Fractal and Fractional

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19 pages, 1037 KB

Open AccessArticle

Fixed-Time Synchronization of Fractional-Order Hopfield Neural Networks with Unbounded Proportional Delay and Bounded Parameter Uncertainties

by Zizhao Guo, Jiayi Cai, Hongguang Fan, Jiyong Tan and Jianxiao Zou

Fractal Fract. 2025, 9(12), 798; https://doi.org/10.3390/fractalfract9120798 - 5 Dec 2025

Viewed by 602

Abstract

This paper investigates the fixed-time synchronization of fractional-order proportional delay Hopfield neural networks (PDHNNs) with bounded parameter uncertainties. Unlike constant delay and bounded variable delay, proportional delay has time-varying and unbounded characteristics, which pose challenges for the synchronization control of primary–secondary fractional neural [...] Read more.

This paper investigates the fixed-time synchronization of fractional-order proportional delay Hopfield neural networks (PDHNNs) with bounded parameter uncertainties. Unlike constant delay and bounded variable delay, proportional delay has time-varying and unbounded characteristics, which pose challenges for the synchronization control of primary–secondary fractional neural networks. To achieve fixed-time synchronization, we propose a new nonlinear multi-module feedback controller. It consists of three key functional modules: eliminating the impact of proportional delay on system stability; ensuring convergence within a fixed time frame without being limited by initial conditions; and expanding the selectable range of parameters. Combining the stability lemma and inequality techniques, synchronization criteria of PDHNNs are derived based on the construction of a Lyapunov function with a negative fractional derivative. The settling time can be effectively estimated, which depends on the control parameters and is independent of initial values. Two numerical experiments verify the effectiveness of the theorem and corollary in this study. Full article

(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)

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27 pages, 643 KB

Open AccessArticle

Fractional Modeling and Stability Analysis of Tomato Yellow Leaf Curl Virus Disease: Insights for Sustainable Crop Protection

by Mansoor Alsulami, Ali Raza, Marek Lampart, Umar Shafique and Eman Ghareeb Rezk

Fractal Fract. 2025, 9(12), 754; https://doi.org/10.3390/fractalfract9120754 - 21 Nov 2025

Viewed by 708

Abstract

Tomato Yellow Leaf Curl Virus (TYLCV) has recently caused severe economic losses in global tomato production. According to the International Plant Protection Convention (IPPC), yield reductions of 50–60% have been reported in several regions, including the Caribbean, Central America, and South Asia, with [...] Read more.

Tomato Yellow Leaf Curl Virus (TYLCV) has recently caused severe economic losses in global tomato production. According to the International Plant Protection Convention (IPPC), yield reductions of 50–60% have been reported in several regions, including the Caribbean, Central America, and South Asia, with losses in sensitive cultivars reaching up to 90–100%. In developing countries, TYLCV and mixed infections affect more than seven million hectares of tomato-growing land annually. In this study, we construct and analyze a nonlinear dynamic model describing the transmission of TYLCV, incorporating the Caputo fractional-order derivative operator. The existence and uniqueness of solutions to the proposed model are rigorously established. Equilibrium points are identified, and the Jacobian determinant approach is applied to compute the basic reproduction number,

R_{0}

. Suitable Lyapunov functions are formulated to analyze the global asymptotic stability of both the disease-free and endemic equilibria. The model is numerically solved using the Grünwald–Letnikov-based nonstandard finite difference method, and simulations assess how the memory index and preventive strategies influence disease propagation. The results reveal critical factors governing TYLCV transmission and suggest effective intervention measures to guide sustainable crop protection policies. Full article

(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)

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24 pages, 13931 KB

Open AccessArticle

Iterative Investigation of the Nonlinear Fractional Cahn–Allen and Fractional Clannish Random Walker’s Parabolic Equations by Using the Hybrid Decomposition Method

by Sarfaraz Ahmed, Ibtisam Aldawish, Syed T. R. Rizvi and Aly R. Seadawy

Fractal Fract. 2025, 9(10), 656; https://doi.org/10.3390/fractalfract9100656 - 11 Oct 2025

Cited by 8 | Viewed by 886

Abstract

In this work, we numerically investigate the fractional clannish random walker’s parabolic equations (FCRWPEs) and the nonlinear fractional Cahn–Allen (NFCA) equation using the Hybrid Decomposition Method (HDM). The analysis uses the Atangana–Baleanu fractional derivative (ABFD) in the Caputo sense, which has a nonsingular [...] Read more.

In this work, we numerically investigate the fractional clannish random walker’s parabolic equations (FCRWPEs) and the nonlinear fractional Cahn–Allen (NFCA) equation using the Hybrid Decomposition Method (HDM). The analysis uses the Atangana–Baleanu fractional derivative (ABFD) in the Caputo sense, which has a nonsingular and nonlocal Mittag–Leffler kernel (MLk) and provides a more accurate depiction of memory and heredity effects, to examine the dynamic behavior of the models. Using nonlinear analysis, the uniqueness of the suggested models is investigated, and distinct wave profiles are created for various fractional orders. The accuracy and effectiveness of the suggested approach are validated by a number of example cases, which also support the approximate solutions of the nonlinear FCRWPEs. This work provides significant insights into the modeling of anomalous diffusion and complex dynamic processes in fields such as phase transitions, biological transport, and population dynamics. The inclusion of the ABFD enhances the model’s ability to capture nonlocal effects and long-range temporal correlations, making it a powerful tool for simulating real-world systems where classical derivatives may be inadequate. Full article

(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)

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25 pages, 5123 KB

Open AccessArticle

Analytical and Numerical Treatment of Evolutionary Time-Fractional Partial Integro-Differential Equations with Singular Memory Kernels

by Kamel Al-Khaled, Isam Al-Darabsah, Amer Darweesh and Amro Alshare

Fractal Fract. 2025, 9(6), 392; https://doi.org/10.3390/fractalfract9060392 - 19 Jun 2025

Cited by 3 | Viewed by 1419

Abstract

Evolution equations with fractional-time derivatives and singular memory kernels are used for modeling phenomena exhibiting hereditary properties, as they effectively incorporate memory effects into their formulation. Time-fractional partial integro-differential equations (FPIDEs) represent a significant class of such evolution equations and are widely used [...] Read more.

Evolution equations with fractional-time derivatives and singular memory kernels are used for modeling phenomena exhibiting hereditary properties, as they effectively incorporate memory effects into their formulation. Time-fractional partial integro-differential equations (FPIDEs) represent a significant class of such evolution equations and are widely used in diverse scientific and engineering fields. In this study, we use the

sinc

-collocation and iterative Laplace transform methods to solve a specific FPIDE with a weakly singular kernel. Specifically, the

sinc

-collocation method is applied to discretize the spatial domain, while a combination of numerical techniques is utilized for temporal discretization. Then, we prove the convergence analytically. To compare the two methods, we provide two examples. We notice that both the

sinc

-collocation and iterative Laplace transform methods provide good approximations. Moreover, we find that the accuracy of the methods is influenced by fractional order

α \in (0, 1)

and the memory-kernel parameter

β \in (0, 1)

. We observe that the error decreases as

β

increases, where the kernel becomes milder, which extends the single-value study of

β = 1 / 2

in the literature. Full article

(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)

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33 pages, 1387 KB

Open AccessArticle

Design of Non-Standard Finite Difference and Dynamical Consistent Approximation of Campylobacteriosis Epidemic Model with Memory Effects

by Ali Raza, Feliz Minhós, Umar Shafique, Emad Fadhal and Wafa F. Alfwzan

Fractal Fract. 2025, 9(6), 358; https://doi.org/10.3390/fractalfract9060358 - 29 May 2025

Cited by 2 | Viewed by 1019

Abstract

Campylobacteriosis has been described as an ever-changing disease and health issue that is rather dangerous for different population groups all over the globe. The World Health Organization (WHO) reports that 33 million years of healthy living are lost annually, and nearly one in [...] Read more.

Campylobacteriosis has been described as an ever-changing disease and health issue that is rather dangerous for different population groups all over the globe. The World Health Organization (WHO) reports that 33 million years of healthy living are lost annually, and nearly one in ten persons have foodborne illnesses, including Campylobacteriosis. This explains why there is a need to develop new policies and strategies in the management of diseases at the intergovernmental level. Within this framework, an advanced stochastic fractional delayed model for Campylobacteriosis includes new stochastic, memory, and time delay factors. This model adopts a numerical computational technique called the Grunwald–Letnikov-based Nonstandard Finite Difference (GL-NSFD) scheme, which yields an exponential fitted solution that is non-negative and uniformly bounded, which are essential characteristics when working with compartmental models in epidemic research. Two equilibrium states are identified: the first is an infectious Campylobacteriosis-free state, and the second is a Campylobacteriosis-present state. When stability analysis with the help of the basic reproduction number

R_{0}

is performed, the stability of both equilibrium points depends on the

R_{0}

value. This is in concordance with the actual epidemiological data and the research conducted by the WHO in recent years, with a focus on the tendency to increase the rate of infections and the necessity to intervene in time. The model goes further to analyze how a delay in response affects the band of Campylobacteriosis spread, and also agrees that a delay in response is a significant factor. The first simulations of the current state of the system suggest that certain conditions can be achieved, and the eradication of the disease is possible if specific precautions are taken. The outcomes also indicate that enhancing the levels of compliance with the WHO-endorsed SOPs by a significant margin can lower infection rates significantly, which can serve as a roadmap to respond to this public health threat. Unlike most analytical papers, this research contributes actual findings and provides useful recommendations for disease management approaches and policies. Full article

(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)

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23 pages, 3829 KB

Open AccessArticle

Dissipativity Constraints in Zener-Type Time Dispersive Electromagnetic Materials of the Fractional Type

by Teodor M. Atanacković, Marko Janev, Milan Narandžić and Stevan Pilipović

Fractal Fract. 2025, 9(6), 342; https://doi.org/10.3390/fractalfract9060342 - 26 May 2025

Viewed by 685

Abstract

Thermodynamic constraints must be satisfied for the parameters of a constitutive relation, particularly for a model describing an electromagnetic (or any other) material with the intention of giving that model a physical meaning. We present sufficient conditions for the parameters of the constitutive [...] Read more.

Thermodynamic constraints must be satisfied for the parameters of a constitutive relation, particularly for a model describing an electromagnetic (or any other) material with the intention of giving that model a physical meaning. We present sufficient conditions for the parameters of the constitutive relation of an electromagnetic Zener-type fractional 2D and 3D anisotropic model so that a weak form of the thermodynamic (entropy) inequality is satisfied. Moreover, for such models, we analyze the corresponding thermodynamic constraints for field reconstruction and regularity in the 2D anisotropic case. This is carried out by the use of the matrix version of the Bochner theorem in the most general form, including generalized functions as elements of a matrix, which appear in that theorem. The given numerical results confirm the calculus presented in the paper. Full article

(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)

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16 pages, 402 KB

Open AccessArticle

A Simple Fractional Model with Unusual Dynamics in the Derivative Order

by Guillermo Fernández-Anaya, Francisco A. Godínez, Rogelio Valdés, Luis Alberto Quezada-Téllez and M. A. Polo-Labarrios

Fractal Fract. 2025, 9(4), 264; https://doi.org/10.3390/fractalfract9040264 - 21 Apr 2025

Cited by 1 | Viewed by 1351

Abstract

Fractional variable order systems with unusual dynamics in the order are a little-studied topic. In this study, we present three examples of very simple fractional systems with unusual dynamics in the derivative order. These cases involve different approaches to define the variable-order dynamics: [...] Read more.

Fractional variable order systems with unusual dynamics in the order are a little-studied topic. In this study, we present three examples of very simple fractional systems with unusual dynamics in the derivative order. These cases involve different approaches to define the variable-order dynamics: (1) an integer-order differential equation that includes the state variable, (2) a differential equation that incorporates the state variable and features both integer- and fractional-order derivatives, and (3) fractional variable-order differential equations nested in the derivative orders. We prove a result that shows how the extended recursion of the last case is generalized. These examples illustrate the richness that simple dynamical systems can reveal through the order of their derivatives. Full article

(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)

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27 pages, 1455 KB

Open AccessArticle

Neutral Delayed Fractional Models in Financial Time Series: Insights into Borsa Istanbul Sectors Affected by the Kahramanmaraş Earthquake

by Ömer Akgüller, Mehmet Ali Balcı, Larissa Margareta Batrancea, Dilara Altan Koç and Anca Nichita

Fractal Fract. 2025, 9(3), 141; https://doi.org/10.3390/fractalfract9030141 - 24 Feb 2025

Cited by 2 | Viewed by 1484

Abstract

This study examines the impact of the Kahramanmaraş Earthquake on four key sectors of Borsa Istanbul: Basic Metal, Insurance, Non-Metallic Mineral Products, and Wholesale and Retail Trade using neutral delayed fractional differential equations. Employing the Chebyshev collocation method, we numerically solved the neutral [...] Read more.

This study examines the impact of the Kahramanmaraş Earthquake on four key sectors of Borsa Istanbul: Basic Metal, Insurance, Non-Metallic Mineral Products, and Wholesale and Retail Trade using neutral delayed fractional differential equations. Employing the Chebyshev collocation method, we numerically solved the neutral delayed fractional differential equations with initial conditions scaled by each sector’s log difference standard deviation to accurately reflect market volatility. Fractional orders were derived from the Hurst exponent, and time delays were identified using average mutual information, autocorrelation function, and partial autocorrelation function methods. The results reveal significant changes post-earthquake, including reduced market persistence and increased volatility in the Basic Metal and Insurance sectors, contrasted by enhanced stability in the Non-Metallic Mineral Products sector. Neutral delayed fractional differential equations demonstrated superior performance over traditional models by effectively capturing memory and delay effects. This work underscores the efficacy of neutral delayed fractional differential equations in modeling financial resilience amid external shocks. Full article

(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)

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