Fractal and Fractional

23 pages, 5786 KB

Open AccessArticle

Fractal Characteristics and Heterogeneity Evaluation of Shale Reservoirs Based on MIP and Gas Adsorption: A Case Study of Marine Shale in the Sichuan Basin

by Meng Wang, Shu Liu, Yuxi Wang, Xinan Yu, Jun Lang, Yulin Cheng, Xingming Duan and Jingjing Guo

Fractal Fract. 2026, 10(5), 349; https://doi.org/10.3390/fractalfract10050349 - 21 May 2026

Cited by 2 | Viewed by 598

Abstract

The deep marine shale of the Wufeng–Longmaxi (WF–LMX) Formation in the Sichuan Basin is characterized by laterally continuous thickness, high porosity, and significant gas content, making it a representative shale reservoir with considerable resource potential. This study investigates the heterogeneity of pore structures [...] Read more.

The deep marine shale of the Wufeng–Longmaxi (WF–LMX) Formation in the Sichuan Basin is characterized by laterally continuous thickness, high porosity, and significant gas content, making it a representative shale reservoir with considerable resource potential. This study investigates the heterogeneity of pore structures and their controlling factors using shale samples from three representative wells, based on low-temperature nitrogen adsorption and mercury intrusion data. The reservoir can be classified into three main lithofacies: mixed siliceous shale (MSS), clay-rich siliceous shale (CSS), and siliceous clay mixed shale (SMS). The results show that siliceous shales (MSS and CSS) exhibit higher total organic carbon and quartz contents, with more developed pore systems. Among them, the CSS exhibits the highest specific surface area and the largest mesopore and macropore volumes, indicating a greater development of larger pores and superior reservoir quality. All three shale facies exhibit clear single and multifractal characteristics. The average D₁ and D₂ values (fractal dimensions from nitrogen adsorption at P/P₀ < 0.45 and >0.45, respectively) are higher than D_Hg, (fractal dimension from mercury intrusion), indicating greater pore-surface roughness than internal pore structure complexity and stronger heterogeneity in larger pores. The D(q)–q spectrum shows a left-wide/right-narrow pattern, whereas the α–f(α) spectrum exhibits the opposite trend. The branch-width ratios S_kd and S_ka (indices of pore-size distribution complexity and heterogeneity) are both <0.1, suggesting that heterogeneity is more pronounced in low-probability regions. Fractal and multifractal analyses reveal significant pore structure heterogeneity across different lithofacies, with CSS showing relatively more homogeneous pore structures, whereas MSS exhibits stronger heterogeneity and poorer connectivity. The heterogeneity of shale reservoirs is primarily controlled by pore development, especially micropores and mesopores, and is strongly influenced by total organic carbon and quartz content. Full article

(This article belongs to the Special Issue Pore Structure and Fractal Characteristics in Unconventional Oil and Gas Reservoirs, 2nd Edition)

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16 pages, 319 KB

Open AccessArticle

On the Existence of Solutions for ψ-Caputo Fractional Integro-Differential Boundary Value Problems

by Panjaiyan Karthikeyan, Ambigapathi Manikandan, Mohammed Rabih and Marappan Sathish Kumar

Fractal Fract. 2026, 10(5), 348; https://doi.org/10.3390/fractalfract10050348 - 21 May 2026

Viewed by 403

Abstract

In this article, we investigate the existence of solutions for a class of fractional integro-differential equations (FIDE’s) involving the

ψ

-Caputo fractional derivative (

ψ

-CFD) subject to

ψ

-Caputo boundary conditions. The analysis is carried out in an appropriate Banach space setting [...] Read more.

In this article, we investigate the existence of solutions for a class of fractional integro-differential equations (FIDE’s) involving the

ψ

-Caputo fractional derivative (

ψ

-CFD) subject to

ψ

-Caputo boundary conditions. The analysis is carried out in an appropriate Banach space setting using the Mönch fixed-point theorem. Furthermore, sufficient conditions ensuring the existence and uniqueness of solutions are derived by employing tools from nonlinear functional analysis. In addition, the obtained results contribute to the current literature by extending existing works on fractional differential equations (FDE’s) involving generalized Caputo-type operators. The novelty of this study lies in the incorporation of

ψ

-CFD’s together with

ψ

-Caputo boundary conditions under the framework of Mönch fixed-point theory. An illustrative example is provided to verify the applicability and effectiveness of the theoretical findings. Full article

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

15 pages, 1138 KB

Open AccessArticle

A Compact Finite Difference Scheme with Second-Order Temporal and Sixth-Order Spatial Accuracy for Variable-Order Time-Fractional Sub-Diffusion Equations

by Yu Bo, Xin Zhang, Yu Wang and Yuanfeng Jin

Fractal Fract. 2026, 10(5), 347; https://doi.org/10.3390/fractalfract10050347 - 21 May 2026

Viewed by 488

Abstract

In this paper, a finite difference scheme is proposed for the variable-order time-fractional sub-diffusion equation, achieving second-order accuracy in time and sixth-order accuracy in space. For spatial discretization, a newly constructed operator

A

is employed to obtain a sixth-order compact approximation of the [...] Read more.

In this paper, a finite difference scheme is proposed for the variable-order time-fractional sub-diffusion equation, achieving second-order accuracy in time and sixth-order accuracy in space. For spatial discretization, a newly constructed operator

A

is employed to obtain a sixth-order compact approximation of the second derivative. Using an energy analysis method, a priori estimates of the scheme are derived, and the unconditional stability and convergence are rigorously proved. Numerical examples are provided to verify the theoretical accuracy of the scheme. Full article

(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)

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27 pages, 2976 KB

Open AccessArticle

A Fractional-Order Model for Chikungunya Virus Transmission with Optimal Control and Artificial Neural Network Validation

by Zakirullah, Chen Lu, Nouf Abdulrahman Alqahtani and Mohammadi Begum Jeelani

Fractal Fract. 2026, 10(5), 346; https://doi.org/10.3390/fractalfract10050346 - 20 May 2026

Viewed by 748

Abstract

In this study, a fractional-order epidemic compartmental model is formulated using the Caputo derivative to account for the memory effects of the chikungunya virus. Based on Banach contractions, fixed-point theorems are used to prove existence and uniqueness, and fundamental properties such as positivity [...] Read more.

In this study, a fractional-order epidemic compartmental model is formulated using the Caputo derivative to account for the memory effects of the chikungunya virus. Based on Banach contractions, fixed-point theorems are used to prove existence and uniqueness, and fundamental properties such as positivity and boundedness are established. Normalized forward sensitivity indices are employed to evaluate the relative impact of model parameters on the transmission dynamics and control of the disease. To reduce the spreading of infection, an optimal control problem is formulated by introducing time-dependent control measures with four control strategies that include public health prevention, treatment enhancement, and vector-control measures. Necessary conditions for optimality are derived using Pontryagin’s Maximum Principle. The predictor–corrector Adams–Bashforth–Moulton scheme is applied across different fractional orders and effectively reduces infection levels. The influence of the fractional order

ξ

on the epidemic dynamics is investigated, showing that lower values of

ξ

slow disease progression through a memory effect inherent in the Caputo operator. Moreover, an artificial neural network (ANN) trained via the Levenberg–Marquardt algorithm independently validates the numerical solutions. Full article

(This article belongs to the Special Issue Fractional Order Modelling of Dynamical Systems)

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20 pages, 473 KB

Open AccessArticle

Data-Driven Event-Triggered Scheme for Model-Unknown Fractional-Order Networked Control Systems: A Parametrization Transform Method

by Meixuan Li

Fractal Fract. 2026, 10(5), 345; https://doi.org/10.3390/fractalfract10050345 - 19 May 2026

Viewed by 400

Abstract

This paper proposes a parametrization transform method for model-unknown networked control systems by using a data-driven event-triggered scheme. The key contribution is that an easy-to-apply parametrization transform method is proposed to convert the model-based linear matrix inequality (LMI) conditions into data-driven ones. Compared [...] Read more.

This paper proposes a parametrization transform method for model-unknown networked control systems by using a data-driven event-triggered scheme. The key contribution is that an easy-to-apply parametrization transform method is proposed to convert the model-based linear matrix inequality (LMI) conditions into data-driven ones. Compared with existing ones, using the proposed transform method is without requirements on the specified sizes, structures, and unknown system matrices’ positions of model-based LMI conditions. On this basis, by using Lyapunov theory and some inequality techniques, some data-driven condition are derived to guarantee stability. Without considering model dynamics, the controller gain and trigger parameters can be easily derived by learning from collecting offline data packets. Finally, an illustrative example is presented to showcase the outcomes. Full article

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

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29 pages, 1925 KB

Open AccessArticle

Practical Exponential Stability of Tempered ϖ-Fractional Systems: Lyapunov Criteria and Applications to Perturbed and Controlled Systems

by Ayed R. A. Alanzi, Raouf Fakhfakh, Abdellatif Ben Makhlouf and Omar Naifar

Fractal Fract. 2026, 10(5), 344; https://doi.org/10.3390/fractalfract10050344 - 19 May 2026

Viewed by 690

Abstract

In this paper, we investigate the practical exponential stability of a class of nonlinear systems governed by the tempered

ϖ

-Caputo fractional derivative. A new Lyapunov-based criterion is established to derive sufficient conditions ensuring

ϖ

-practical exponential stability. The obtained result is formulated [...] Read more.

In this paper, we investigate the practical exponential stability of a class of nonlinear systems governed by the tempered

ϖ

-Caputo fractional derivative. A new Lyapunov-based criterion is established to derive sufficient conditions ensuring

ϖ

-practical exponential stability. The obtained result is formulated in a general framework involving suitable growth bounds on the Lyapunov function together with a tempered fractional derivative inequality and a boundedness condition on a weighted integral term. The proposed theorem provides an explicit practical exponential estimate for the system trajectories and extends existing stability results that are available for standard fractional and tempered fractional systems. To demonstrate the applicability of the developed theory, two applications are presented. First, the general criterion is applied to a class of perturbed tempered

ϖ

-fractional systems, for which verifiable sufficient conditions are derived in terms of quadratic Lyapunov functions and perturbation bounds. Second, a state-feedback stabilization result is established for a class of nonlinear tempered fractional control systems, showing that the proposed theorem can be used as an effective tool for closed-loop practical exponential stabilization. Finally, numerical examples are provided to validate the theoretical developments and to illustrate the effectiveness of the proposed approach. An additional test case with

η_{3} > 0

is included to demonstrate the nontrivial range of Theorem 1. Furthermore, a socio-economic tempered fractional cobweb model is incorporated to show how the proposed criterion applies to price-adjustment dynamics with memory and persistent market perturbations. Full article

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

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24 pages, 2259 KB

Open AccessArticle

Fractional-Order Adaptive Resilient Cluster Synchronization Control of Heterogeneous Unmanned Systems Under Deception Attacks and DoS Attacks

by Mengna Li, Ziquan Yu, Ruifeng Zhou and Youmin Zhang

Fractal Fract. 2026, 10(5), 343; https://doi.org/10.3390/fractalfract10050343 - 19 May 2026

Viewed by 325

Abstract

The security of heterogeneous unmanned systems (HUSs) operating in open environments has become a key concern. Therefore, this paper focuses on the fractional-order adaptive resilient clustering synchronization control for a class of networked HUSs composed of multiple unmanned surface vehicles and unmanned aerial [...] Read more.

The security of heterogeneous unmanned systems (HUSs) operating in open environments has become a key concern. Therefore, this paper focuses on the fractional-order adaptive resilient clustering synchronization control for a class of networked HUSs composed of multiple unmanned surface vehicles and unmanned aerial vehicles subject to deception attacks and denial-of-service (DoS) attacks. First, a distributed cluster trajectory generator is designed for each vehicle in a networked HUS to estimate the output trajectory of the leader in their respective clusters in the presence of DoS attacks on the communication layer. Then, by combining backstepping control and fractional calculus, and immersion and invariance (I&I) theory, a fractional-order adaptive synchronization tracking controller is developed to form the desired cluster formation configuration under disturbances and actuator attacks. Among them, the I&I adaptive strategy is designed to estimate the lumped uncertainty caused by attacks and disturbances. Finally, stability analysis and simulation experiments demonstrate the effectiveness of the proposed control scheme. Full article

(This article belongs to the Special Issue Fractional-Order Modeling in Data-Driven Intelligent Systems: From Optimization to AI/ML Applications)

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32 pages, 12648 KB

Open AccessArticle

Fractional-Order-Enhanced Dual-View Representation and VibrMamba–VMamba Collaborative Modeling for Gearbox Fault Diagnosis

by Fengyun Xie, Kang Niu, Zeyan Song, Shulei Wang, Huihang Chen and Ying Cao

Fractal Fract. 2026, 10(5), 342; https://doi.org/10.3390/fractalfract10050342 - 19 May 2026

Viewed by 402

Abstract

Gearbox fault diagnosis under controlled bench-test conditions with known speed variations and noise interference remains challenging because nonstationarity, background noise, and operating-condition fluctuations can easily submerge weak localized fault features. To address this issue, this study proposes a fault diagnosis method based on [...] Read more.

Gearbox fault diagnosis under controlled bench-test conditions with known speed variations and noise interference remains challenging because nonstationarity, background noise, and operating-condition fluctuations can easily submerge weak localized fault features. To address this issue, this study proposes a fault diagnosis method based on a fractional-order-enhanced dual-view representation and VibrMamba–VMamba collaborative modeling. First, this study introduces a Grünwald–Letnikov fractional-order differential enhancement module with a fractional order of

α = 0.6

to strengthen fault-sensitive impulsive components and improve the representation of nonstationary vibration signals. The framework then uses the enhanced signal to construct dual-view inputs: a fractional-order-enhanced one-dimensional vibration sequence and a fractional-order-enhanced synchrosqueezing transform (SST) time–frequency image. Subsequently, the framework constructs a VibrMamba temporal branch and a VMamba visual branch to extract dynamic temporal features and global structural features, respectively. Instead of using simple feature concatenation, this study designs a sample-adaptive collaborative fusion mechanism with gated weighting and cross-branch residual enhancement to integrate complementary temporal–visual representations. Bench-level experiments show that the proposed method achieves 98.90% diagnostic accuracy under clean test conditions and maintains 91.52% accuracy at −5 dB signal-to-noise ratio (SNR). These results should be interpreted as bench-level validation under controlled laboratory conditions rather than as direct evidence of field-level generalization. This framework provides a methodological solution that integrates fractional-order signal enhancement, dual-view representation, and Mamba-style collaborative state-space modeling for gearbox fault classification under controlled laboratory conditions with known speed variations and noise disturbances. Full article

(This article belongs to the Special Issue Implementations and Applications of Algorithms Based on Fractional Calculus to Engineering Problems)

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18 pages, 794 KB

Open AccessArticle

Computational Analysis of Newton-Type Inequalities for Differentiable Strongly Convex Functions via RL-Integrals

by Ghulam Abbas, Saima Riaz, Tamador Alihia, Khuram Ali Khan, Saba Yasmin and Ramy M. Hafez

Fractal Fract. 2026, 10(5), 341; https://doi.org/10.3390/fractalfract10050341 - 18 May 2026

Cited by 1 | Viewed by 840

Abstract

In this paper, new generalizations of Newton-type inequalities for the class of strongly convex functions by utilizing Riemann–Liouville fractional integrals are established. New estimates are obtained for functions that are strongly convex. The established inequalities are further improved. Examples, along with graphs, are [...] Read more.

In this paper, new generalizations of Newton-type inequalities for the class of strongly convex functions by utilizing Riemann–Liouville fractional integrals are established. New estimates are obtained for functions that are strongly convex. The established inequalities are further improved. Examples, along with graphs, are provided to demonstrate the validity of the newly established inequalities and comparisons with existing results. It is expected that the results of this paper will open up new avenues of research and may be generalized to other types of fractional operators and generalized convex functions. Full article

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25 pages, 841 KB

Open AccessArticle

A Unified Caputo—ABC Fractional Framework for High-Order Iterative Methods in Nonlinear Equations

by Mudassir Shams and Bruno Carpentieri

Fractal Fract. 2026, 10(5), 340; https://doi.org/10.3390/fractalfract10050340 - 18 May 2026

Viewed by 288

Abstract

Nonlinear equations arise extensively in engineering and applied sciences, where efficient and reliable iterative solvers are required. This study introduces two fractional-order iterative schemes based on a common predictor–corrector structure: a Caputo-based method, NCFS₁, and an Atangana–Baleanu–Caputo (ABC)-based variant, [...] Read more.

Nonlinear equations arise extensively in engineering and applied sciences, where efficient and reliable iterative solvers are required. This study introduces two fractional-order iterative schemes based on a common predictor–corrector structure: a Caputo-based method, NCFS₁, and an Atangana–Baleanu–Caputo (ABC)-based variant,

{NFS}_{1}^{a b c}

. The proposed schemes incorporate a fractional order and two tunable parameters to improve flexibility in the iterative process. The local convergence behavior of the Caputo-based method is analyzed by means of fractional Taylor expansions, yielding an explicit error equation and convergence order, while analogous asymptotic considerations are discussed for the ABC-based variant. A dynamical-systems analysis is also performed through basins of attraction, the Convergence Area Index, and the Wada measure. Numerical experiments on application-motivated nonlinear models indicate that the proposed methods can provide faster error reduction, smaller residuals, and lower computational cost than selected existing fractional iterative schemes. These results suggest that the proposed framework is a flexible and effective approach for nonlinear root-finding problems, combining local convergence analysis with global dynamical assessment. Full article

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

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18 pages, 2047 KB

Open AccessArticle

Efficient Runge–Kutta Scheme Combined with Richardson Extrapolation for Nonlinear Fractional Partial Differential Equations Involving the Fractional Laplacian

by Yifei Hao, Yiyin Liang and Shichao Yi

Fractal Fract. 2026, 10(5), 339; https://doi.org/10.3390/fractalfract10050339 - 18 May 2026

Viewed by 384

Abstract

In this paper, an efficient numerical framework combined with RK4 method and Richardson extrapolation is proposed to solve nonlinear time-dependent partial differential equations involving the Riesz fractional Laplacian operator

{(- Δ)}^{s}

. The RK4 method guarantees fourth-order temporal accuracy and [...] Read more.

In this paper, an efficient numerical framework combined with RK4 method and Richardson extrapolation is proposed to solve nonlinear time-dependent partial differential equations involving the Riesz fractional Laplacian operator

{(- Δ)}^{s}

. The RK4 method guarantees fourth-order temporal accuracy and L-stability, whereas the spatial fractional operator is discretized using a second-order central finite difference scheme. Based on the consistency conditions of the underlying spatial discretization, and by constructing a Vandermonde matrix to determine the extrapolation coefficients, novel high-order Richardson extrapolation formulas are derived, achieving a maximum convergence order of

O (h^{2 n})

. Numerical experiments, covering 1D variable-coefficient cases, 2D cases with equal/unequal spatial steps, and 3D equidistant differencing cases, demonstrate that the proposed method stably upgrades the convergence order from second-order to fourth-order and further to sixth-order under oscillatory and nonlinear variable-coefficient conditions, with the extrapolated numerical errors reduced to the magnitude of

10^{- 13}

. Asynchronous convergence observed in 2D unequal-step cases validates Theorem 3, while fourth-order convergence is achieved via extrapolation in 3D complex domains. This method possesses prominent advantages of high accuracy, strong robustness, and high efficiency, breaking through the dimensionality and convergence order limitations of traditional high-precision numerical algorithms. Full article

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17 pages, 774 KB

Open AccessArticle

Fractional-Order Orthogonal Jacobi Function-Based Operational Approach for Multi-Term Diffusion-Wave Equations of Fractional Order

by Amal Alshabanat, Mohamed A. Saker, Hanaa Moussa and Samer S. Ezz-Eldien

Fractal Fract. 2026, 10(5), 338; https://doi.org/10.3390/fractalfract10050338 - 18 May 2026

Viewed by 527

Abstract

Solving fractional differential equations using spectral collocation methods based on classical orthogonal polynomials often leads to a reduced convergence rate due to the limited regularity of the solutions. Therefore, spectral collocation methods that employ non-smooth orthogonal functions are frequently preferred for solving various [...] Read more.

Solving fractional differential equations using spectral collocation methods based on classical orthogonal polynomials often leads to a reduced convergence rate due to the limited regularity of the solutions. Therefore, spectral collocation methods that employ non-smooth orthogonal functions are frequently preferred for solving various fractional differential equations. This study focuses on solving one- and two-dimensional time-fractional diffusion-wave equations (DWEs). A spectral collocation technique is developed based on fractional-order orthogonal Jacobi functions to approximate the time-fractional derivatives and orthogonal Jacobi polynomials in the spatial directions. For the first time, a fractional-order orthogonal Jacobi functions-based operational matrix is derived and combined with an orthogonal Jacobi polynomials-based operational matrix of second-order derivatives to solve one- and two-dimensional time-fractional DWEs. Three test problems are conducted to evaluate the efficiency of the proposed numerical technique. Full article

(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)

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35 pages, 449 KB

Open AccessArticle

Approximate Controllability of Higher-Order Hilfer Fractional Neutral Stochastic Systems Driven by Fractional Brownian Motion, Poisson Jumps, and Non-Instantaneous Impulses

by A. M. Sayed Ahmed, Taha Radwan, M. Elsaid Ramadan and Hamdy M. Ahmed

Fractal Fract. 2026, 10(5), 337; https://doi.org/10.3390/fractalfract10050337 - 16 May 2026

Cited by 1 | Viewed by 560

Abstract

This paper addresses the existence of mild solutions and the approximate controllability of a class of higher-order Hilfer fractional semi-linear neutral stochastic differential equations with non-instantaneous impulses in Hilbert spaces. The system is driven by both fractional Brownian motion and Poisson jumps, thereby [...] Read more.

This paper addresses the existence of mild solutions and the approximate controllability of a class of higher-order Hilfer fractional semi-linear neutral stochastic differential equations with non-instantaneous impulses in Hilbert spaces. The system is driven by both fractional Brownian motion and Poisson jumps, thereby capturing long-range dependence as well as random discontinuities. By combining techniques from fractional calculus, stochastic analysis, and operator theory, we establish sufficient conditions for the existence of mild solutions. The analysis is carried out through the construction of suitable solution operator families and the application of Sadovskii’s fixed point theorem in an appropriate phase space framework. In addition, we investigate the controllability properties of the system and derive criteria ensuring approximate controllability of the underlying fractional neutral dynamics. The proposed approach relies on the structural properties of the higher-order Hilfer fractional derivative, estimates for stochastic integrals with respect to fractional Brownian motion, and compactness arguments adapted to non-instantaneous impulsive effects. The inclusion of Poisson jumps and neutral terms introduces significant analytical difficulties, which are overcome using refined resolvent operator techniques and fractional power estimates. An illustrative example is presented to demonstrate the applicability of the theoretical results. The results obtained generalize and unify several recent developments in the theory of fractional stochastic systems and provide a flexible framework for analyzing controlled dynamical models with memory, randomness, and impulsive behavior. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications, Second Edition)

16 pages, 970 KB

Open AccessArticle

Refined Hermite–Hadamard Type Inequalities via the Extended Atangana–Baleanu Fractional Integral

by Mehmet Zeki Sarikaya, Nadiyah Hussain Alharthi and Rubayyi T. Alqahtani

Fractal Fract. 2026, 10(5), 336; https://doi.org/10.3390/fractalfract10050336 - 15 May 2026

Viewed by 463

Abstract

In this study, we obtain new Hermite–Hadamard type inequalities involving an extended form of the Atangana–Baleanu fractional integral operator having Mittag-Leffler kernels. The approach is based on a suitable integral identity for differentiable functions together with the convexity of the absolute value of [...] Read more.

In this study, we obtain new Hermite–Hadamard type inequalities involving an extended form of the Atangana–Baleanu fractional integral operator having Mittag-Leffler kernels. The approach is based on a suitable integral identity for differentiable functions together with the convexity of the absolute value of the first derivative. Within this framework, we extend the classical Hermite–Hadamard inequality to a fractional setting governed by the parameters

α \in (0, 1)

,

β \in (0, 1]

, and

λ > 0

. Full article

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13 pages, 1177 KB

Open AccessArticle

Bifurcation Analysis and Chaotic Behaviors of and a Traveling-Wave Solution to the Zhiber–Shabat Equation with a Truncated M-Fractional Derivative

by Zhao Li and Ejaz Hussain

Fractal Fract. 2026, 10(5), 335; https://doi.org/10.3390/fractalfract10050335 - 15 May 2026

Cited by 13 | Viewed by 737

Abstract

In this article, we use truncated M-fractional derivatives to analyze the bifurcation and chaotic behavior of and traveling-wave solutions to the Zhiber–Shabat equation. By introducing truncated M-fractional derivatives, the equation exhibits richer dynamic properties. Based on phase diagram analysis and dynamical system theory, [...] Read more.

In this article, we use truncated M-fractional derivatives to analyze the bifurcation and chaotic behavior of and traveling-wave solutions to the Zhiber–Shabat equation. By introducing truncated M-fractional derivatives, the equation exhibits richer dynamic properties. Based on phase diagram analysis and dynamical system theory, the bifurcation behavior of the equilibrium point of a two-dimensional dynamical system is discussed. At the same time, the dynamical behavior of a two-dimensional dynamical system with periodic disturbances is considered, revealing the complex chaotic phenomena of the system under specific parameters. A planar phase diagram, a three-dimensional phase diagram, a sensitivity analysis, and a maximum Lyapunov exponent diagram of the perturbed two-dimensional dynamical system were employed. Furthermore, various forms of accurate analytical solutions were obtained through traveling-wave transformation and numerical simulation. The three-dimensional, two-dimensional, density, and polar coordinates of the solutions were plotted using mathematical software. The results indicate that the fractional order and system parameters have a significant impact on the morphology and chaotic characteristics of the solution. This study provides new theoretical insights into the nonlinear dynamics of fractional-order Zhiber–Shabat equations. Full article

(This article belongs to the Special Issue Fractional Nonlinear Dynamics in Science and Engineering)

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26 pages, 7636 KB

Open AccessArticle

Dynamics and Efficient Numerical Simulation of a Fractional-Order T System

by Liping Yu and Hongyi Zhu

Fractal Fract. 2026, 10(5), 334; https://doi.org/10.3390/fractalfract10050334 - 14 May 2026

Viewed by 527

Abstract

In this paper, we propose and numerically investigate a fractional T system. As a fractional generalization of the classical T model, the fractional order serves as a memory parameter governing the system dynamics. By employing the fractional stability criterion, the local stability of [...] Read more.

In this paper, we propose and numerically investigate a fractional T system. As a fractional generalization of the classical T model, the fractional order serves as a memory parameter governing the system dynamics. By employing the fractional stability criterion, the local stability of the equilibrium points is analyzed, and the existence of Hopf bifurcation is characterized. To efficiently simulate the long-time dynamics induced by fractional memory, a linear semi-implicit numerical scheme accelerated by a sum-of-exponentials approximation of the Caputo derivative is developed. The proposed scheme is shown to be stable and enables a significant reduction in computational cost compared with classical L1 and Grünwald–Letnikov methods. Numerical experiments, including time series, phase portraits, Lyapunov exponent computations, and bifurcation diagrams, demonstrate that varying the fractional order leads to transitions among stable, periodic, and chaotic regimes. In particular, pronounced transient dynamics are observed as the fractional order approaches its critical value, highlighting the memory-induced effects inherent in fractional-order systems. Full article

(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)

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40 pages, 472 KB

Open AccessArticle

Fractional Fuzzy Tensor-Based Bonferroni Aggregation Operators and Their Application in Cloudburst Disaster Management in Northern Pakistan

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

Fractal Fract. 2026, 10(5), 333; https://doi.org/10.3390/fractalfract10050333 - 14 May 2026

Viewed by 700

Abstract

The growing complexity of modern decision-making environments, characterized by multi-dimensional data, uncertainty, and dynamic behavior, demands advanced mathematical frameworks for effective information aggregation. Although fractional fuzzy tensor (FFT) models provide a powerful tool for representing such complex systems by integrating fuzzy logic, tensor [...] Read more.

The growing complexity of modern decision-making environments, characterized by multi-dimensional data, uncertainty, and dynamic behavior, demands advanced mathematical frameworks for effective information aggregation. Although fractional fuzzy tensor (FFT) models provide a powerful tool for representing such complex systems by integrating fuzzy logic, tensor structures, and fractional dynamics, the lack of suitable aggregation mechanisms significantly limits their practical applicability. To address this challenge, this paper proposes a novel family of Bonferroni mean-based aggregation operators within the fractional fuzzy tensor environment. The proposed framework extends the classical Bonferroni mean to multi-dimensional fractional fuzzy settings, enabling the effective modeling of interrelationships among criteria while preserving the structural and dynamic properties of FFTs. Specifically, four aggregation operators—namely, the fractional fuzzy tensor Bonferroni mean (FFT-BM), weighted Bonferroni mean (FFT-WBM), ordered Bonferroni mean (FFT-OBM), and hybrid Bonferroni mean (FFT-HBM)—are systematically developed. A comprehensive theoretical analysis is conducted to investigate fundamental properties such as idempotency, monotonicity, boundedness, commutativity, and stability, thereby establishing the mathematical consistency and reliability of the proposed operators. Furthermore, a structured multi-criteria decision-making (MCDM) algorithm is formulated, incorporating tensor construction, aggregation, evaluation, and sensitivity analysis phases to handle complex uncertain information effectively. To demonstrate the practical applicability of the proposed framework, a real-world case study related to disaster management decision-making is presented. The results are further validated through quantitative comparative analysis with classical and recent aggregation operators, revealing improved discrimination power, robustness, and ranking consistency. Additionally, sensitivity analysis confirms the stability of the proposed approach under varying parameters. The findings indicate that the proposed Bonferroni mean-based aggregation framework significantly enhances the capability of FFT models in handling high-dimensional, uncertain, and dynamic decision-making problems. This study not only strengthens the theoretical foundation of aggregation in tensor-based fuzzy environments but also provides a flexible and reliable decision-support tool for complex real-world applications. Full article

(This article belongs to the Section Complexity)

34 pages, 396 KB

Open AccessArticle

Existence and Uniqueness of Solutions to Fractional Differential Equations in Complex-Valued Suprametric Spaces

by Hanadi Zahed

Fractal Fract. 2026, 10(5), 332; https://doi.org/10.3390/fractalfract10050332 - 13 May 2026

Cited by 1 | Viewed by 634

Abstract

This study focuses on establishing the existence and uniqueness of solutions for nonlinear fractional differential equations through the application of fixed-point methods in complex-valued suprametric spaces. In order to accomplish this, novel cyclic and interpolative contractive conditions are formulated within the complex-valued suprametric [...] Read more.

This study focuses on establishing the existence and uniqueness of solutions for nonlinear fractional differential equations through the application of fixed-point methods in complex-valued suprametric spaces. In order to accomplish this, novel cyclic and interpolative contractive conditions are formulated within the complex-valued suprametric setting, leading to the derivation of several common fixed-point theorems. The obtained results extend and encompass a variety of known fixed-point theorems in complex-valued metric spaces as particular instances. In addition, meaningful and non-trivial examples are presented to highlight the effectiveness and practical relevance of the developed theoretical framework. Full article

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

31 pages, 4209 KB

Open AccessArticle

Tempered Fractional Deterministic Learning for Online Battery State-of-Health Estimation: A Cycle-Level Benchmark Study

by Omar Kahouli, Younès Bahou, Moawia Farah and Imed Bouzida

Fractal Fract. 2026, 10(5), 331; https://doi.org/10.3390/fractalfract10050331 - 12 May 2026

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Abstract

Accurate battery state-of-health (SOH) estimation is essential for safe and reliable electric-vehicle operation. This paper presents a Tempered Fractional Deterministic Learning (TF-DL) framework that combines deterministic learning with a tempered fractional adaptation law to introduce tunable memory and graceful forgetting into SOH estimation. [...] Read more.

Accurate battery state-of-health (SOH) estimation is essential for safe and reliable electric-vehicle operation. This paper presents a Tempered Fractional Deterministic Learning (TF-DL) framework that combines deterministic learning with a tempered fractional adaptation law to introduce tunable memory and graceful forgetting into SOH estimation. Two realizations are considered: an exact truncated variant (TF-DL-T) and an embedded low-memory variant (TF-DL-E). The framework is evaluated on a NASA-derived cycle-level battery aging benchmark with capacity-based SOH labels and a battery-level train/test split. After filtering, 14 batteries were retained, of which 9 were used for training and 5 unseen batteries were used for testing. Random Forest achieved the best overall performance (MAE = 0.0436, RMSE = 0.0496), while LSTM was the strongest sequence baseline (MAE = 0.0757, RMSE = 0.0920). Among the online RBF-based methods, TF-DL-E achieved the best performance (MAE = 0.0966, RMSE = 0.1077), outperforming GD-DL and TF-DL-T. Unlike offline methods, TF-DL-E operates online with constant memory, which makes it suitable for embedded battery management systems. The results indicate that TF-DL-E is the more robust and practically relevant tempered variant, whereas TF-DL-T remains more fragile and parameter-sensitive. Full article

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

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16 pages, 2449 KB

Open AccessArticle

Straightforward Design of a Robust Fractional-Order Controller

by Robin De Keyser, Marcian D. Mihai, Isabela R. Birs and Cristina I. Muresan

Fractal Fract. 2026, 10(5), 330; https://doi.org/10.3390/fractalfract10050330 - 12 May 2026

Cited by 1 | Viewed by 867

Abstract

Fractional-order controllers have emerged as robust alternatives to conventional PID controllers. Existing tuning methods generally focus solely on robustness to process gain variations. This paper introduces a design method for fractional-order PI controllers, specifically resilient to time constant changes by shaping the loop [...] Read more.

Fractional-order controllers have emerged as robust alternatives to conventional PID controllers. Existing tuning methods generally focus solely on robustness to process gain variations. This paper introduces a design method for fractional-order PI controllers, specifically resilient to time constant changes by shaping the loop frequency response. This work simplifies the design method by replacing the separate magnitude and phase derivative calculations used in prior techniques with a unified, single partial derivative approach. Instead of using cumbersome optimization routines and graphical analysis used in existing fractional-order controller tuning methods, the proposed approach uses a direct, simple, and efficient 1-step algorithm. Numerical simulations for lag- and delay-dominant processes are included to highlight the efficiency of the proposed approach. Traditional integer order controllers are designed for comparative purposes. The proposed approach achieves a constant overshoot despite time constant variations, an advantage compared to classical controllers. Full article

(This article belongs to the Special Issue Novel and Effective Applications of Fractional-Order Models)

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24 pages, 7453 KB

Open AccessArticle

Fractal Metrics and Pore Architecture as Determinants of Diffusion in High-Rank Coal Reservoirs of the Mengjin Coalfield, Henan Province

by Zixuan Liu, Detian Yan, Shangbin Chen and Derek Elsworth

Fractal Fract. 2026, 10(5), 329; https://doi.org/10.3390/fractalfract10050329 - 11 May 2026

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Abstract

Understanding the pore structure of high-rank coals is essential in evaluating gas storage and transport. Here, twelve semianthracite samples from the early Permian Shanxi Formation were investigated by proximate analysis, optical microscopy, low-temperature N₂ adsorption, and fractal analysis, coupled with diffusion coefficient [...] Read more.

Understanding the pore structure of high-rank coals is essential in evaluating gas storage and transport. Here, twelve semianthracite samples from the early Permian Shanxi Formation were investigated by proximate analysis, optical microscopy, low-temperature N₂ adsorption, and fractal analysis, coupled with diffusion coefficient modeling. The coals exhibit diverse pore types (plant-cellular, interparticle, and dissolution pores) shaped by coalification and minerals and show Type IV (a) isotherms with H4 hysteresis loops, indicating complex pore networks. Pore-size partitioning reveals that mesopores and macropores dominate total pore volume, whereas mesopores contribute most of the specific surface area. The pore structure exhibits strong fractal characteristics with an average comprehensive fractal dimension (F_c) of 2.628. The calculated gas diffusion coefficient decreases monotonically with increasing pressure from 1 MPa to 5.8 MPa, with a more pronounced decline at low pressure, indicating a clear pressure-dependent attenuation effect. Diffusion capacity is weakly related to average pore diameter but shows positive correlations with total pore volume and, particularly, macropore volume. Multiple linear regression further demonstrates that pore volume structure is the dominant control on diffusion under both low- and high-pressure conditions, with the relative importance ranked as macropores > mesopores > micropores. Macropores provide the main low-resistance transport framework, mesopores serve as transitional pathways linking storage and transport domains, whereas micropores mainly contribute to gas storage and may even suppress apparent diffusion when overly developed. These results reveal a clear functional differentiation of multiscale pore systems and highlight that gas migration in semianthracite is jointly governed by pore size distribution, connectivity, tortuosity, and fractal network topology. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs, 2nd Edition)

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26 pages, 557 KB

Open AccessArticle

Perturbed Hybrid Pantograph Systems with Deformable Derivatives: Well-Posedness, Stability, Numerical Sensitivity, and a Delay-Feedback Toy Example

by Rafik Zeraoulia, Souad Ayadi, Amina Boucenna, Meltem Erden Ege, Ozgur Ege and Mohammed Rabih

Fractal Fract. 2026, 10(5), 328; https://doi.org/10.3390/fractalfract10050328 - 11 May 2026

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Abstract

We study a perturbed coupled system of generalized hybrid pantograph equations involving the deformable derivative of Zulfeqarr–Ujlayan–Ahuja. A central point of the revision is made explicit: for classically differentiable functions this derivative is local and satisfies [...] Read more.

We study a perturbed coupled system of generalized hybrid pantograph equations involving the deformable derivative of Zulfeqarr–Ujlayan–Ahuja. A central point of the revision is made explicit: for classically differentiable functions this derivative is local and satisfies

D^{τ} u = (1 - τ) u + τ u^{'}

. Therefore, in the present differentiable setting the memory or aftereffect is produced by the proportional pantograph delays, while the deformable order

τ

supplies an order-dependent local relaxation/drift term. After rewriting the system as an equivalent integral equation on

X = C (I, R^{2})

, we establish invariant-ball conditions, existence and uniqueness within invariant balls, generalized Ulam–Hyers stability, and Lipschitz continuous dependence on the perturbation amplitude

ε

. The assumptions and constants are stated so that the restrictive roles of the Lipschitz bounds, the interval length, and

| ε |

are transparent. We then provide numerical parameter sensitivity diagrams for illustrative pantograph systems and include step-size refinement checks and performance indices. The numerical and plasma-inspired sections are deliberately framed as exploratory delay-feedback examples rather than as first-principles plasma models or rigorous bifurcation theory. 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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16 pages, 337 KB

Open AccessArticle

A Fractional Differential Equation Model and Dynamic Analysis of Animal Avoidance Learning

by Kaihong Zhao

Fractal Fract. 2026, 10(5), 327; https://doi.org/10.3390/fractalfract10050327 - 11 May 2026

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Abstract

This article employs a fractional differential equation model to probe the dynamic mechanism of animal avoidance learning and memory retention. This model encompasses both linear and nonlinear scenarios. We first obtain the series-type analytical solution for the linear scenario and its absolute uniform [...] Read more.

This article employs a fractional differential equation model to probe the dynamic mechanism of animal avoidance learning and memory retention. This model encompasses both linear and nonlinear scenarios. We first obtain the series-type analytical solution for the linear scenario and its absolute uniform convergence by Laplace transform and Mittag–Leffler function. Secondly, we establish the existence, uniqueness and Ulam–Hyers stability for the nonlinear scenario via the fixed point theorem and analytical techniques. Eventually, some examples and numerical simulations are provided to examine the effectiveness and availability of the main findings. Full article

(This article belongs to the Special Issue Modeling and Dynamic Analysis of Fractional-Order Systems)

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32 pages, 2955 KB

Open AccessArticle

Multifractal Dynamics and Spillover Effects Between China’s Carbon and Energy Markets Under Policy Shocks

by Tian Zhang and Shaohui Zou

Fractal Fract. 2026, 10(5), 326; https://doi.org/10.3390/fractalfract10050326 - 11 May 2026

Viewed by 586

Abstract

Understanding the multifractal dynamics of carbon and energy markets is essential for capturing complex cross-market interactions and policy-induced volatility. This study investigates China’s carbon and energy markets from 16 July 2021 to 30 January 2026, integrating macro policy interventions with nonlinear market evolution. [...] Read more.

Understanding the multifractal dynamics of carbon and energy markets is essential for capturing complex cross-market interactions and policy-induced volatility. This study investigates China’s carbon and energy markets from 16 July 2021 to 30 January 2026, integrating macro policy interventions with nonlinear market evolution. We first employ a Generalized Autoregressive Conditional Heteroskedasticity-Dynamic Conditional Correlation (GARCH-DCC) model with exogenous policy variables to quantify volatility spillovers and dynamic correlations under policy shocks. Then, a rolling-window multifractal detrended cross-correlation analysis (MF-DCCA) is applied to reveal multiscale dependencies, characteristic periods, and complex fractal structures in cross-market linkages. The results indicate: (1) pronounced spillover effects exist among carbon and energy markets, with policy interventions amplifying short-term contagion; (2) policy shocks exert a “green-squeezing” effect, particularly in the coal market, while endogenous volatility structures exhibit long-term resilience; (3) cross-market linkages display multifractal characteristics, with turning points between the carbon market and electricity, new energy, and coal markets at approximately 6.28, 5.58, and 6.96 months, respectively. These findings provide insights for policymakers in designing differentiated energy regulations and for investors in multiscale risk management and asset allocation. Full article

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19 pages, 2645 KB

Open AccessArticle

A Cyclic Constitutive Model Based on Fractional Derivative for Rate-Dependent Ratcheting of EA4T Axle Steel

by Xuehong Ren, Chenzhuo Qu, Jiujian Wang, Wenjie Zhao, Shaopu Yang and Yongqiang Liu

Fractal Fract. 2026, 10(5), 325; https://doi.org/10.3390/fractalfract10050325 - 11 May 2026

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Abstract

Within the framework of elastoplastic theory, this study develops and improves a fractional cyclic constitutive model capable of describing rate-dependent ratcheting behavior by defining the ratcheting parameter as a function of the cumulative plastic strain rate and describing the plastic strain rate and [...] Read more.

Within the framework of elastoplastic theory, this study develops and improves a fractional cyclic constitutive model capable of describing rate-dependent ratcheting behavior by defining the ratcheting parameter as a function of the cumulative plastic strain rate and describing the plastic strain rate and back stress in fractional-order forms. Additionally, a brief introduction is provided on the numerical implementation process and parameter determination method of this model. The newly improved fractional-order model was subsequently employed to simulate and predict the cyclic deformation of the cyclically softening material, EA4T axle steel. The following conclusions can be drawn: owing to the incorporation of fractional calculus, the newly improved model can predict both the monotonic tensile curves and the cyclic softening behavior of materials under different strain rates—capabilities that are not achievable with conventional elastic–plastic cyclic constitutive models. By defining the ratcheting parameter as a function of the cumulative plastic strain rate, the improved fractional model can reasonably predict the evolution laws of both uniaxial and non-proportional multiaxial ratcheting. By describing the evolution of plastic strain rate and back stress in fractional-order forms, the newly improved fractional model can provide a relatively accurate prediction of the rate-dependent uniaxial and multiaxial ratcheting behaviors. Full article

(This article belongs to the Special Issue Fractional Modeling and Dynamics Analysis of Complex Systems)

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27 pages, 1676 KB

Open AccessArticle

A Space–Time Spectral Method for Nonlinear Fractional Convection–Diffusion Equations with Viscosity Terms

by Zhe Yu, Shanshan Guo, Xinming Zhang and Baohe Zhang

Fractal Fract. 2026, 10(5), 324; https://doi.org/10.3390/fractalfract10050324 - 10 May 2026

Viewed by 641

Abstract

We develop a high-order space-time spectral method for nonlinear convection–diffusion equations with a Riemann–Liouville time-fractional derivative and a spectrally defined space-fractional Laplacian. The spatial discretization uses a Fourier spectral method that diagonalizes the fractional Laplacian under periodic boundary conditions. The temporal discretization employs [...] Read more.

We develop a high-order space-time spectral method for nonlinear convection–diffusion equations with a Riemann–Liouville time-fractional derivative and a spectrally defined space-fractional Laplacian. The spatial discretization uses a Fourier spectral method that diagonalizes the fractional Laplacian under periodic boundary conditions. The temporal discretization employs a Petrov–Galerkin method based on generalized Jacobi functions which capture the initial singularity exactly. The nonlinear convection term is treated pseudo-spectrally, and the resulting algebraic system is solved with a damped Newton iteration. Rigorous error analysis proves exponential convergence in both space and time. Numerical experiments for various fractional orders confirm the spectral accuracy. Simulations of the fractional Burgers equation demonstrate that increasing the viscosity enhances diffusion and stabilizes the solution, while a nonlinear coefficient that significantly exceeds the viscosity leads to error growth over long time intervals. The method provides an efficient and accurate tool for simulating anomalous transport phenomena. Full article

(This article belongs to the Special Issue Fractional Modeling and Dynamics Analysis of Complex Systems)

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13 pages, 330 KB

Open AccessArticle

Lyapunov-Type and Hartman–Wintner-Type Inequalities for a Class of Composite Fractional Integral Operators

by Rubayyi T. Alqahtani and Mehmet Zeki Sarikaya

Fractal Fract. 2026, 10(5), 323; https://doi.org/10.3390/fractalfract10050323 - 9 May 2026

Cited by 1 | Viewed by 398

Abstract

In this paper, we establish Lyapunov-type and Hartman–Wintner-type integral inequalities for boundary value problems involving a class of composite fractional integral operators. By employing an explicit Green function representation and sharp uniform bounds for the associated kernel, we derive necessary conditions for the [...] Read more.

In this paper, we establish Lyapunov-type and Hartman–Wintner-type integral inequalities for boundary value problems involving a class of composite fractional integral operators. By employing an explicit Green function representation and sharp uniform bounds for the associated kernel, we derive necessary conditions for the existence of nontrivial solutions. A distinctive feature of the obtained results is the higher-order scaling behavior induced by the interaction of left- and right-sided components of the operator, which cannot be observed in classical one-sided fractional models. As applications, we obtain quantitative nonexistence criteria and explicit lower bounds for the principal eigenvalue of the corresponding eigenvalue problem. These estimates extend classical results to a higher-order fractional framework and highlight the influence of the domain geometry on the stability of the solutions. Full article

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19 pages, 4347 KB

Open AccessArticle

Wind Speed Prediction Based on Wavelet Decomposition and the Fractal-Based LSTM Method

by Dandan Xia, Shaokun Shi, Yongchen Peng, Zhiqun Yuan and Li Lin

Fractal Fract. 2026, 10(5), 322; https://doi.org/10.3390/fractalfract10050322 - 9 May 2026

Viewed by 570

Abstract

Accurate wind speed prediction is essential for wind power generation and wind-resistant structural design. In this research, a short-term wind speed prediction model that combines the wavelet decomposition (WD) method and a fractal-based long short-term memory (LSTM) network is proposed. The fractal dimensions [...] Read more.

Accurate wind speed prediction is essential for wind power generation and wind-resistant structural design. In this research, a short-term wind speed prediction model that combines the wavelet decomposition (WD) method and a fractal-based long short-term memory (LSTM) network is proposed. The fractal dimensions of the dataset at each level, which are decomposed using the WD method, are calculated using the box-counting method. With the dynamic learning rate in the loss function updated by fractal dimensions, the gradient-enhanced LSTM network is applied for wind speed prediction. Experimental wind speed data collected during wind field measurement experiments in Pingtan, Fujian Province, China, were used to validate the proposed wind speed prediction model. The predicted wind speed results at different considered time intervals are compared with those of the traditional LSTM method. The results suggest that the proposed method significantly improves the accuracy of wind speed forecasting. Full article

(This article belongs to the Section Engineering)

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51 pages, 5561 KB

Open AccessArticle

A Unified Comparative Framework for Fractional-Order Operator Approximation with Pareto-Based Analysis

by Abebe Alemu Wendimu, Radek Matušů, Ibrahim Shaikh, Mihret Kochito Wolde, Meron Tadele Roba and Feleke Tsegaye Yareshe

Fractal Fract. 2026, 10(5), 321; https://doi.org/10.3390/fractalfract10050321 - 9 May 2026

Viewed by 744

Abstract

Fractional-order operators play a fundamental role in the modeling and control of complex dynamical systems; however, their infinite-dimensional nature necessitates rational approximation for practical implementation. This paper presents a unified comparative framework for evaluating widely used approximation methods, including standard and refined Oustaloup [...] Read more.

Fractional-order operators play a fundamental role in the modeling and control of complex dynamical systems; however, their infinite-dimensional nature necessitates rational approximation for practical implementation. This paper presents a unified comparative framework for evaluating widely used approximation methods, including standard and refined Oustaloup filters, continued fraction expansion (CFE), Matsuda, curve-fitting, and modified stability boundary locus (M-SBL) approaches. A systematic evaluation methodology is developed to assess these methods based on frequency-domain accuracy, time-domain performance, and robustness. Furthermore, a Pareto-based multi-objective analysis is introduced to explicitly capture the trade-offs among conflicting performance criteria, enabling the identification of non-dominated solutions without relying on weighted-sum formulations. Extensive simulations are conducted over a wide frequency range to evaluate approximation accuracy and control-oriented performance. The results reveal that different methods exhibit distinct trade-offs between accuracy, robustness, and complexity. In particular, the Oustaloup and M-SBL approaches demonstrate strong overall performance across multiple criteria, while methods such as CFE and curve-fitting show limitations under wideband conditions. The proposed framework provides a systematic and reproducible basis for selecting appropriate approximation techniques in fractional-order control applications, offering valuable insights into their practical implementation and performance trade-offs. Full article

(This article belongs to the Section Engineering)

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34 pages, 847 KB

Open AccessArticle

Mathematical and Numerical Analysis of a Fractional Diabetes Model with Singular Operator

by Pratibha Verma and Wojciech Sumelka

Fractal Fract. 2026, 10(5), 320; https://doi.org/10.3390/fractalfract10050320 - 9 May 2026

Cited by 1 | Viewed by 1308

Abstract

Diabetes mellitus is a chronic disease with complex progression dynamics. This study introduces a fractional order compartmental model based on the Caputo derivative, a singular-kernel derivative, to describe disease progression across four compartments: susceptible, insulin-resistant, diabetic without complications, and diabetic with complications. The [...] Read more.

Diabetes mellitus is a chronic disease with complex progression dynamics. This study introduces a fractional order compartmental model based on the Caputo derivative, a singular-kernel derivative, to describe disease progression across four compartments: susceptible, insulin-resistant, diabetic without complications, and diabetic with complications. The model is novel for integrating memory effects into disease-stage transitions while maintaining dimensional consistency. Key mathematical properties, including existence, uniqueness, positivity, boundedness, equilibrium analysis, and both local and global stability, are established. Ulam–Hyers stability is also examined to evaluate the robustness of the model solutions. Numerical approximations are obtained using the Adomian Decomposition Method and its Laplace variant. Simulations indicate that lower fractional orders enhance memory effects, slow disease progression, and influence long-term dynamics. These results demonstrate that the proposed approach provides a flexible and robust framework for studying chronic disease progression and makes a meaningful contribution to the literature on fractional diabetes models. Full article

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

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