Fractal and Fractional

13 pages, 1850 KiB

Open AccessArticle

Investigation of a Spatio-Temporal Fractal Fractional Coupled Hirota System

by Obaid J. Algahtani

Fractal Fract. 2024, 8(3), 178; https://doi.org/10.3390/fractalfract8030178 - 21 Mar 2024

Viewed by 682

This article aims to examine the nonlinear excitations in a coupled Hirota system described by the fractal fractional order derivative. By using the Laplace transform with Adomian decomposition (LADM), the numerical solution for the considered system is derived. It has been shown that [...] Read more.

This article aims to examine the nonlinear excitations in a coupled Hirota system described by the fractal fractional order derivative. By using the Laplace transform with Adomian decomposition (LADM), the numerical solution for the considered system is derived. It has been shown that the suggested technique offers a systematic and effective method to solve complex nonlinear systems. Employing the Banach contraction theorem, it is confirmed that the LADM leads to a convergent solution. The numerical analysis of the solutions demonstrates the confinement of the carrier wave and the presence of confined wave packets. The dispersion nonlinear parameter reduction equally influences the wave amplitude and spatial width. The localized internal oscillations in the solitary waves decreased the wave collapsing effect at comparatively small dispersion. Furthermore, it is also shown that the amplitude of the solitary wave solution increases by reducing the fractal derivative. It is evident that decreasing the order

α

modifies the nature of the solitary wave solutions and marginally decreases the amplitude. The numerical and approximation solutions correspond effectively for specific values of time (t). However, when the fractal or fractional derivative is set to one by increasing time, the wave amplitude increases. The absolute error analysis between the obtained series solutions and the accurate solutions are also presented. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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13 pages, 395 KiB

Open AccessArticle

The Soliton Solutions for Nonlocal Multi-Component Higher-Order Gerdjikov–Ivanov Equation via Riemann–Hilbert Problem

by Jinshan Liu, Huanhe Dong, Yong Fang and Yong Zhang

Fractal Fract. 2024, 8(3), 177; https://doi.org/10.3390/fractalfract8030177 - 19 Mar 2024

Viewed by 660

Abstract

The Lax pairs of the higher-order Gerdjikov–Ivanov (HOGI) equation are extended to the multi-component formula. Then, we first derive four different types of nonlocal group reductions to this new system. To construct the solution of these four nonlocal equations, we utilize the Riemann–Hilbert [...] Read more.

The Lax pairs of the higher-order Gerdjikov–Ivanov (HOGI) equation are extended to the multi-component formula. Then, we first derive four different types of nonlocal group reductions to this new system. To construct the solution of these four nonlocal equations, we utilize the Riemann–Hilbert method. Compared to the local HOGI equation, the solutions of nonlocal equations not only depend on the local spatial and time variables, but also the nonlocal variables. To exhibit the dynamic behavior, we consider the reverse-spacetime multi-component HOGI equation and its Riemann–Hilbert problem. When the Riemann–Hilbert problem is regular, the integral form solution can be given. Conversely, the exact solutions can be obtained explicitly. Finally, as concrete examples, the periodic solutions of the two-component nonlocal HOGI equation are given, which is different from the local equation. Full article

(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)

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17 pages, 5398 KiB

Open AccessArticle

High-Frequency Fractional Predictions and Spatial Distribution of the Magnetic Loss in a Grain-Oriented Magnetic Steel Lamination

by Benjamin Ducharne, Hamed Hamzehbahmani, Yanhui Gao, Patrick Fagan and Gael Sebald

Fractal Fract. 2024, 8(3), 176; https://doi.org/10.3390/fractalfract8030176 - 19 Mar 2024

Viewed by 803

Abstract

Grain-oriented silicon steel (GO FeSi) laminations are vital components for efficient energy conversion in electromagnetic devices. While traditionally optimized for power frequencies of 50/60 Hz, the pursuit of higher frequency operation (f ≥ 200 Hz) promises enhanced power density. This paper introduces [...] Read more.

Grain-oriented silicon steel (GO FeSi) laminations are vital components for efficient energy conversion in electromagnetic devices. While traditionally optimized for power frequencies of 50/60 Hz, the pursuit of higher frequency operation (f ≥ 200 Hz) promises enhanced power density. This paper introduces a model for estimating GO FeSi laminations’ magnetic behavior under these elevated operational frequencies. The proposed model combines the Maxwell diffusion equation and a material law derived from a fractional differential equation, capturing the viscoelastic characteristics of the magnetization process. Remarkably, the model’s dynamical contribution, characterized by only two parameters, achieves a notable 4.8% Euclidean relative distance error across the frequency spectrum from 50 Hz to 1 kHz. The paper’s initial section offers an exhaustive description of the model, featuring comprehensive comparisons between simulated and measured data. Subsequently, a methodology is presented for the localized segregation of magnetic losses into three conventional categories: hysteresis, classical, and excess, delineated across various tested frequencies. Further leveraging the model’s predictive capabilities, the study extends to investigating the very high-frequency regime, elucidating the spatial distribution of loss contributions. The application of proportional–iterative learning control facilitates the model’s adaptation to standard characterization conditions, employing sinusoidal imposed flux density. The paper deliberates on the implications of GO FeSi behavior under extreme operational conditions, offering insights and reflections essential for understanding and optimizing magnetic core performance in high-frequency applications. Full article

(This article belongs to the Special Issue Advances in Fractional Integral and Derivative Operators with Applications)

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15 pages, 2277 KiB

Open AccessArticle

Structural Characterization of Toxoplasma gondii Brain Cysts in a Model of Reactivated Toxoplasmosis Using Computational Image Analysis

by Neda Bauman, Jelena Srbljanović, Ivana Čolović Čalovski, Olivera Lijeskić, Vladimir Ćirković, Jelena Trajković, Branko Bobić, Andjelija Ž. Ilić and Tijana Štajner

Fractal Fract. 2024, 8(3), 175; https://doi.org/10.3390/fractalfract8030175 - 18 Mar 2024

Viewed by 845

Abstract

Toxoplasma gondii is an obligate intracellular parasite existing in three infectious life stages—tachyzoites, bradyzoites, and sporozoites. Rupture of tissue cysts and re-conversion of bradyzoites to tachyzoites leads to reactivated toxoplasmosis (RT) in an immunocompromised host. The aim of this study was to apply [...] Read more.

Toxoplasma gondii is an obligate intracellular parasite existing in three infectious life stages—tachyzoites, bradyzoites, and sporozoites. Rupture of tissue cysts and re-conversion of bradyzoites to tachyzoites leads to reactivated toxoplasmosis (RT) in an immunocompromised host. The aim of this study was to apply ImageJ software for analysis of T. gondii brain cysts obtained from a newly established in vivo model of RT. Mice chronically infected with T. gondii (BGD1 and BGD26 strains) were treated with cyclophosphamide and hydrocortisone (experimental group—EG) or left untreated as infection controls (ICs). RT in mice was confirmed by qPCR (PCR+); mice remaining chronically infected were PCR−. A total of 90 images of cysts were analyzed for fractal dimension (FD), lacunarity (L), diameter (D), circularity (C), and packing density (PD). Circularity was significantly higher in PCR+ compared to IC mice (p < 0.05 for BGD1, p < 0.001 for the BGD26 strain). A significant negative correlation between D and PD was observed only in IC for the BGD1 strain (ρ = −0.384, p = 0.048), while fractal parameters were stable. Significantly higher D, C, and PD and lower lacunarity, L, were noticed in the BGD1 compared to the more aggressive BGD26 strain. In conclusion, these results demonstrate the complexity of structural alterations of T. gondii cysts in an immunocompromised host and emphasize the application potential of ImageJ in the experimental models of toxoplasmosis. Full article

(This article belongs to the Section Life Science, Biophysics)

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17 pages, 18858 KiB

Open AccessArticle

Quasi-P-Wave Reverse Time Migration in TTI Media with a Generalized Fractional Convolution Stencil

by Shanyuan Qin, Jidong Yang, Ning Qin, Jianping Huang and Kun Tian

Fractal Fract. 2024, 8(3), 174; https://doi.org/10.3390/fractalfract8030174 - 18 Mar 2024

Viewed by 757

Abstract

In seismic modeling and reverse time migration (RTM), incorporating anisotropy is crucial for accurate wavefield modeling and high-quality images. Due to the trade-off between computational cost and simulation accuracy, the pure quasi-P-wave equation has good accuracy to describe wave propagation in tilted transverse [...] Read more.

In seismic modeling and reverse time migration (RTM), incorporating anisotropy is crucial for accurate wavefield modeling and high-quality images. Due to the trade-off between computational cost and simulation accuracy, the pure quasi-P-wave equation has good accuracy to describe wave propagation in tilted transverse isotropic (TTI) media. However, it involves a fractional pseudo-differential operator that depends on the anisotropy parameters, making it unsuitable for resolution using conventional solvers for fractional operators. To address this issue, we propose a novel pure quasi-P-wave equation with a generalized fractional convolution operator in TTI media. First, we decompose the conventional pure quasi-P-wave equation into an elliptical anisotropy equation and a fractional pseudo-differential correction term. Then, we use a generalized fractional convolution stencil to approximate the spatial-domain pseudo-differential term through the solution of an inverse problem. The proposed approximation method is accurate, and the wavefield modeling method based on it also accurately describes quasi-P-wave propagation in TTI media. Moreover, it only increases the computational cost for calculating mixed partial derivatives compared to those in vertical transverse isotropic (VTI) media. Finally, the proposed wavefield modeling method is utilized in RTM to correct the anisotropic effects in seismic imaging. Numerical RTM experiments demonstrate the flexibility and viability of the proposed method. Full article

(This article belongs to the Special Issue Fractional Equations and Calculation Methods in Exploration Seismology)

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23 pages, 437 KiB

Open AccessArticle

Monotone Positive Radial Solution of Double Index Logarithm Parabolic Equations

by Mengru Liu and Lihong Zhang

Fractal Fract. 2024, 8(3), 173; https://doi.org/10.3390/fractalfract8030173 - 16 Mar 2024

Viewed by 757

Abstract

This article mainly studies the double index logarithmic nonlinear fractional

g -

Laplacian parabolic equations with the Marchaud fractional time derivatives

\partial_{t}^{α}

. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double [...] Read more.

This article mainly studies the double index logarithmic nonlinear fractional

g -

Laplacian parabolic equations with the Marchaud fractional time derivatives

\partial_{t}^{α}

. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double non-locality of space-time and the nonlinearity of the fractional

g -

Laplacian, we establish the unbounded narrow domain principle, which provides a starting point for the moving plane method. Meanwhile, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of a non-local operator are established; then, these averaging effects are applied twice to ensure that the plane can be continuously moved toward infinity. Based on the above, the monotonicity of a positive solution for the above fractional

g -

Laplacian parabolic equations is studied. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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19 pages, 1872 KiB

Open AccessArticle

Fractional Fuzzy Neural System: Fractional Differential-Based Compensation Prediction for Reputation Infringement Cases

by Ni Zhang, Wu-Yang Zhu, Peng Jin, Guo Huang and Yi-Fei Pu

Fractal Fract. 2024, 8(3), 172; https://doi.org/10.3390/fractalfract8030172 - 16 Mar 2024

Viewed by 674

Abstract

With the rise of social media and the internet, the rapid dissemination of information has increased the likelihood of reputation infringement. This study utilizes judicial big data and AI to analyze intrinsic connections in reputation infringement cases, aiding judges in delivering consistent rulings. [...] Read more.

With the rise of social media and the internet, the rapid dissemination of information has increased the likelihood of reputation infringement. This study utilizes judicial big data and AI to analyze intrinsic connections in reputation infringement cases, aiding judges in delivering consistent rulings. The challenge lies in balancing freedom of speech with the right to reputation and addressing the ambiguity and subjectivity in infringement cases. This research constructs a structured reputation infringement case dataset from Chinese Judgments Online. It introduces a Fractional Fuzzy Neural System (FFNS) to tackle the vagueness in reputation infringement acts and judicial language, enhancing prediction accuracy for case outcomes. The FFNS, integrating fractional calculus, fuzzy logic, and neural networks, excels in adaptability and nonlinear modeling. It uses fractional order fuzzy membership functions to depict the extent and severity of reputation infringement accurately, combining these outputs with neural networks for predictive analysis. The result is a more precise adjudication tool, demonstrating significant potential for judicial application. Full article

(This article belongs to the Special Issue Fractional Calculus in Signal, Imaging Processing and Machine Learning)

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21 pages, 4049 KiB

Open AccessArticle

Identification of the Dynamic Trade Relationship between China and the United States Using the Quantile Grey Lotka–Volterra Model

by Zheng-Xin Wang, Yue-Ting Li and Ling-Fei Gao

Fractal Fract. 2024, 8(3), 171; https://doi.org/10.3390/fractalfract8030171 - 15 Mar 2024

Viewed by 801

Abstract

The quantile regression technique is introduced into the Lotka–Volterra ecosystem analysis framework. The quantile grey Lotka–Volterra model is established to reveal the dynamic trade relationship between China and the United States. An optimisation model is constructed to solve optimum quantile parameters. The empirical [...] Read more.

The quantile regression technique is introduced into the Lotka–Volterra ecosystem analysis framework. The quantile grey Lotka–Volterra model is established to reveal the dynamic trade relationship between China and the United States. An optimisation model is constructed to solve optimum quantile parameters. The empirical results show that the quantile grey Lotka–Volterra model shows higher fitting accuracy and reveals the trade relationships at different quantiles based on quarterly data on China–US trade from 1999 to 2019. The long-term China–US trade relationship presents a prominent predator–prey relationship because exports from China to the US inhibited China’s imports from the United States. Moreover, we divide samples into five stages according to four key events, China’s accession to the WTO, the 2008 global financial crisis, the weak global economic recovery in 2015, and the 2018 China–US trade war, recognising various characteristics at different stages. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)

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13 pages, 415 KiB

Open AccessArticle

Accelerated Gradient Descent Driven by Lévy Perturbations

by Yuquan Chen, Zhenlong Wu, Yixiang Lu, Yangquan Chen and Yong Wang

Fractal Fract. 2024, 8(3), 170; https://doi.org/10.3390/fractalfract8030170 - 14 Mar 2024

Viewed by 746

Abstract

In this paper, we mainly consider two kinds of perturbed accelerated gradient descents driven by Lévy perturbations, which is of great importance for enhancing the global search ability. By using Lévy representation, Lévy perturbations can be divided into two parts: small jumps and [...] Read more.

In this paper, we mainly consider two kinds of perturbed accelerated gradient descents driven by Lévy perturbations, which is of great importance for enhancing the global search ability. By using Lévy representation, Lévy perturbations can be divided into two parts: small jumps and large jumps, whose properties are then carefully discussed. By introducing the concept of attraction domain for local minima, Makovian transition properties are proven for the proposed two perturbed accelerated gradient descents with different infinitesimal matrices. Finally, all the results are extended to the vector case and two simulation examples are provided to validate all the conclusions. Full article

(This article belongs to the Special Issue Fractional Calculus in Signal, Imaging Processing and Machine Learning)

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10 pages, 406 KiB

Open AccessArticle

Utilizing a Fractional-Order Grey Model to Predict the Development Trends of China’s Electronic Commerce Service Industry

by Jianhong Guo, Che-Jung Chang and Yingyi Huang

Fractal Fract. 2024, 8(3), 169; https://doi.org/10.3390/fractalfract8030169 - 14 Mar 2024

Viewed by 760

Abstract

Electronic commerce plays a vital role in the digital age, and the creation of a good electronic commerce ecosystem is crucial to maintaining economic growth. The electronic commerce service industry is a leading indicator of electronic commerce development, and its possible changes imply [...] Read more.

Electronic commerce plays a vital role in the digital age, and the creation of a good electronic commerce ecosystem is crucial to maintaining economic growth. The electronic commerce service industry is a leading indicator of electronic commerce development, and its possible changes imply the future trends and innovation directions of the electronic commerce industry. An accurate grasp of the possible future revenue scale of the electronic commerce service industry can provide decision-making information for government policy formulation. Electronic commerce companies must formulate operational plans based on the latest information to determine strategic directions that are reasonable and consistent with the actual situation. Although there exist many prediction methods, they often fail to produce ideal results when the number of observations is insufficient. The fractional-order grey model is a common method used to deal with small data set prediction problems. This study therefore proposes a new modeling procedure for the fractional-order grey model to predict the revenue scale of China’s electronic commerce service industry. The results of experiments demonstrate that the proposed procedure can yield robust outputs under the condition of small data sets to reduce decision-making risks. Therefore, it can be regarded as a practical small data set analysis tool for managers. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)

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21 pages, 1350 KiB

Open AccessArticle

Convergence Analysis of Iterative Learning Control for Initialized Fractional Order Systems

by Xiaofeng Xu, Jiangang Lu and Jinshui Chen

Fractal Fract. 2024, 8(3), 168; https://doi.org/10.3390/fractalfract8030168 - 14 Mar 2024

Viewed by 719

Abstract

Iterative learning control is widely applied to address the tracking problem of dynamic systems. Although this strategy can be applied to fractional order systems, most existing studies neglected the impact of the system initialization on operation repeatability, which is a critical issue since [...] Read more.

Iterative learning control is widely applied to address the tracking problem of dynamic systems. Although this strategy can be applied to fractional order systems, most existing studies neglected the impact of the system initialization on operation repeatability, which is a critical issue since memory effect is inherent for fractional operators. In response to the above deficiencies, this paper derives robust convergence conditions for iterative learning control under non-repetitive initialization functions, where the bound of the final tracking error depends on the shift degree of the initialization function. Model nonlinearity, initial error, and channel noises are also discussed in the derivation. On this basis, a novel initialization learning strategy is proposed to obtain perfect tracking performance and desired initialization trajectory simultaneously, providing a new approach for fractional order system design. Finally, two numerical examples are presented to illustrate the theoretical results and their potential applications. Full article

(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems)

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13 pages, 12732 KiB

Open AccessArticle

The Classification and Evaluation of an Interlayer Shale Oil Reservoir Based on the Fractal Characteristics of Pore Systems: A Case Study in the HSN Area, China

by Changsheng Lu, Xixin Wang, Shuwei Ma, Shaohua Li, Ting Xue and Qiangqiang Li

Fractal Fract. 2024, 8(3), 167; https://doi.org/10.3390/fractalfract8030167 - 14 Mar 2024

Viewed by 748

Abstract

The evaluation of shale reservoir quality is of great significance for the exploration and development of shale oil. To more effectively study the distribution characteristics of shale reservoir quality, thin-section observation, scanning electron microscopy and pressure-controlled porosimetry were used to obtain the pore [...] Read more.

The evaluation of shale reservoir quality is of great significance for the exploration and development of shale oil. To more effectively study the distribution characteristics of shale reservoir quality, thin-section observation, scanning electron microscopy and pressure-controlled porosimetry were used to obtain the pore structure characteristics of shale in Chang 7, including pore types, pore size distribution, etc. In addition, the fractal dimensions of the shale samples were calculated based on pressure-controlled porosimetry data. The results show that residual interparticle pores, dissolution pores and clay-dominated pores were the main pore types. The overall pore size was mainly distributed between 3 nm and 50 μm. The pore system was divided into four types using fractal features, and the shale reservoir was divided into four types based on the proportion of different types of pore system. In different types of reservoirs, the production capacity of exploration wells varies significantly, as does the production capacity of horizontal wells. The classification of shale reservoirs using mercury intrusion fractal analysis proved to be suited for the efficient development of Chang 7 shale oil reservoirs. Full article

(This article belongs to the Special Issue Flow and Transport in Fractal Models of Rock Mechanics)

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19 pages, 342 KiB

Open AccessArticle

On a Faster Iterative Method for Solving Fractional Delay Differential Equations in Banach Spaces

by James Abah Ugboh, Joseph Oboyi, Mfon Okon Udo, Hossam A. Nabwey, Austine Efut Ofem and Ojen Kumar Narain

Fractal Fract. 2024, 8(3), 166; https://doi.org/10.3390/fractalfract8030166 - 14 Mar 2024

Viewed by 783

Abstract

In this paper, we consider a faster iterative method for approximating the fixed points of generalized

α

-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our [...] Read more.

In this paper, we consider a faster iterative method for approximating the fixed points of generalized

α

-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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17 pages, 2931 KiB

Open AccessArticle

New Results on

(r, k, μ)

-Riemann–Liouville Fractional Operators in Complex Domain with Applications

by Adel Salim Tayyah and Waggas Galib Atshan

Fractal Fract. 2024, 8(3), 165; https://doi.org/10.3390/fractalfract8030165 - 13 Mar 2024

Viewed by 751

Abstract

This paper introduces fractional operators in the complex domain as generalizations for the Srivastava–Owa operators. Some properties for the above operators are also provided. We discuss the convexity and starlikeness of the generalized Libera integral operator. A condition for the convexity and starlikeness [...] Read more.

This paper introduces fractional operators in the complex domain as generalizations for the Srivastava–Owa operators. Some properties for the above operators are also provided. We discuss the convexity and starlikeness of the generalized Libera integral operator. A condition for the convexity and starlikeness of the solutions of fractional differential equations is provided. Finally, a fractional differential equation is converted into an ordinary differential equation by wave transformation; illustrative examples are provided to clarify the solution within the complex domain. Full article

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

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13 pages, 323 KiB

Open AccessArticle

An α-Robust Galerkin Spectral Method for the Nonlinear Distributed-Order Time-Fractional Diffusion Equations with Initial Singularity

by Haiyu Liu and Shujuan Lü

Fractal Fract. 2024, 8(3), 164; https://doi.org/10.3390/fractalfract8030164 - 13 Mar 2024

Viewed by 746

Abstract

In this paper, we numerically solve the nonlinear time-fractional diffusion equation of distributed order on an unbounded domain with a weak singularity. A fully discrete implicit scheme is developed based on the L1 formula on graded meshes in time and the Galerkin spectral [...] Read more.

In this paper, we numerically solve the nonlinear time-fractional diffusion equation of distributed order on an unbounded domain with a weak singularity. A fully discrete implicit scheme is developed based on the L1 formula on graded meshes in time and the Galerkin spectral method using the Laguerre function in space. We obtained an

α

-robust discrete Gronwall inequality and the a priori error estimation of the numerical solution. Then, the existence and uniqueness of the numerical solution are discussed. Next, we present the

α

-robust stability and convergence of the fully discrete scheme, where the convergence was obtained based on the regularity conditions of the exact solution. A numerical example demonstrates the validity of the theoretical results. Full article

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

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12 pages, 3122 KiB

Open AccessArticle

Numerical Simulation of Soliton Propagation Behavior for the Fractional-in-Space NLSE with Variable Coefficients on Unbounded Domain

by Fengzhou Tian, Yulan Wang and Zhiyuan Li

Fractal Fract. 2024, 8(3), 163; https://doi.org/10.3390/fractalfract8030163 - 12 Mar 2024

Viewed by 972

Abstract

The soliton propagation of the fractional-in-space nonlinear Schrodinger equation (NLSE) is much more complicated than that of the corresponding integer NLSE. The aim of this paper is to discover some novel fractal soliton propagation behaviors (FSPBs) of this fractional-in-space NLSE. Firstly, the exact [...] Read more.

The soliton propagation of the fractional-in-space nonlinear Schrodinger equation (NLSE) is much more complicated than that of the corresponding integer NLSE. The aim of this paper is to discover some novel fractal soliton propagation behaviors (FSPBs) of this fractional-in-space NLSE. Firstly, the exact solution is compared with the present numerical solution, and the validity and accuracy of the present numerical method are verified. Secondly, the effect of fractional derivatives on soliton propagation is explored through the present numerical simulation results. At the same time, the present method is extended to the three-dimensional fractional-order NLSE. Finally, some novel FSPBs of the fractional-in-space NLSE are given. Full article

(This article belongs to the Topic AI and Computational Methods for Modelling, Simulations and Optimizing of Advanced Systems: Innovations in Complexity)

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27 pages, 2722 KiB

Open AccessArticle

A New Approach to Multiroot Vectorial Problems: Highly Efficient Parallel Computing Schemes

by Mudassir Shams, Naila Rafiq, Bruno Carpentieri and Nazir Ahmad Mir

Fractal Fract. 2024, 8(3), 162; https://doi.org/10.3390/fractalfract8030162 - 12 Mar 2024

Viewed by 910

Abstract

In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous [...] Read more.

In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous iterative algorithms are thoroughly examined. Using self-similar features, fractal-based simultaneous schemes can converge to solutions faster, saving computational time and resources necessary for solving nonlinear equations. Fractals analysis illustrates the newly developed method’s global convergence behavior when compared to single root-finding procedures for solving fractional order polynomials that arise in complex engineering applications. Some real problems from various branches of engineering along with some higher degree polynomials are considered as test examples to show the global convergence property of simultaneous methods, performance and efficiency of the proposed family of methods. Further computational efficiencies, CPU time and residual graphs are also drawn to validate the efficiency, robustness of the newly introduced family of methods as compared to the existing methods in the literature. Full article

(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)

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19 pages, 9522 KiB

Open AccessArticle

Analysis of Pore Characterization and Energy Evolution of Granite by Microwave Radiation

by Keping Zhou, Yifan Zhang, Chun Yang, Niange Yang and Zheng Pan

Fractal Fract. 2024, 8(3), 161; https://doi.org/10.3390/fractalfract8030161 - 12 Mar 2024

Viewed by 806

Abstract

To study the dynamic response of granite to different levels of microwave power, an intelligent microwave rock-breaking instrument is used to irradiate different power from three directions. The servo universal testing machine is used to carry out a uniaxial compression test on the [...] Read more.

To study the dynamic response of granite to different levels of microwave power, an intelligent microwave rock-breaking instrument is used to irradiate different power from three directions. The servo universal testing machine is used to carry out a uniaxial compression test on the granite after microwave damage to analyze the strength damage characteristics and the degree of pore damage. Pore fractal characteristics are analyzed based on nuclear magnetic resonance to establish the microwave damage degradation model. In parallel, the energy evolution process of granite under the influence of various power levels is analyzed using the theory of energy dissipation. Simultaneously, based on the energy dissipation theory, we analyze the energy evolution process of granite under the action of different powers. The results show that with higher microwave power, the peak strength and modulus of elasticity show a linear decreasing law. The degree of fragmentation is more obvious, showing the damage characteristics with two big ends and little in the middle. The higher the power, the greater the porosity and the more sensitive the micropore becomes to microwaves. Additionally, the damage degradation model established to evaluate the microwave damage of the rock showed that it was feasible. The higher the power, the lower the total energy, elastic energy, and dissipation energy, and the granite is gradually transformed from elastic deformation to plastic deformation. The elastic energy ratio decreases, the dissipation energy ratio increases, and the degree of damage becomes more and more serious. This study provides theoretical support for exploring the mechanical behavior and mechanism of microwave-assisted rock breaking and is of great practical significance. Full article

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

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15 pages, 1459 KiB

Open AccessArticle

Event-Triggered Adaptive Fuzzy Control for Strict-Feedback Nonlinear FOSs Subjected to Finite-Time Full-State Constraints

by Changhui Wang, Wencheng Li and Mei Liang

Fractal Fract. 2024, 8(3), 160; https://doi.org/10.3390/fractalfract8030160 - 12 Mar 2024

Viewed by 841

Abstract

In this article, an event-triggered adaptive fuzzy finite-time dynamic surface control (DSC) is presented for a class of strict-feedback nonlinear fractional-order systems (FOSs) with full-state constraints. The fuzzy logic systems (FLSs) are employed to approximate uncertain nonlinear functions in the backstepping process, the [...] Read more.

In this article, an event-triggered adaptive fuzzy finite-time dynamic surface control (DSC) is presented for a class of strict-feedback nonlinear fractional-order systems (FOSs) with full-state constraints. The fuzzy logic systems (FLSs) are employed to approximate uncertain nonlinear functions in the backstepping process, the dynamic surface method is applied to overcome the inherent computational complexity from the virtual controller and its fractional-order derivative, and the barrier Lyapunov function (BLF) is used to handle the full-state constraints. By introducing the finite-time stability criteria from fractional-order Lyapunov method, it is verified that the tracking error converges to a small neighborhood near the zero and the full-state constraints are satisfied within a predetermined finite time. Moreover, reducing the communication burden can be guaranteed without the occurrence of Zeno behavior, and the example is given to demonstrate the effectiveness of the proposed controller. Full article

(This article belongs to the Special Issue Fractional Order Controllers for Non-linear Systems)

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18 pages, 3066 KiB

Open AccessArticle

A Fractal-Based Quantitative Method for the Study of Fracture Evolution of Coal under Different Confining Pressures

by Ancheng Wang and Lei Wang

Fractal Fract. 2024, 8(3), 159; https://doi.org/10.3390/fractalfract8030159 - 11 Mar 2024

Viewed by 881

Abstract

To study the dynamic crack evolution process of loaded coal from the perspective of fractals, we carried out in situ industrial CT scanning tests of loaded coal under different confining pressures, visualizing loaded coal fracturing. Combined with fractal theory, the temporal and spatial [...] Read more.

To study the dynamic crack evolution process of loaded coal from the perspective of fractals, we carried out in situ industrial CT scanning tests of loaded coal under different confining pressures, visualizing loaded coal fracturing. Combined with fractal theory, the temporal and spatial evolution law of coal cracks is described quantitatively. The results provide two findings: (1) from the perspective of two-dimensional images and three-dimensional space, the evolution characteristics of cracks in coal under different confining pressures were basically the same in each loading stage. During the loading stages, the cracks exhibited a change rule of a slow reduction, initiation/development, rapid increase, expansion, and penetration. (2) The fractal dimension of coal was calculated by introducing fractal theory, and its change law was in good agreement with the dynamic changes of the cracks, which can explain the influence of the confining pressure on the loaded coal. The fractal dimension showed three stages: a slight decrease, a stable increase, and then a significant increase. The larger the confining pressure, the more obvious the limiting effect. Thus, our approach provides a more accurate method for evaluating the spatial and temporal evolution of cracks in loaded coal. This study can be used to predict the instability failure of loaded coal samples. Full article

(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)

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18 pages, 3891 KiB

Open AccessArticle

Fuzzy Mandelbric Set and Its Perturbations by Dynamical Noises

by Nikola Popović, Soley Ersoy, İbrahim İnce, Ana Savić and Vladimir Baltić

Fractal Fract. 2024, 8(3), 158; https://doi.org/10.3390/fractalfract8030158 - 11 Mar 2024

Viewed by 770

Abstract

In this paper, we introduce a membership function used to form the fuzzy Mandelbric set and investigate the structural effects of additive and multiplicative dynamic noises on it. The newly defined membership function of this fuzzy set and its perturbations is a generalization [...] Read more.

In this paper, we introduce a membership function used to form the fuzzy Mandelbric set and investigate the structural effects of additive and multiplicative dynamic noises on it. The newly defined membership function of this fuzzy set and its perturbations is a generalization of the indicator function for the classical Mandelbric set. We present an algorithm for detecting each complex number’s fuzzy membership degree. Through the use of the membership degrees of each complex number and experimental mathematics based on the visualizations of a variety of versions by utilizing computer-aided design, we gain a deep foresight for the structure characteristics of the additive and multiplicative perturbed fuzzy Mandelbric sets. Our novel approach allows us to identify the symmetry states of the Mandelbric set and its perturbations by the membership degrees of complex numbers, unlike the existing methods described in the literature. Full article

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19 pages, 11261 KiB

Open AccessArticle

Smooth and Efficient Path Planning for Car-like Mobile Robot Using Improved Ant Colony Optimization in Narrow and Large-Size Scenes

by Likun Li, Liyu Jiang, Wenzhang Tu, Liquan Jiang and Ruhan He

Fractal Fract. 2024, 8(3), 157; https://doi.org/10.3390/fractalfract8030157 - 10 Mar 2024

Viewed by 949

Abstract

Car-like mobile robots (CLMRs) are extensively utilized in various intricate scenarios owing to their exceptional maneuverability, stability, and adaptability, in which path planning is an important technical basis for their autonomous navigation. However, path planning methods are prone to inefficiently generate unsmooth paths [...] Read more.

Car-like mobile robots (CLMRs) are extensively utilized in various intricate scenarios owing to their exceptional maneuverability, stability, and adaptability, in which path planning is an important technical basis for their autonomous navigation. However, path planning methods are prone to inefficiently generate unsmooth paths in narrow and large-size scenes, especially considering the chassis model complexity of CLMRs with suspension. To this end, instead of traditional path planning based on an integer order model, this paper proposes fractional-order enhanced path planning using an improved Ant Colony Optimization (ACO) for CLMRs with suspension, which can obtain smooth and efficient paths in narrow and large-size scenes. On one hand, to improve the accuracy of the kinematic model construction of CLMRs with suspension, an accurate fractional-order-based kinematic modelling method is proposed, which considers the dynamic adjustment of the angle constraints. On the other hand, an improved ACO-based path planning method using fractional-order models is introduced by adopting a global multifactorial heuristic function with dynamic angle constraints, adaptive pheromone adjustment, and fractional-order state-transfer models, which avoids easily falling into a local optimum and unsmooth problem in a narrow space while increasing the search speed and success rate in large-scale scenes. Finally, the proposed method’s effectiveness is validated in both large-scale and narrow scenes, confirming its capability to handle various challenging scenarios. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Calculus in Robotics)

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18 pages, 353 KiB

Open AccessArticle

Variable-Order Fractional Linear Systems with Distributed Delays—Existence, Uniqueness and Integral Representation of the Solutions

by Hristo Kiskinov, Mariyan Milev, Milena Petkova and Andrey Zahariev

Fractal Fract. 2024, 8(3), 156; https://doi.org/10.3390/fractalfract8030156 - 10 Mar 2024

Viewed by 757

Abstract

In this work, we study a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. We propose an approach consisting of finding an associated one-parameter family of constant-order fractional systems, which is “almost” equivalent to the [...] Read more.

In this work, we study a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. We propose an approach consisting of finding an associated one-parameter family of constant-order fractional systems, which is “almost” equivalent to the considered variable-order system in an appropriate sense. This approach allows us to replace the study of the initial problem (IP) for variable-order fractional systems with the study of an IP for these one-parameter families of constant-order fractional systems. We prove that the initial problem for the variable-order fractional system with a discontinuous initial function possesses a unique continuous solution on the half-axis when the function describing the variable order of differentiation is locally bounded, Lebesgue integrable and has an appropriate decomposition similar to the Lebesgue decomposition of functions with bounded variation. The obtained results lead to the existence and uniqueness of a fundamental matrix for the studied variable-order fractional homogeneous system. As an application of the obtained results, we establish an integral representation of the solutions of the studied IP. Full article

(This article belongs to the Special Issue Advances in Variable-Order Fractional Calculus and Its Applications)

19 pages, 667 KiB

Open AccessArticle

Novel Controller Design for Finite-Time Synchronization of Fractional-Order Nonidentical Complex Dynamical Networks under Uncertain Parameters

by Xiliang He, Yu Wang, Tianzeng Li, Rong Kang and Yu Zhao

Fractal Fract. 2024, 8(3), 155; https://doi.org/10.3390/fractalfract8030155 - 10 Mar 2024

Cited by 1 | Viewed by 761

Abstract

The synchronization of complex networks, as an important and captivating dynamic phenomenon, has been investigated across diverse domains ranging from social activities to ecosystems and power systems. Furthermore, the synchronization of networks proves instrumental in solving engineering quandaries, such as cryptography and image [...] Read more.

The synchronization of complex networks, as an important and captivating dynamic phenomenon, has been investigated across diverse domains ranging from social activities to ecosystems and power systems. Furthermore, the synchronization of networks proves instrumental in solving engineering quandaries, such as cryptography and image encryption. And finite-time synchronization (FTS) controls exhibit substantial resistance to interference, accelerating network convergence speed and heightening control efficiency. In this paper, finite-time synchronization (FTS) is investigated for a class of fractional-order nonidentical complex networks under uncertain parameters (FONCNUPs). Firstly, some new FTS criteria for FONCNUPs are proposed based on Lyapunov theory and fractional calculus theory. Then, the new controller is designed based on inequality theory. Compared to the general controller, it controls all nodes and adds additional control to some of them. When compared to other controllers, it has lower control costs and higher efficiency. Finally, a numerical example is presented to validate the effectiveness and rationality of the obtained results. Full article

(This article belongs to the Special Issue Fractional Modelling, Analysis and Control for Power System)

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17 pages, 355 KiB

Open AccessArticle

Discrete Octonion Linear Canonical Transform: Definition and Properties

by Wen-Biao Gao

Fractal Fract. 2024, 8(3), 154; https://doi.org/10.3390/fractalfract8030154 - 08 Mar 2024

Viewed by 939

Abstract

In this paper, the discrete octonion linear canonical transform (DOCLCT) is defined. According to the definition of the DOCLCT, some properties associated with the DOCLCT are explored, such as linearity, scaling, boundedness, Plancherel theorem, inversion transform and shift transform. Then, the relationship between [...] Read more.

In this paper, the discrete octonion linear canonical transform (DOCLCT) is defined. According to the definition of the DOCLCT, some properties associated with the DOCLCT are explored, such as linearity, scaling, boundedness, Plancherel theorem, inversion transform and shift transform. Then, the relationship between the DOCLCT and the three-dimensional (3-D) discrete linear canonical transform (DLCT) is obtained. Moreover, based on a new convolution operator, we derive the convolution theorem of the DOCLCT. Finally, the correlation theorem of the DOCLCT is established. Full article

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18 pages, 6626 KiB

Open AccessArticle

Optically Transparent Dual-Band Metamaterial Absorber Using Ag Nanowire Screen-Printed Second-Order Cross-Fractal Structures

by Sumin Bark, Junghyeon Kim, Minjae Lee and Sungjoon Lim

Fractal Fract. 2024, 8(3), 153; https://doi.org/10.3390/fractalfract8030153 - 08 Mar 2024

Viewed by 936

Abstract

In this paper, we propose an optically transparent dual-band metamaterial absorber (MMA) that uses Ag nanowire screen-printed fractal structures. The proposed MMA exhibits near-perfect absorption in the C- and K-bands. This dual-band absorption property is achieved through two inductive–capacitive (L-C) resonances located at [...] Read more.

In this paper, we propose an optically transparent dual-band metamaterial absorber (MMA) that uses Ag nanowire screen-printed fractal structures. The proposed MMA exhibits near-perfect absorption in the C- and K-bands. This dual-band absorption property is achieved through two inductive–capacitive (L-C) resonances located at 6.45 and 21.14 GHz, which are generated by the second-order fractal structures. We analyzed the microwave absorbing mechanisms through the distributions of the surface current and electromagnetic field on the top and bottom layers. The MMA demonstrates an optical transmittance of 63.1% at a wavelength of 550 nm. This high optical transmittance is attained by screen printing transparent Ag nanowire ink onto a transparent PET substrate. Since screen printing is a simple and low-cost fabrication method, the proposed MMA offers the advantages of being low cost while having the properties of optical transparency and effective dual-band absorption. Consequently, it holds great potential for the radar stealth application of C- and K-bands in that it can be attached to the windows of stealth aircraft due to its optical transparency and dual-band near-perfect absorption property. Full article

(This article belongs to the Special Issue Advances in Fractal Antennas: Design, Modeling and Applications)

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20 pages, 4189 KiB

Open AccessArticle

Statistical Study of the Bias and Precision for Six Estimation Methods for the Fractal Dimension of Randomly Rough Surfaces

by Jorge Luis Flores Alarcón, Carlos Gabriel Figueroa, Víctor Hugo Jacobo, Fernando Velázquez Villegas and Rafael Schouwenaars

Fractal Fract. 2024, 8(3), 152; https://doi.org/10.3390/fractalfract8030152 - 07 Mar 2024

Viewed by 1027

Abstract

The simulation and characterisation of randomly rough surfaces is an important topic in surface science, tribology, geo- and planetary sciences, image analysis and optics. Extensions to general random processes with two continuous variables are straightforward. Several surface generation algorithms are available, and preference [...] Read more.

The simulation and characterisation of randomly rough surfaces is an important topic in surface science, tribology, geo- and planetary sciences, image analysis and optics. Extensions to general random processes with two continuous variables are straightforward. Several surface generation algorithms are available, and preference for one or another method often depends on the specific scientific field. The same holds for the methods to estimate the fractal dimension D. This work analyses six algorithms for the determination of D as a function of the size of the domain, variance, and the input value for D, using surfaces generated by Fourier filtering techniques and the random midpoint displacement algorithm. Several of the methods to determine fractal dimension are needlessly complex and severely biased, whereas simple and computationally efficient methods produce better results. A fine-tuned analysis of the power spectral density is very precise and shows how the different surface generation algorithms deviate from ideal fractal behaviour. For large datasets defined on equidistant two-dimensional grids, it is clearly the most sensitive and precise method to determine fractal dimension. Full article

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

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14 pages, 1510 KiB

Open AccessArticle

Convolution Kernel Function and Its Invariance Properties of Bone Fractal Operators

by Zhimo Jian, Gang Peng, Chaoqian Luo, Tianyi Zhou and Yajun Yin

Fractal Fract. 2024, 8(3), 151; https://doi.org/10.3390/fractalfract8030151 - 06 Mar 2024

Viewed by 813

Abstract

This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing [...] Read more.

This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators:

\sqrt{p + α^{2}}

-type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators. Full article

(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)

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18 pages, 6865 KiB

Open AccessArticle

A Class of Fifth-Order Chebyshev–Halley-Type Iterative Methods and Its Stability Analysis

by Xiaofeng Wang and Shaonan Guo

Fractal Fract. 2024, 8(3), 150; https://doi.org/10.3390/fractalfract8030150 - 06 Mar 2024

Viewed by 862

Abstract

In this paper, a family of fifth-order Chebyshev–Halley-type iterative methods with one parameter is presented. The convergence order of the new iterative method is analyzed. By obtaining rational operators associated with iterative methods, the stability of the iterative method is studied by using [...] Read more.

In this paper, a family of fifth-order Chebyshev–Halley-type iterative methods with one parameter is presented. The convergence order of the new iterative method is analyzed. By obtaining rational operators associated with iterative methods, the stability of the iterative method is studied by using fractal theory. In addition, some strange fixed points and critical points are obtained. By using the parameter space related to the critical points, some parameters with good stability are obtained. The dynamic plane corresponding to these parameters is plotted, visualizing the stability characteristics. Finally, the fractal diagrams of several iterative methods on different polynomials are compared. Both numerical results and fractal graphs show that the new iterative method has good convergence and stability when

α = \frac{1}{2}

. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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25 pages, 2779 KiB

Open AccessArticle

Hybrid LSTM-Based Fractional-Order Neural Network for Jeju Island’s Wind Farm Power Forecasting

by Bhukya Ramadevi, Venkata Ramana Kasi and Kishore Bingi

Fractal Fract. 2024, 8(3), 149; https://doi.org/10.3390/fractalfract8030149 - 05 Mar 2024

Cited by 1 | Viewed by 998

Abstract

Efficient integration of wind energy requires accurate wind power forecasting. This prediction is critical in optimising grid operation, energy trading, and effectively harnessing renewable resources. However, the wind’s complex and variable nature poses considerable challenges to achieving accurate forecasts. In this context, the [...] Read more.

Efficient integration of wind energy requires accurate wind power forecasting. This prediction is critical in optimising grid operation, energy trading, and effectively harnessing renewable resources. However, the wind’s complex and variable nature poses considerable challenges to achieving accurate forecasts. In this context, the accuracy of wind parameter forecasts, including wind speed and direction, is essential to enhancing the precision of wind power predictions. The presence of missing data in these parameters further complicates the forecasting process. These missing values could result from sensor malfunctions, communication issues, or other technical constraints. Addressing this issue is essential to ensuring the reliability of wind power predictions and the stability of the power grid. This paper proposes a long short-term memory (LSTM) model to forecast missing wind speed and direction data to tackle these issues. A fractional-order neural network (FONN) with a fractional arctan activation function is also developed to enhance generated wind power prediction. The predictive efficacy of the FONN model is demonstrated through two comprehensive case studies. In the first case, wind direction and forecast wind speed data are used, while in the second case, wind speed and forecast wind direction data are used for predicting power. The proposed hybrid neural network model improves wind power forecasting accuracy and addresses data gaps. The model’s performance is measured using mean errors and R² values. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Calculus in Robotics)

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Fractal Fract., Volume 8, Issue 3 (March 2024) – 55 articles

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