily in successive time intervals, of first-principal component analysis, correlation analysis, and parametric models of point process intensity. The relationships between changes in seismic noise properties and the response of noise properties to the irregularity of the Earth’s rotation with the sequence of strong earthquakes, including those of a predictive nature, are investigated. Full article

(This article belongs to the Special Issue Fractals in Earthquake and Atmospheric Science)

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33 pages, 910 KB

Open AccessArticle

(p,q,r)-Fractional Fuzzy Similarity and Dissimilarity Measures with an Inferior Ratio Decision Framework

by Muhammad Jabir Khan, Kanikar Muangchoo, Nasser Aedh Alreshidi and Sakulbuth Ekvittayaniphon

Fractal Fract. 2026, 10(4), 266; https://doi.org/10.3390/fractalfract10040266 - 17 Apr 2026

Viewed by 327

Abstract

This paper develops novel similarity and dissimilarity measures for

(p, q, r)

-fractional fuzzy sets to enhance information discrimination and decision-making under complex uncertainty. We first introduce axiomatic dissimilarity measures and establish their fundamental mathematical properties, including boundedness, symmetry, [...] Read more.

This paper develops novel similarity and dissimilarity measures for

(p, q, r)

-fractional fuzzy sets to enhance information discrimination and decision-making under complex uncertainty. We first introduce axiomatic dissimilarity measures and establish their fundamental mathematical properties, including boundedness, symmetry, monotonicity, and identity conditions. Based on these, we derive corresponding similarity measures that improve discrimination capability. We further propose a multi-criteria group decision-making framework to facilitate robust, accurate ranking of alternatives by integrating the developed measures into a

(p, q, r)

-fractional fuzzy inferior ratio method. The approach evaluates alternatives using relative inferiority relationships and provides stable, reliable rankings in uncertain environments. Illustrative examples demonstrate the proposed method’s effectiveness and applicability, and sensitivity analysis examines decision robustness. Comparative analysis with existing methods confirms the superiority of the proposed framework, showing that it offers stronger discrimination ability and serves as a flexible, reliable tool for complex multi-criteria group decision problems under

(p, q, r)

-fractional fuzzy environments. Full article

(This article belongs to the Section Optimization, Big Data, and AI/ML)

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18 pages, 1586 KB

Open AccessArticle

Fractal Duffing Oscillators with Two Degrees of Freedom and Cubic–Quintic Nonlinear Stiffness

by Guozhong Xiu, Jihuan He, Yusry O. El-Dib and Haifa A. Alyousef

Fractal Fract. 2026, 10(4), 265; https://doi.org/10.3390/fractalfract10040265 - 17 Apr 2026

Viewed by 410

Abstract

The harmonic equivalent method is a non-perturbative approach to nonlinear vibration issues, aiming to create linearly coupled systems from coupled vibrations. However, there is still much to be discovered about managing interconnected nonlinear components. This paper examines the nonlinear components of a fractal-connected [...] Read more.

The harmonic equivalent method is a non-perturbative approach to nonlinear vibration issues, aiming to create linearly coupled systems from coupled vibrations. However, there is still much to be discovered about managing interconnected nonlinear components. This paper examines the nonlinear components of a fractal-connected system and offers suggestions. This paper explores insights into the principles and uses of nonlinear systems in science and engineering by investigating the dynamic behavior of a connected cubic–quintic damping fractal system analytically using an innovative approach to analytical examination. A two-scale transformation and reformulation of the system into fractal form simplify its governing equations for dynamic and stability analysis. Two analytical scopes are presented: one decouples nonlinear systems using weighted averaging functions, and the other converts even nonlinearities into odd terms using El-Dib’s frequency formulas for linear representation, enabling an equivalent linear representation of the system. The resilience of the decoupled system is verified by numerical simulations using Mathematica, which shows high agreement and minimal relative errors. It also accurately reflects dynamic behavior. Additionally, the work uses the bridging techniques of El-Dib and Elgazery to convert a linear damping fractal coupled system into a classical continuous-space form. A scaling fractal factor is made possible by re-expressing the fractal structure using pseudo-dimensional parameters. The linearly linked damping system has an exact analytical solution. The paper provides valuable insights into the design and control of coupled nonlinear oscillatory systems by validating analytical solutions through numerical simulations using Mathematica. Full article

(This article belongs to the Section Mathematical Physics)

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31 pages, 536 KB

Open AccessArticle

On the Center-Radius Order (P,m)-Superquadratic Interval Valued Functions and Their Fractional Perspective with Applications

by Saad Ihsan Butt, Arshad Yaqoob, Dawood Khan and Youngsoo Seol

Fractal Fract. 2026, 10(4), 264; https://doi.org/10.3390/fractalfract10040264 - 16 Apr 2026

Viewed by 325

Abstract

In this paper, we introduce, for the first time, a novel class of (center-radius order

(P, m)

-superquadratic interval-valued functions)

cr

-

(P, m)

-superquadratic

IVF

s, and systematically investigate their fundamental structural properties. Building upon these [...] Read more.

In this paper, we introduce, for the first time, a novel class of (center-radius order

(P, m)

-superquadratic interval-valued functions)

cr

-

(P, m)

-superquadratic

IVF

s, and systematically investigate their fundamental structural properties. Building upon these properties, we establish new Jensen and Hermite–Hadamard (

HH

) type inequalities, together with their fractional extensions formulated via Riemann–Liouville (

RL

) fractional integral operators within the setting of interval calculus. The validity and sharpness of the derived results are illustrated through numerical examples and graphical representations. Moreover, the theoretical developments are further enriched by applications in information theory, leading to meaningful generalizations and notable improvements over several existing results reported in the literature. Full article

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

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28 pages, 411 KB

Open AccessArticle

Positive Ground State Solutions for Fractional (p, q)-Laplacian Choquard Equation with Singularity and Upper Critical Exponent

by Zhenyu Bai and Chuanzhi Bai

Fractal Fract. 2026, 10(4), 263; https://doi.org/10.3390/fractalfract10040263 - 16 Apr 2026

Viewed by 273

Abstract

We prove the existence of a positive ground state solution for a fractional

(p, q)

-Laplacian Choquard equation that features both a singularity and an upper critical exponent. The proof relies on a combination of the Nehari manifold technique and [...] Read more.

We prove the existence of a positive ground state solution for a fractional

(p, q)

-Laplacian Choquard equation that features both a singularity and an upper critical exponent. The proof relies on a combination of the Nehari manifold technique and Ekeland’s variational principle. Full article

(This article belongs to the Special Issue Theoretical and Numerical Methods for Fractional and Nonlocal Partial Differential Equations)

31 pages, 455 KB

Open AccessArticle

Numerical and Stability Analysis of Hilfer-Type Fuzzy Fractional Control Systems with Infinite Delay

by Aeshah Abdullah Muhammad Al-Dosari

Fractal Fract. 2026, 10(4), 262; https://doi.org/10.3390/fractalfract10040262 - 15 Apr 2026

Viewed by 327

Abstract

This paper presents a unified theoretical and numerical investigation of Hilfer-type fuzzy fractional control systems with infinite continuous delay. By employing contraction mapping principles and compact semigroup theory, we establish rigorous solvability conditions together with Ulam–Hyers–Rassias stability results expressed in terms of Mittag–Leffler [...] Read more.

This paper presents a unified theoretical and numerical investigation of Hilfer-type fuzzy fractional control systems with infinite continuous delay. By employing contraction mapping principles and compact semigroup theory, we establish rigorous solvability conditions together with Ulam–Hyers–Rassias stability results expressed in terms of Mittag–Leffler functions. To complement the analytical framework, we design and implement numerical schemes based on Euler and IMEX approaches, which confirm the theoretical predictions through simulations. The computational experiments demonstrate the robustness of the proposed framework under delayed feedback and fractional memory effects, highlighting its relevance to practical domains such as biological regulation, porous media transport, and intelligent traffic systems. The contribution of this study lies in the bridge between mathematical rigor and computational implementation, thus advancing the theory of fractional differential inclusions and providing a versatile tool for the stability analysis and control of complex systems with uncertainty and hereditary dynamics. Full article

(This article belongs to the Special Issue Fractional-Order Modeling and Control for Complex Dynamic Systems: Methodologies and Applications)

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20 pages, 406 KB

Open AccessArticle

Fixed Point Results in Extended ℱ-Metric Spaces with Applications to Caputo Fractional Differential Equations

by Badriah Alamri

Fractal Fract. 2026, 10(4), 261; https://doi.org/10.3390/fractalfract10040261 - 15 Apr 2026

Viewed by 296

Abstract

The purpose of this research work is to propose and develop the notion of

(α, ψ)

-contractions in the setting of extended

F

-metric spaces and to establish corresponding fixed point results. Using these results, we derive fixed point results for graphic [...] Read more.

The purpose of this research work is to propose and develop the notion of

(α, ψ)

-contractions in the setting of extended

F

-metric spaces and to establish corresponding fixed point results. Using these results, we derive fixed point results for graphic contractions in extended

F

-metric spaces as well as for mappings in partially ordered extended

F

-metric spaces. To demonstrate the validity and novelty of the proposed results, a non-trivial example is provided. Moreover, the constructed framework serves as a tool to investigate the existence of solutions for Caputo fractional differential equations, thereby highlighting both its effectiveness and practical significance. Full article

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

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28 pages, 677 KB

Open AccessArticle

Mathematical Investigation of Cancer-Immune-Angiogenesis Model Using Fuzzy Piecewise Fractional Derivatives

by Rabeb Sidaoui, Ashraf A. Qurtam, Mohammed Almalahi, Habeeb Ibrahim, Khaled Aldwoah, Amer Alsulami and Mohammed Messaoudi

Fractal Fract. 2026, 10(4), 260; https://doi.org/10.3390/fractalfract10040260 - 15 Apr 2026

Viewed by 281

Abstract

This work develops a fuzzy piecewise fractional derivative (FPFD) model for cancer-immune-angiogenesis dynamics under uncertainty. Five fuzzy state variables track tumor cells, immune effectors, vessel density, oxygen, and drug concentration. We employ fuzzy triangular numbers with

α

-cut interval arithmetic using constrained fuzzy [...] Read more.

This work develops a fuzzy piecewise fractional derivative (FPFD) model for cancer-immune-angiogenesis dynamics under uncertainty. Five fuzzy state variables track tumor cells, immune effectors, vessel density, oxygen, and drug concentration. We employ fuzzy triangular numbers with

α

-cut interval arithmetic using constrained fuzzy arithmetic model parametric uncertainty, with numerical values. Oxygen-dependent carrying capacity follows a Hill-type function; hypoxia-induced angiogenesis follows a decreasing Michaelis–Menten function. The model transitions at

t_{1} = 50

days from memoryless fuzzy classical derivative to fuzzy ABC fractional derivative of order

ψ

. The transition time

t_{1} = 50

days is biologically justified based on experimental observations of the angiogenic switch in solid tumors, which typically occurs within 4–8 weeks post-inoculation. Positivity, boundedness, Lipschitz continuity, existence, and uniqueness of fuzzy solutions are proved via Banach fixed-point theorem in a weighted norm. A basic reproduction number interval

R_{0} = [{\underset{̲}{R}}_{0}, {\bar{R}}_{0}]

is derived; local and global stability conditions are established for disease-free and endemic equilibria using fuzzy differential inclusions. Global sensitivity analysis using latin hypercube sampling with

N = 500

samples explores the range of possible outcomes across the fuzzy parameter support. In the numerical implementation, we use a fourth-order fuzzy Runge–Kutta method (Phase I), and a fractional Adams–Bashforth–Moulton predictor-corrector method (Phase II), ensuring preservation of fuzzy number characteristics. Full article

(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus and Mathematical Modelling)

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15 pages, 2682 KB

Open AccessArticle

Pore Structure and Multifractal Characteristics of Tight Sandstone: A Case Study of the Jurassic Sangonghe Formation in Northern Turpan-Hami Basin, NW China

by Jiacheng Huang, Zongbao Liu, Bin Hao and Zhiwen Dong

Fractal Fract. 2026, 10(4), 259; https://doi.org/10.3390/fractalfract10040259 - 15 Apr 2026

Viewed by 335

Abstract

Pore structure and multifractal characteristics are two critical indicators for evaluating the heterogeneity of tight sandstone reservoirs. An integrated analysis comprising physical property tests, X-ray diffraction, casting thin sections, scanning electron microscopy, high-pressure mercury intrusion (HPMI), and constant-rate mercury intrusion (CRMI) is conducted [...] Read more.

Pore structure and multifractal characteristics are two critical indicators for evaluating the heterogeneity of tight sandstone reservoirs. An integrated analysis comprising physical property tests, X-ray diffraction, casting thin sections, scanning electron microscopy, high-pressure mercury intrusion (HPMI), and constant-rate mercury intrusion (CRMI) is conducted on five samples from the Jurassic Sangonghe Formation in the northern Turpan-Hami Basin to investigate the full-scale pore size distribution (FPSD) and its multifractal characteristics. The results indicate that the pores in tight sandstone are mainly residual intergranular pores, dissolution pores, intercrystalline pores, and microfractures. The FPSD exhibits a bimodal or trimodal pattern, with dominant pore sizes ranging from 0.00516 μm to 1.15

μ m

. Two key multifractal parameters, the multifractal dimension range (

D_{m i n} - D_{m a x}

) and the relative dispersion (

R_{d}

), were utilized to effectively characterize pore structure heterogeneity and asymmetry. Higher

D_{m i n} - D_{m a x}

values correspond to stronger heterogeneity, whereas lower

R_{d}

values indicate a dominance of nanoscale pores. Furthermore,

D_{m i n} - D_{m a x}

and

R_{d}

exhibit negative correlations with permeability and clay mineral content, and positive correlations with feldspar content. This study demonstrates the utility of FPSD in characterizing pore structure and highlights the applicability of multifractal theory in assessing the heterogeneity of tight sandstone reservoirs. Full article

(This article belongs to the Special Issue Fractal Analysis in Unconventional Reservoirs: Theory and Applications)

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26 pages, 479 KB

Open AccessArticle

Fixed Point Theorems in Complex-Valued b-Suprametric Spaces with Applications to Fractional Differential Equations

by Maha Noorwali and Afrah Ahmad Noman Abdou

Fractal Fract. 2026, 10(4), 258; https://doi.org/10.3390/fractalfract10040258 - 14 Apr 2026

Viewed by 282

Abstract

In this research article, we introduce and develop the notion of complex-valued b-suprametric spaces as a natural generalization of existing metric-type structures. Fundamental concepts, including convergence, Cauchy sequences, and completeness, are examined in this new setting. We establish new common fixed point [...] Read more.

In this research article, we introduce and develop the notion of complex-valued b-suprametric spaces as a natural generalization of existing metric-type structures. Fundamental concepts, including convergence, Cauchy sequences, and completeness, are examined in this new setting. We establish new common fixed point theorems for generalized and cyclic rational contractive mappings. The obtained results extend and unify various known fixed point theorems available in the current literature. To demonstrate the applicability and effectiveness of our theoretical findings, illustrative nontrivial examples are provided. As an application, we investigate the existence and uniqueness of solutions for Caputo fractional differential equations, which naturally arise in systems with hereditary and memory effects, particularly in biomedical modeling of viscoelastic biological tissues such as arteries, cartilage, and brain tissue. This demonstrates both the mathematical strength and the practical relevance of the proposed framework. Full article

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

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20 pages, 3679 KB

Open AccessArticle

Coupled Fractal–Fractional Modeling of Coal Creep Behavior Under Mining-Induced Stress

by Wenhao Jia, Eryi Hu, Shukai Jin, Shuai Zhang, Shuai Yang, Lu An and Senlin Xie

Fractal Fract. 2026, 10(4), 257; https://doi.org/10.3390/fractalfract10040257 - 14 Apr 2026

Viewed by 217

Abstract

Understanding the evolution of coal pore–fracture structures under coupled stress paths and creep deformation is critical for enhancing coalbed methane extraction and preventing coal and gas outbursts. In this study, coal samples from the Ningtiaota Mine were investigated using online Nuclear Magnetic Resonance [...] Read more.

Understanding the evolution of coal pore–fracture structures under coupled stress paths and creep deformation is critical for enhancing coalbed methane extraction and preventing coal and gas outbursts. In this study, coal samples from the Ningtiaota Mine were investigated using online Nuclear Magnetic Resonance (NMR) technology combined with triaxial loading–creep coupled experiments. The dynamic evolution of pore–fracture structures (PFSs) under different deviatoric stress levels was characterized and visualized in real time and across multiple scales. The results reveal a pronounced stress-dependent pore evolution during creep. Under low-stress conditions, seepage pores were compressed and gradually transformed into adsorption pores, whereas under high-stress conditions, seepage pores expanded and interconnected, dominating deformation and failure. Fractal theory was employed to quantify pore structure complexity, and repeated experiments demonstrated a significant positive correlation between the fractal dimension and the fractional order. Based on these findings, a fractal-dimension-based fractional creep model was developed by introducing a Riemann–Liouville fractional dashpot. The proposed model accurately captures the nonlinear creep behavior of coal and provides a microstructural interpretation of the fractional order. This study provides theoretical and experimental support for long-term stability assessment of deep coal–rock masses and prediction of coalbed methane migration. Full article

(This article belongs to the Special Issue Fractional Calculus with Applications in Rock Mechanics and Mining Engineering)

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15 pages, 331 KB

Open AccessArticle

Non-Decreasing Solutions for (k,Υ)-Fractional Quadratic Integral Equations of Urysohn–Volterra Type

by Shahenda S. El-Malty, Mahmoud M. El-Borai, Wagdy G. El-Sayed and Mohamed I. Abbas

Fractal Fract. 2026, 10(4), 256; https://doi.org/10.3390/fractalfract10040256 - 13 Apr 2026

Viewed by 394

Abstract

In this paper, we investigate a

(k, Υ)

fractional quadratic integral equation in the Banach space of real-valued continuous functions on

[0, 1]

. By using a measure of noncompactness associated with monotonicity and Darbo’s fixed point [...] Read more.

In this paper, we investigate a

(k, Υ)

fractional quadratic integral equation in the Banach space of real-valued continuous functions on

[0, 1]

. By using a measure of noncompactness associated with monotonicity and Darbo’s fixed point theorem, we provide sufficient conditions for the existence of at least one monotonic solution and analyze its stability. Finally, an illustrative example is presented to demonstrate the theoretical results, including several particular cases. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations, 2nd Edition)

22 pages, 2972 KB

Open AccessArticle

Innovative Approximate Solution for Jerk Model of Non-Newtonian Bio-Nanofluid in Fractal Space via Highly Efficient Linear Approximation

by Nasser S. Elgazery and Taghreed H. Al-Arabi

Fractal Fract. 2026, 10(4), 255; https://doi.org/10.3390/fractalfract10040255 - 13 Apr 2026

Viewed by 303

Abstract

In this article, we present a new approximate solution for blood nanofluid having gold nanoparticles as it flows near a stretching porous cylinder in fractal space. A Casson non-Newtonian magneto-bio-nanofluid flowing through a porous medium is considered a potential application in chemotherapy for [...] Read more.

In this article, we present a new approximate solution for blood nanofluid having gold nanoparticles as it flows near a stretching porous cylinder in fractal space. A Casson non-Newtonian magneto-bio-nanofluid flowing through a porous medium is considered a potential application in chemotherapy for eradicating cancer cells. Without the need for the nonperturbative approach, the proposed solution uses an alternative approach to dealing with nonlinear problems. This approach transforms the nonlinear cubic jerk model resulting from the simplification of the governing fractional partial differential equations into an equivalent linear formula. This approach is known as highly efficient linear approximation (HELA) or non-perturbation technique (NPT), and this represents a significant advancement over traditional perturbation methods in the analysis of non-linear systems. As a robust mathematical approach, it excels at handling a wide range of coefficient values, particularly in cases of clear nonlinearity. This study also utilized the masking technique simultaneously with HELA, which played a crucial role, as they simplify the complex dynamics of the system, making it more amenable to analysis. The numerical solution by the Runge–Kutta fourth-order (RK-4) method integrated with a shooting technique compared favorably with graphs drawn for the analytical solution from the proposed strategy HELA. The current results show that an increase in the fractal factors enhances the resistance to fluid motion, leading to a suppression of the velocity field. Physically, this often relates to the complexity of the medium or the fractal nature of the transport process, where higher fractal dimensions or factors can lead to slower diffusion or flow rates, like the role of porous media. Therefore, the current study has significant implications in the promotion of nanotechnology fields in medicine, particularly the use of gold nanoparticles in chemotherapy for the eradication of cancerous cells. Full article

(This article belongs to the Section Mathematical Physics)

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17 pages, 2361 KB

Open AccessArticle

Fractional-Order Modelling of Pneumatic Transmission Dynamics in Soft Robotic Actuation

by Kutlo Popo, Andres San-Millan and Sumeet S. Aphale

Fractal Fract. 2026, 10(4), 254; https://doi.org/10.3390/fractalfract10040254 - 13 Apr 2026

Viewed by 382

Abstract

Pneumatic transmission lines play a critical role in the dynamic performance of soft robotic actuation systems, yet their behaviour is difficult to capture using conventional integer-order (IO) models. In long, slender pipelines, compressibility, viscothermal losses, and wave propagation give rise to distributed damping [...] Read more.

Pneumatic transmission lines play a critical role in the dynamic performance of soft robotic actuation systems, yet their behaviour is difficult to capture using conventional integer-order (IO) models. In long, slender pipelines, compressibility, viscothermal losses, and wave propagation give rise to distributed damping and non-exponential relaxation dynamics that are not well represented by finite-dimensional models. This paper presents a control-oriented, experimentally validated fractional-order (FO) modelling framework for pneumatic pipeline dynamics under closed-end boundary conditions. Models are calibrated using measured step-response data from a 13.2 m pipeline, with all parameters—including the fractional order—identified through a unified optimisation procedure. In addition to global fitting accuracy, model performance is evaluated using control-relevant metrics, including effective delay, initial slope and early transient behaviour, and early-time error. The results show that FO models provide a more compact and structurally consistent representation of long-memory dynamics while improving the accuracy of control-relevant features compared to their IO counterparts. These findings demonstrate that fractional dynamics offer a physically meaningful and practically useful framework for modelling pneumatic transmission lines, with direct implications for high-performance control design in soft robotic systems. Full article

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

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20 pages, 683 KB

Open AccessArticle

Exploring Fixed-Time Synchronization of Fractional-Order Fuzzy Cellular Neural Networks with Information Interactions and Time-Varying Delays via Adaptive Multi-Module Control

by Hongguang Fan, Kaibo Shi, Anran Zhou, Fei Meng and Liang Jiang

Fractal Fract. 2026, 10(4), 253; https://doi.org/10.3390/fractalfract10040253 - 13 Apr 2026

Viewed by 275

Abstract

This article focuses on the fixed-time synchronization problem for fractional-order fuzzy cellular neural networks (FOFCNNs) with information interactions and time-varying delays. To capture the complex dynamics of practical networks, nonlinear activation functions along with fuzzy AND and OR operators are incorporated into the [...] Read more.

This article focuses on the fixed-time synchronization problem for fractional-order fuzzy cellular neural networks (FOFCNNs) with information interactions and time-varying delays. To capture the complex dynamics of practical networks, nonlinear activation functions along with fuzzy AND and OR operators are incorporated into the master–slave systems. To achieve fixed-time synchronization despite these complexities, a novel adaptive multi-module controller is proposed. This controller integrates three functionally distinct components to accelerate the convergence rate, eliminate the effects of delays, and introduce negative feedback during communication, respectively. By employing fractional calculus tools, inequality techniques, and the proposed control law, sufficient criteria for the synchronization of the considered systems are rigorously established. Compared with existing synchronization works, this paper has significant advantages in model generality and controller design. Additionally, an explicit settling-time estimate is derived, which depends solely on control parameters and is independent of the initial conditions. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Control for Nonlinear Systems)

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32 pages, 1103 KB

Open AccessArticle

On the Existence of Solutions to a Nonlinear Atangana–Baleanu Type Fractional Differential Equation

by Hanadi Zahed

Fractal Fract. 2026, 10(4), 252; https://doi.org/10.3390/fractalfract10040252 - 13 Apr 2026

Viewed by 249

Abstract

In this research work, we investigate a nonlinear fractional initial value problem involving the Atangana-Baleanu-Caputo derivative of order

0 < α < 1 .

By means of the associated fractional integral operator, the problem is converted into an equivalent nonlinear integral equation. The [...] Read more.

In this research work, we investigate a nonlinear fractional initial value problem involving the Atangana-Baleanu-Caputo derivative of order

0 < α < 1 .

By means of the associated fractional integral operator, the problem is converted into an equivalent nonlinear integral equation. The existence of solutions is established in the context of extended

F

-metric spaces via a fixed point approach based on an (

α, ψ

)-contractive condition of rational form. Furthermore, we develop the notion of graphic rational contractions in the setting of extended

F

-metric spaces and prove new fixed point results. Our results extend and unify several known results in the existing literature as special cases. Nontrivial examples are provided to demonstrate the applicability of the theoretical findings. These results highlight the effectiveness of extended

F

-metric techniques in the analys. Full article

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

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26 pages, 10912 KB

Open AccessArticle

Study of the Pore Structure Effect on Seepage in Coal Reservoirs Based on Multifractal Analysis

by Bocen Chen, Hongwei Zhou, Zelin Liu, Senlin Xie, Wenhao Jia, Longdan Zhang, Lei Zhang and Yimeng Liu

Fractal Fract. 2026, 10(4), 251; https://doi.org/10.3390/fractalfract10040251 - 12 Apr 2026

Viewed by 370

Abstract

Coalbed methane is vital for the transition toward low-carbon energy systems, yet its recovery efficiency is critically limited by inaccurate classification of movable water during drainage and depressurization due to the complex pore–fracture system. To understand the influence of the pore–fracture structure on [...] Read more.

Coalbed methane is vital for the transition toward low-carbon energy systems, yet its recovery efficiency is critically limited by inaccurate classification of movable water during drainage and depressurization due to the complex pore–fracture system. To understand the influence of the pore–fracture structure on water flow law in coal reservoirs, this study constructed the relationship based on the memory effect of multiscale complex pore–fracture structures on seepage. Nuclear magnetic resonance (NMR) measurements were performed on water-saturated coal samples both before and after centrifugation, enabling the experimental identification of absolute irreducible water, partial movable water, and absolute movable water and yielding dual cutoffs. The complexity of the pore–fracture structure of the samples was quantified by multifractal analysis of the NMR test results. A fractional derivative model was developed to determine dual cutoffs, T_2c1 and T_2c2, based on the memory effect and validated against experimental data. Compared to empirical models, the proposed fractional derivative model improves R² fitting accuracy by 4.2% for T_2c1 and 9.7% for T_2c2, demonstrating its superior capability in translating structural complexity into physically meaningful cutoff determination. This work provides a mechanism-based approach for water typing, presenting a reliable foundation for drainage and depressurization in coalbed methane reservoir. Full article

(This article belongs to the Special Issue Fractional Calculus with Applications in Rock Mechanics and Mining Engineering)

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14 pages, 40289 KB

Open AccessArticle

Fractal Analysis of Thermally Induced Damage in Volcanic Rocks: Linking Mechanical Behavior and Mineralogical Controls

by Özge Dinç Göğüş, Enes Zengin, Mehmet Korkut, Mehmet Mert Doğu, Mustafa Avcıoğlu, Ömer Ündül and Emin Çiftçi

Fractal Fract. 2026, 10(4), 250; https://doi.org/10.3390/fractalfract10040250 - 11 Apr 2026

Viewed by 309

Abstract

Moderate thermal exposure can significantly influence the mechanical behavior of volcanic rocks by inducing microcrack development and altering crack network characteristics. However, quantifying such damage processes remains challenging when relying solely on conventional mechanical parameters. In this study, the evolution of crack network [...] Read more.

Moderate thermal exposure can significantly influence the mechanical behavior of volcanic rocks by inducing microcrack development and altering crack network characteristics. However, quantifying such damage processes remains challenging when relying solely on conventional mechanical parameters. In this study, the evolution of crack network complexity in andesite and andesitic–basaltic rocks subjected to moderate thermal exposure (200 °C) is investigated using fractal analysis integrated with mechanical and mineralogical observations. Six core specimens were tested under uniaxial compression, including three natural specimens and three specimens thermally treated at 200 °C prior to loading. After failure, crack surfaces were digitized and fractal dimensions (D) were calculated using the box-counting method. Petrographic observations and X-ray powder diffraction (XRPD) analyses were conducted to characterize the mineralogical composition and microstructural features controlling crack development. The results indicate that thermal exposure primarily reduces rock stiffness rather than peak strength. While the uniaxial compressive strength (UCS) of two specimens remains nearly unchanged after heating, the elastic modulus (E) decreases in all thermally treated specimens. Mineralogical observations reveal a heterogeneous volcanic fabric dominated by plagioclase and pyroxene within a fine-grained groundmass, with secondary calcite phases occurring in veins and pocket fillings. Fractal analysis shows generally lower D values in thermally treated specimens, suggesting crack redistribution and coalescence rather than increased network complexity, consistent with the observed reduction in stiffness and a tendency toward more ductile deformation behavior. Full article

(This article belongs to the Section Engineering)

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39 pages, 509 KB

Open AccessArticle

Solvability of Generalized Hilfer Fractional p-Laplacian Differential Problems in Orlicz Spaces

by Mieczysław Cichoń, Masouda M. A. Al-Fadel and Hussein A. H. Salem

Fractal Fract. 2026, 10(4), 249; https://doi.org/10.3390/fractalfract10040249 - 10 Apr 2026

Viewed by 310

Abstract

This paper investigates non-fractional operators, a type of nonlocal operator, within the framework of Orlicz spaces. Using inclusions between certain function spaces, we prove the continuity and/or compactness of generalized operators in Orlicz spaces and show that solutions exist for integral equations of [...] Read more.

This paper investigates non-fractional operators, a type of nonlocal operator, within the framework of Orlicz spaces. Using inclusions between certain function spaces, we prove the continuity and/or compactness of generalized operators in Orlicz spaces and show that solutions exist for integral equations of fractional order. We also introduce a generalized Hilfer-type derivative and examine the equivalence of differential and integral problems. Finally, we relate these results to the study of compositional p-Laplacian fractional problems involving generalized Hilfer fractional derivatives. Among other things, we prove the existence of solutions to such problems in Orlicz and Orlicz–Sobolev spaces. Full article

(This article belongs to the Special Issue Recent Advances in Nonlocal Problems Involving the Fractional Laplacian Operators)

30 pages, 716 KB

Open AccessArticle

Stability of a Fractional HIV/AIDS Epidemic Model with Drug Control by Continuous-Time Random Walk

by Jiao Li, Yongguang Yu, Zhenzhen Lu and Weiyi Xu

Fractal Fract. 2026, 10(4), 248; https://doi.org/10.3390/fractalfract10040248 - 9 Apr 2026

Viewed by 250

Abstract

In recent years, fractional HIV models have received increasing attention. This study derives a fractional HIV model using the continuous-time random walk (CTRW) method, endowing the mathematical model with physical significance. Based on the transmission characteristics of HIV, the proposed model considers extrinsic [...] Read more.

In recent years, fractional HIV models have received increasing attention. This study derives a fractional HIV model using the continuous-time random walk (CTRW) method, endowing the mathematical model with physical significance. Based on the transmission characteristics of HIV, the proposed model considers extrinsic infectivity, intrinsic infectivity, and drug control, specifically as follows: the extrinsic infectivity is a constant independent of the infection time; the intrinsic infectivity is a power-law function that depends on drug efficacy and infection time; the drug efficacy rate follows a Mittag–Leffler distribution with a long-term effect. Based on these considerations, a fractional HIV model with drug control is established in this paper. In addition, the global asymptotic stability of the equilibrium and the sensitivity analysis of the basic reproduction number

R_{0}

are studied, and the theoretical results are verified by numerical simulations. The results show that reducing extrinsic infectivity, controlling intrinsic infectivity, and the drug efficacy rate are crucial in controlling the spread of HIV. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Analysis: Theory and Applications)

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30 pages, 922 KB

Open AccessArticle

A Comprehensive Analysis of Proportional Caputo-Hybrid Fractional Inequalities and Numerical Verification via Artificial Neural Networks

by Ayed R. A. Alanzi, Mariem Al-Hazmy, Raouf Fakhfakh, Wedad Saleh, Abdellatif Ben Makhlouf and Abdelghani Lakhdari

Fractal Fract. 2026, 10(4), 247; https://doi.org/10.3390/fractalfract10040247 - 8 Apr 2026

Viewed by 367

Abstract

Accuracy in fractional numerical integration is often limited by the regularity of the integrand. This work proposes a flexible error estimation framework for proportional Caputo-hybrid integral operators based on

s

-convexity. We introduce a parametric Newton–Cotes formula ( [...] Read more.

Accuracy in fractional numerical integration is often limited by the regularity of the integrand. This work proposes a flexible error estimation framework for proportional Caputo-hybrid integral operators based on

s

-convexity. We introduce a parametric Newton–Cotes formula (

ν \in [0, 1]

) that bridges the gap between classical quadrature rules, recovering the fractional Trapezoidal, Midpoint, and Simpson’s methods as specific instances. In order to confirm the correctness of our results, we provide an illustrative example with graphical representations. Furthermore, we provide some additional results using Hölder’s and power mean inequalities and employ a verification strategy based on an Artificial Neural Networks (ANNs) model. The ANN approach allows for high-dimensional parameter space exploration, demonstrating that the proposed inequalities provide robust and precise error estimates. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)

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27 pages, 23751 KB

Open AccessArticle

A Mathematical Framework for Retinal Vessel Segmentation: Fractional Hessian-Based Curvature Analysis

by Priyanka Harjule, Mukesh Delu, Rajesh Kumar and Pilani Nkomozepi

Fractal Fract. 2026, 10(4), 246; https://doi.org/10.3390/fractalfract10040246 - 8 Apr 2026

Viewed by 340

Abstract

This study proposes an improved retinal blood vessel segmentation method to enhance the diagnosis of microvascular retinal complications. The proposed method extracts local shape features from retinal images utilizing a fractional Hessian matrix, which models blood vessels as surface structures characterized by ridges [...] Read more.

This study proposes an improved retinal blood vessel segmentation method to enhance the diagnosis of microvascular retinal complications. The proposed method extracts local shape features from retinal images utilizing a fractional Hessian matrix, which models blood vessels as surface structures characterized by ridges and valleys resulting from variations in curvature. The methodology integrates adaptive principal curvature estimation with a new framework leveraging the fractional Hessian matrix with nonsingular and nonlocal kernels. The effectiveness of the suggested method is assessed using publicly accessible datasets, including DRIVE, HRF, STARE, and some real images obtained from a local hospital. The proposed segmentation achieves 96.77% accuracy and 98.82% specificity on the DRIVE database, 96.91% accuracy and 98.69% specificity on STARE, and 95.90% accuracy and 98.36% specificity on the HRF database. Optimal parameters for the fractional order and Gaussian standard deviation were empirically determined by maximizing segmentation accuracy. Our findings show that the proposed approach achieves competitive performance compared to the listed methods, including several deep learning approaches, while maintaining significant computational efficiency. The output of the suggested method can be further utilized with deep learning techniques, which will be applied in the clinical context of diabetic retinopathy and glaucoma to identify abnormalities likely related to disease progression and different stages. Full article

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25 pages, 43751 KB

Open AccessArticle

A Computational Framework for Escape Dynamics and Fractal Structures in Transcendental Complex Maps

by Asifa Tassaddiq, Muhammad Tanveer, Rabab Alharbi, Aiman Albarakati, Ruhaila Md Kasmani and Sania Qureshi

Fractal Fract. 2026, 10(4), 245; https://doi.org/10.3390/fractalfract10040245 - 7 Apr 2026

Viewed by 399

Abstract

This study offers a computational framework that analyzes the escape characteristics of transcendental complex maps by utilizing the AK iteration scheme. The well-known polynomial map of the form

z^{n} + c

is generalized to the form [...] Read more.

This study offers a computational framework that analyzes the escape characteristics of transcendental complex maps by utilizing the AK iteration scheme. The well-known polynomial map of the form

z^{n} + c

is generalized to the form

z^{n} + \sin (z) + \log (c^{m})

, with

m \geq 1

and

c \in C \ {0}

, allowing the creation of complex fractal structures. A precise escape criterion is developed for the AK iteration scheme, ensuring the numerical stability of the scheme when applied to the construction of the Mandelbrot set and the Julia set. In order to validate the effectiveness of the developed framework, a comparative analysis is performed between the AK iteration scheme and the CR iteration scheme, focusing on the first parametric case of the Mandelbrot set and the Julia set. The average escape time, average number of iterations, non-escaping area index, and fractal dimension are analyzed with respect to the two iteration schemes. The numerical results indicate that the fractal structure obtained by the AK iteration scheme is different from the fractal structure obtained by the CR iteration scheme, showing the effectiveness of the AK iteration scheme as a powerful tool in the study of complex systems. Full article

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

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Fractal Fract., Volume 10, Issue 4 (April 2026) – 65 articles

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