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17 pages, 719 KiB

Open AccessArticle

Difference Approximation for 2D Time-Fractional Integro-Differential Equation with Given Initial and Boundary Conditions

by Xindong Zhang, Ziyang Luo, Quan Tang, Leilei Wei and Juan Liu

Fractal Fract. 2024, 8(8), 495; https://doi.org/10.3390/fractalfract8080495 - 22 Aug 2024

Viewed by 394

Abstract

In this investigation, a new algorithm based on the compact difference method is proposed. The purpose of this investigation is to solve the 2D time-fractional integro-differential equation. The Riemann–Liouville derivative was utilized to define the time-fractional derivative. Meanwhile, the weighted and shifted Grünwald [...] Read more.

In this investigation, a new algorithm based on the compact difference method is proposed. The purpose of this investigation is to solve the 2D time-fractional integro-differential equation. The Riemann–Liouville derivative was utilized to define the time-fractional derivative. Meanwhile, the weighted and shifted Grünwald difference operator and product trapezoidal formula were utilized to construct a high-order numerical scheme. Also, we analyzed the stability and convergence. The convergence order was

O (τ^{2} + h_{x}^{4} + h_{y}^{4})

, where

τ

is the time step size,

h_{x}

and

h_{y}

are the spatial step sizes. Furthermore, several examples were provided to verify the correctness of our theoretical reasoning. Full article

(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)

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23 pages, 1728 KiB

Open AccessArticle

Fractional Fourier Series on the Torus and Applications

by Chen Wang, Xianming Hou, Qingyan Wu, Pei Dang and Zunwei Fu

Fractal Fract. 2024, 8(8), 494; https://doi.org/10.3390/fractalfract8080494 - 21 Aug 2024

Viewed by 342

Abstract

This paper introduces the fractional Fourier series on the fractional torus and proceeds to investigate several fundamental aspects. Our study includes topics such as fractional convolution, fractional approximation, fractional Fourier inversion, and the Poisson summation formula. We also explore the relationship between the [...] Read more.

This paper introduces the fractional Fourier series on the fractional torus and proceeds to investigate several fundamental aspects. Our study includes topics such as fractional convolution, fractional approximation, fractional Fourier inversion, and the Poisson summation formula. We also explore the relationship between the decay of its fractional Fourier coefficients and the smoothness of a function. Additionally, we establish the convergence of the fractional Féjer means and Bochner–Riesz means. Finally, we demonstrate the practical applications of the fractional Fourier series, particularly in the context of solving fractional partial differential equations with periodic boundary conditions, and showcase the utility of approximation methods on the fractional torus for recovering non-stationary signals. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications, 2nd Edition)

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17 pages, 588 KiB

Open AccessArticle

Fractals of Interpolative Kannan Mappings

by Xiangting Shi, Umar Ishtiaq, Muhammad Din and Mohammad Akram

Fractal Fract. 2024, 8(8), 493; https://doi.org/10.3390/fractalfract8080493 - 21 Aug 2024

Viewed by 363

Abstract

In 2018, Erdal Karapinar introduced the concept of interpolative Kannan operators, a novel adaptation of the Kannan mapping originally defined in 1969 by Kannan. This new mapping condition is more lenient than the basic contraction condition. In this paper, we study the concept [...] Read more.

In 2018, Erdal Karapinar introduced the concept of interpolative Kannan operators, a novel adaptation of the Kannan mapping originally defined in 1969 by Kannan. This new mapping condition is more lenient than the basic contraction condition. In this paper, we study the concept by introducing the IKC-iterated function/multi-function system using interpolative Kannan operators, including a broader area of mappings. Moreover, we establish the Collage Theorem endowed with the iterated function system (IFS) based on the IKC, and show the well-posedness of the IKC-IFS. Interpolative Kannan contractions are meaningful due to their applications in fractals, offering a more versatile framework for creating intricate geometric structures with potentially fewer constraints compared to classical approaches. Full article

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20 pages, 3351 KiB

Open AccessArticle

A Delayed Fractional-Order Predator–Prey Model with Three-Stage Structure and Cannibalism for Prey

by Hui Zhang and Ahmadjan Muhammadhaji

Fractal Fract. 2024, 8(8), 492; https://doi.org/10.3390/fractalfract8080492 - 21 Aug 2024

Viewed by 314

Abstract

In this study, we investigate a delayed fractional-order predator–prey model with a stage structure and cannibalism. The model is characterized by a three-stage structure of the prey population and incorporates cannibalistic interactions. Our main objective is to analyze the existence, uniqueness, boundedness, and [...] Read more.

In this study, we investigate a delayed fractional-order predator–prey model with a stage structure and cannibalism. The model is characterized by a three-stage structure of the prey population and incorporates cannibalistic interactions. Our main objective is to analyze the existence, uniqueness, boundedness, and local stability of the equilibrium points of the proposed system. In addition, we investigate the Hopf bifurcation of the system, taking the digestion delay of the predator as the branch parameter, and clarify the necessary conditions for the existence of the Hopf bifurcation. To confirm our theoretical analysis, we provide a numerical example to validate the accuracy of our research results. In the conclusion section, we carefully review the results of the numerical simulation and propose directions for future research. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing, 2nd Edition)

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15 pages, 4863 KiB

Open AccessArticle

Enhanced Thermal and Mass Diffusion in Maxwell Nanofluid: A Fractional Brownian Motion Model

by Ming Shen, Yihong Liu, Qingan Yin, Hongmei Zhang and Hui Chen

Fractal Fract. 2024, 8(8), 491; https://doi.org/10.3390/fractalfract8080491 - 21 Aug 2024

Viewed by 367

Abstract

This paper introduces fractional Brownian motion into the study of Maxwell nanofluids over a stretching surface. Nonlinear coupled spatial fractional-order energy and mass equations are established and solved numerically by the finite difference method with Newton’s iterative technique. The quantities of physical interest [...] Read more.

This paper introduces fractional Brownian motion into the study of Maxwell nanofluids over a stretching surface. Nonlinear coupled spatial fractional-order energy and mass equations are established and solved numerically by the finite difference method with Newton’s iterative technique. The quantities of physical interest are graphically presented and discussed in detail. It is found that the modified model with fractional Brownian motion is more capable of explaining the thermal conductivity enhancement. The results indicate that a reduction in the fractional parameter leads to thinner thermal and concentration boundary layers, accompanied by higher local Nusselt and Sherwood numbers. Consequently, the introduction of a fractional Brownian model not only enriches our comprehension of the thermal conductivity enhancement phenomenon but also amplifies the efficacy of heat and mass transfer within Maxwell nanofluids. This achievement demonstrates practical application potential in optimizing the efficiency of fluid heating and cooling processes, underscoring its importance in the realm of thermal management and energy conservation. Full article

(This article belongs to the Section Mathematical Physics)

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11 pages, 1749 KiB

Open AccessArticle

Artificial Intelligence in Chromatin Analysis: A Random Forest Model Enhanced by Fractal and Wavelet Features

by Igor Pantic and Jovana Paunovic Pantic

Fractal Fract. 2024, 8(8), 490; https://doi.org/10.3390/fractalfract8080490 - 21 Aug 2024

Viewed by 403

Abstract

In this study, we propose an innovative concept that applies an AI-based approach using the random forest algorithm integrated with fractal and discrete wavelet transform features of nuclear chromatin. This strategy could be employed to identify subtle structural changes in cells that are [...] Read more.

In this study, we propose an innovative concept that applies an AI-based approach using the random forest algorithm integrated with fractal and discrete wavelet transform features of nuclear chromatin. This strategy could be employed to identify subtle structural changes in cells that are in the early stages of programmed cell death. The code for the random forest model is developed using the Scikit-learn library in Python and includes hyperparameter tuning and cross-validation to optimize performance. The suggested input data for the model are chromatin fractal dimension, fractal lacunarity, and three wavelet coefficient energies obtained through high-pass and low-pass filtering. Additionally, the code contains several methods to assess the performance metrics of the model. This model holds potential as a starting point for designing simple yet advanced AI biosensors capable of detecting apoptotic cells that are not discernible through conventional microscopy techniques. Full article

(This article belongs to the Special Issue Fractals in Biophysics and Their Applications)

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21 pages, 7954 KiB

Open AccessArticle

A Pareto-Optimal-Based Fractional-Order Admittance Control Method for Robot Precision Polishing

by Haotian Wu, Jianzhong Yang, Si Huang and Xiao Ning

Fractal Fract. 2024, 8(8), 489; https://doi.org/10.3390/fractalfract8080489 - 20 Aug 2024

Viewed by 304

Abstract

Traditional integer-order admittance control is widely used in industrial scenarios requiring force control, but integer-order models often struggle to accurately depict fractional-order-controlled objects, leading to precision bottlenecks in the field of precision machining. For robotic precision polishing scenarios, to enhance the stability of [...] Read more.

Traditional integer-order admittance control is widely used in industrial scenarios requiring force control, but integer-order models often struggle to accurately depict fractional-order-controlled objects, leading to precision bottlenecks in the field of precision machining. For robotic precision polishing scenarios, to enhance the stability of the control process, we propose a more physically accurate five-parameter fractional-order admittance control model. To reduce contact impact, we introduce a method combining the rear fastest tracking differential with fractional-order admittance control. The optimal parameter identification for the fractional-order system is completed through Pareto optimality and a time–frequency domain fusion analysis of the control system. We completed the optimal parameter identification in a simulation, which is applied to the robotic precision polishing scenario. This method significantly enhanced the force control precision, reducing the error margin from 15% to 5%. Full article

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21 pages, 7691 KiB

Open AccessArticle

Research on Efficiency and Multifractality of Gold Market under Major Events

by Feifei Wang, Jiaxin Chang, Weizhen Zuo and Weijie Zhou

Fractal Fract. 2024, 8(8), 488; https://doi.org/10.3390/fractalfract8080488 - 20 Aug 2024

Viewed by 479

Abstract

(1) Background: As a safe-haven asset, gold attracts a large number of investors during major events. Uncertainties caused by global economic and market conditions lead to increased purchases of gold by investors, which in turn affects the efficiency of the gold market. (2) [...] Read more.

(1) Background: As a safe-haven asset, gold attracts a large number of investors during major events. Uncertainties caused by global economic and market conditions lead to increased purchases of gold by investors, which in turn affects the efficiency of the gold market. (2) Methods: This article focuses on Shanghai gold and explores the potential impacts of different major events on the effectiveness and risk of China’s gold market using the Multifractal Detrended Fluctuation Analysis (MFDFA) method. (3) Results: the Chinese gold spot market is anti-persistence and exhibits significant multifractal characteristics, suggesting that the gold spot market possesses predictability and has significant volatility and high risk. Furthermore, the study conducts stage analysis based on different major events, and the results are as follows. The empirical results show that the gold market exhibits anti-persistence and multifractality features for three major events, i.e., the US–China trade war, the COVID-19 pandemic, and the Russia–Ukraine War. (4) Conclusions: for the COVID-19 pandemic, the intensity of anti-persistence is the highest. In addition, it is also found that the stronger the anti-persistence in the gold markets under a given event, the greater the corresponding risk. Finally, the article provides relevant decision-making suggestions for investors and risk managers. Full article

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32 pages, 17354 KiB

Open AccessArticle

Logging Evaluation of Irreducible Water Saturation: Fractal Theory and Data-Driven Approach—Case Study of Complex Porous Carbonate Reservoirs in Mishrif Formation

by Jianhong Guo, Zhansong Zhang, Xin Nie, Qing Zhao and Hengyang Lv

Fractal Fract. 2024, 8(8), 487; https://doi.org/10.3390/fractalfract8080487 - 19 Aug 2024

Viewed by 596

Abstract

Evaluating irreducible water saturation is crucial for estimating reservoir capacity and developing effective extraction strategies. Traditional methods for predicting irreducible water saturation are limited by their reliance on specific logging data, which affects accuracy and applicability. This study introduces a predictive method based [...] Read more.

Evaluating irreducible water saturation is crucial for estimating reservoir capacity and developing effective extraction strategies. Traditional methods for predicting irreducible water saturation are limited by their reliance on specific logging data, which affects accuracy and applicability. This study introduces a predictive method based on fractal theory and deep learning for assessing irreducible water saturation in complex carbonate reservoirs. Utilizing the Mishrif Formation of the Halfaya oilfield as a case study, a new evaluation model was developed using the nuclear magnetic resonance (NMR) fractal permeability model and validated with surface NMR and mercury injection capillary pressure (MICP) data. The relationship between the logarithm mean of the transverse relaxation time (T_2lm) and physical properties was explored through fractal theory and the Thomeer Function. This relationship was integrated with conventional logging curves and an advanced deep learning algorithm to construct a T_2lm prediction model, offering a robust data foundation for irreducible water saturation evaluation. The results show that the new method is applicable to wells with and without specialized NMR logging data. For the Mishrif Formation, the predicted irreducible water saturation achieved a coefficient of determination of 0.943 compared to core results, with a mean absolute error of 2.37% and a mean relative error of 8.46%. Despite introducing additional errors with inverted T_2lm curves, it remains within acceptable limits. Compared to traditional methods, this approach provides enhanced predictive accuracy and broader applicability. Full article

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

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29 pages, 368 KiB

Open AccessArticle

On Hybrid and Non-Hybrid Discrete Fractional Difference Inclusion Problems for the Elastic Beam Equation

by Faycal Alili, Abdelkader Amara, Khaled Zennir and Taha Radwan

Fractal Fract. 2024, 8(8), 486; https://doi.org/10.3390/fractalfract8080486 - 19 Aug 2024

Viewed by 375

Abstract

The results in this paper are related to the existence of solutions to hybrid and non-hybrid discrete fractional three-point boundary value inclusion problems for the elastic beam equation. The development of our results is attributed to the use of the Caputo and difference [...] Read more.

The results in this paper are related to the existence of solutions to hybrid and non-hybrid discrete fractional three-point boundary value inclusion problems for the elastic beam equation. The development of our results is attributed to the use of the Caputo and difference operators. The existence results for the non-hybrid discrete fractional inclusion problem are established by using fixed point theory for multi-valued upper semi-continuous maps, and the case of the hybrid discrete fractional inclusion problem is treated by Dhage’s fixed point theory. Additionally, we present two examples to illustrate our main results. Full article

(This article belongs to the Special Issue Fractal and Multifractal Analysis in Econometric Models and Empirical Finance)

18 pages, 5746 KiB

Open AccessArticle

Remaining Useful Life Prediction for Power Storage Electronic Components Based on Fractional Weibull Process and Shock Poisson Model

by Wanqing Song, Xianhua Yang, Wujin Deng, Piercarlo Cattani and Francesco Villecco

Fractal Fract. 2024, 8(8), 485; https://doi.org/10.3390/fractalfract8080485 - 19 Aug 2024

Viewed by 384

Abstract

For lithium-ion batteries and supercapacitors in hybrid power storage facilities, both steady degradation and random shock contribute to their failure. To this end, in this paper, we propose to introduce the degradation-threshold-shock (DTS) model for their remaining useful life (RUL) prediction. Non-homogeneous compound [...] Read more.

For lithium-ion batteries and supercapacitors in hybrid power storage facilities, both steady degradation and random shock contribute to their failure. To this end, in this paper, we propose to introduce the degradation-threshold-shock (DTS) model for their remaining useful life (RUL) prediction. Non-homogeneous compound Poisson process (NHCP) is proposed to simulate the shock effect in the DTS model. Considering the long-range dependence and heavy-tailed characteristics of the degradation process, fractional Weibull process (fWp) is employed in the diffusion term of the stochastic degradation model. Furthermore, the drift and diffusion coefficients are constantly updated to describe the environmental interference. Prior to the model training, steady degradation and shock data must be separated, based on the three-sigma principle. Degradation data for the lithium-ion batteries (LIBs) and ultracapacitors are employed for model verification under different operation protocols in the power system. Recent deep learning models and stochastic process-based methods are utilized for model comparison, and the proposed model shows higher prediction accuracy. Full article

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

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12 pages, 918 KiB

Open AccessArticle

Wind Turbine Blade Fault Diagnosis: Approximate Entropy as a Tool to Detect Erosion and Mass Imbalance

by Salim Lahmiri

Fractal Fract. 2024, 8(8), 484; https://doi.org/10.3390/fractalfract8080484 - 19 Aug 2024

Viewed by 354

Abstract

Wind energy is a clean, sustainable, and renewable source. It is receiving a large amount of attention from governments and energy companies worldwide as it plays a significant role as an alternative source of energy in reducing carbon emissions. However, due to long-term [...] Read more.

Wind energy is a clean, sustainable, and renewable source. It is receiving a large amount of attention from governments and energy companies worldwide as it plays a significant role as an alternative source of energy in reducing carbon emissions. However, due to long-term operation in reduced and difficult weather conditions, wind turbine blades are always seriously damaged. Hence, damage detection in blade structure is essential to evaluate its operational condition and ensure its structural integrity and safety. We aim to use fractal, entropy, and chaos concepts as descriptors for the diagnosis of wind turbine blade condition. They are, respectively, estimated by the correlation dimension, approximate entropy, and the Lyapunov exponent. Formal statistical tests are performed to check how they are different across wind turbine blade conditions. The experimental results follow. First, the correlation dimension is not able to distinguish between all conditions of wind turbine blades. Second, approximate entropy is suitable to distinguish between healthy and erosion conditions and between healthy and mass imbalance conditions. Third, chaos is not a discriminative feature to distinguish between wind turbine blade conditions. Fourth, wind turbine blades with either erosion or mass imbalance exhibit less irregularity in their respective signals than healthy wind turbine blades. Full article

(This article belongs to the Special Issue Fractal Dimension and Fractional Calculus in Mechanical Signal Processing)

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22 pages, 8850 KiB

Open AccessArticle

Analysis of Fractal Properties of Atmospheric Turbulence and the Practical Applications

by Zihan Liu, Hongsheng Zhang, Zuntao Fu, Xuhui Cai and Yu Song

Fractal Fract. 2024, 8(8), 483; https://doi.org/10.3390/fractalfract8080483 - 19 Aug 2024

Viewed by 437

Abstract

Atmospheric turbulence, recognized as a quintessential space–time chaotic system, can be characterized by its fractal properties. The characteristics of the time series of multiple orders of fractal dimensions, together with their relationships with stability parameters, are examined using the data from an observational [...] Read more.

Atmospheric turbulence, recognized as a quintessential space–time chaotic system, can be characterized by its fractal properties. The characteristics of the time series of multiple orders of fractal dimensions, together with their relationships with stability parameters, are examined using the data from an observational station in Horqin Sandy Land to explore how the diurnal variation, synoptic process, and stratification conditions can affect the fractal characteristics. The findings reveal that different stratification conditions can disrupt the quasi-three-dimensional state of atmospheric turbulence in different manners within different scales of motion. Two aspects of practical applications of fractal dimensions are explored. Firstly, fractal properties can be employed to refine similarity relationships, thereby offering prospects for revealing more information and expanding the scope of application of similarity theories. Secondly, utilizing different orders of fractal dimensions, a systematic algorithm is developed. This algorithm distinguishes and eliminates non-turbulent motions from observational data, which are shown to exhibit slow-changing features and result in a universal overestimation of turbulent fluxes. This overestimation correlates positively with the boundary frequency between turbulent and non-turbulent motions. The evaluation of these two aspects of applications confirms that fractal properties hold promise for practical studies on atmospheric turbulence. Full article

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13 pages, 4488 KiB

Open AccessArticle

Application of Fractional Calculus in Predicting the Temperature-Dependent Creep Behavior of Concrete

by Jiecheng Chen, Lingwei Gong and Ruifan Meng

Fractal Fract. 2024, 8(8), 482; https://doi.org/10.3390/fractalfract8080482 - 18 Aug 2024

Viewed by 336

Abstract

Creep is an essential aspect of the durability and longevity of concrete structures. Based on fractional-order viscoelastic theory, this study investigated a creep model for predicting the temperature-dependent creep behavior of concrete. The order of the proposed fractional-order creep model can intuitively reflect [...] Read more.

Creep is an essential aspect of the durability and longevity of concrete structures. Based on fractional-order viscoelastic theory, this study investigated a creep model for predicting the temperature-dependent creep behavior of concrete. The order of the proposed fractional-order creep model can intuitively reflect the evolution of the material characteristics between solids and fluids, which provides a quantitative way to directly reveal the influence of loading conditions on the temperature-dependent mechanical properties of concrete during creep. The effectiveness of the model was verified using the experimental data of lightweight expansive shale concrete under various temperature and stress conditions, and the comparison of the results with those of the model in the literature showed that the proposed model has good accuracy while maintaining simplicity. Further analysis of the fractional order showed that temperature, not stress level, is the key factor affecting the creep process of concrete. At the same temperature, the fractional order is almost a fixed value and increases with the increase in temperature, reflecting the gradual softening of the mechanical properties of concrete at higher temperature. Finally, a novel prediction formula containing the average fractional-order value at each temperature was established, and the creep deformation of concrete can be predicted only by changing the applied stress, which provides a simple and practical method for predicting the temperature-dependent creep behavior of concrete. Full article

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

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21 pages, 866 KiB

Open AccessArticle

Feedback Control Design Strategy for Stabilization of Delayed Descriptor Fractional Neutral Systems with Order 0 < ϱ < 1 in the Presence of Time-Varying Parametric Uncertainty

by Zahra Sadat Aghayan, Alireza Alfi, Seyed Mehdi Abedi Pahnehkolaei and António M. Lopes

Fractal Fract. 2024, 8(8), 481; https://doi.org/10.3390/fractalfract8080481 - 17 Aug 2024

Viewed by 359

Abstract

Descriptor systems are more complex than normal systems, which are modeled by differential equations. This paper derives stability and stabilization criteria for uncertain fractional descriptor systems with neutral-type delay. Through the Lyapunov–Krasovskii functional approach, conditions subject to time-varying delay and parametric uncertainty are [...] Read more.

Descriptor systems are more complex than normal systems, which are modeled by differential equations. This paper derives stability and stabilization criteria for uncertain fractional descriptor systems with neutral-type delay. Through the Lyapunov–Krasovskii functional approach, conditions subject to time-varying delay and parametric uncertainty are formulated as linear matrix inequalities. Based on the established criteria, static state- and output-feedback control laws are designed to ensure regularity and impulse-free properties, together with robust stability of the closed-loop system under permissible uncertainties. Numerical examples illustrate the effectiveness of the control methods and show that the results depend on the range of variation in the delays and on the fractional order, leading to stability analysis results that are less conservative than those reported in the literature. Full article

(This article belongs to the Special Issue Delayed and Stochastic Fractional Order Dynamical Systems: Theory, Numerical Methods, Control, and Applications)

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15 pages, 1786 KiB

Open AccessArticle

Numerical Analysis and Computation of the Finite Volume Element Method for the Nonlinear Coupled Time-Fractional Schrödinger Equations

by Xinyue Zhao, Yining Yang, Hong Li, Zhichao Fang and Yang Liu

Fractal Fract. 2024, 8(8), 480; https://doi.org/10.3390/fractalfract8080480 - 17 Aug 2024

Viewed by 324

Abstract

In this article, our aim is to consider an efficient finite volume element method combined with the

L 2 - 1_{σ}

formula for solving the coupled Schrödinger equations with nonlinear terms and time-fractional derivative terms. We design the fully discrete scheme, where [...] Read more.

In this article, our aim is to consider an efficient finite volume element method combined with the

L 2 - 1_{σ}

formula for solving the coupled Schrödinger equations with nonlinear terms and time-fractional derivative terms. We design the fully discrete scheme, where the space direction is approximated using the finite volume element method and the time direction is discretized making use of the

L 2 - 1_{σ}

formula. We then prove the stability for the fully discrete scheme, and derive the optimal convergence result, from which one can see that our scheme has second-order accuracy in both the temporal and spatial directions. We carry out numerical experiments with different examples to verify the optimal convergence result. Full article

(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)

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30 pages, 2530 KiB

Open AccessArticle

Stationary Responses of Seven Classes of Fractional Vibrations Driven by Sinusoidal Force

by Ming Li

Fractal Fract. 2024, 8(8), 479; https://doi.org/10.3390/fractalfract8080479 - 16 Aug 2024

Viewed by 370

Abstract

This paper gives the contributions in three folds. First, we propose fractional phasor motion equations of seven classes of fractional vibrators. Second, we put forward fractional phasor responses to seven classes of fractional vibrators. Third, we bring forward the analytical expressions of stationary [...] Read more.

This paper gives the contributions in three folds. First, we propose fractional phasor motion equations of seven classes of fractional vibrators. Second, we put forward fractional phasor responses to seven classes of fractional vibrators. Third, we bring forward the analytical expressions of stationary responses in time to seven classes of fractional vibration systems driven by sinusoidal force using elementary functions. The present results show that there are obvious effects of fractional orders on the sinusoidal stationary responses to fractional vibrations. Full article

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

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22 pages, 2349 KiB

Open AccessArticle

Valuation of Currency Option Based on Uncertain Fractional Differential Equation

by Weiwei Wang, Dan A. Ralescu and Xiaojuan Xue

Fractal Fract. 2024, 8(8), 478; https://doi.org/10.3390/fractalfract8080478 - 16 Aug 2024

Viewed by 379

Abstract

Uncertain fractional differential equations (UFDEs) are excellent tools for describing complicated dynamic systems. This study analyzes the valuation problems of currency options based on UFDE under the optimistic value criterion. Firstly, a new uncertain fractional currency model is formulated to describe the dynamics [...] Read more.

Uncertain fractional differential equations (UFDEs) are excellent tools for describing complicated dynamic systems. This study analyzes the valuation problems of currency options based on UFDE under the optimistic value criterion. Firstly, a new uncertain fractional currency model is formulated to describe the dynamics of the foreign exchange rate. Then, the pricing formulae of European, American, and Asian currency options are obtained under the optimistic value criterion. Numerical simulations are performed to discuss the properties of the option prices with respect to some parameters. Finally, a real-world example is provided to show that the uncertain fractional currency model is superior to the classical stochastic model. Full article

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20 pages, 5790 KiB

Open AccessArticle

FOMCON Toolbox-Based Direct Approximation of Fractional Order Systems Using Gaze Cues Learning-Based Grey Wolf Optimizer

by Bala Bhaskar Duddeti, Asim Kumar Naskar, Veerpratap Meena, Jitendra Bahadur, Pavan Kumar Meena and Ibrahim A. Hameed

Fractal Fract. 2024, 8(8), 477; https://doi.org/10.3390/fractalfract8080477 - 15 Aug 2024

Viewed by 507

Abstract

This study discusses a new method for the fractional-order system reduction. It offers an adaptable framework for approximating various fractional-order systems (FOSs), including commensurate and non-commensurate. The fractional-order modeling and control (FOMCON) toolbox in MATLAB and the gaze cues learning-based grey wolf optimizer [...] Read more.

This study discusses a new method for the fractional-order system reduction. It offers an adaptable framework for approximating various fractional-order systems (FOSs), including commensurate and non-commensurate. The fractional-order modeling and control (FOMCON) toolbox in MATLAB and the gaze cues learning-based grey wolf optimizer (GGWO) technique form the basis of the recommended method. The fundamental advantage of the offered method is that it does not need intermediate steps, a mathematical substitution, or an operator-based approximation for the order reduction of a commensurate and non-commensurate FOS. The cost function is set up so that the sum of the integral squared differences in step responses and the root mean squared differences in Bode magnitude plots between the original FOS and the reduced models is as tiny as possible. Two case studies support the suggested method. The simulation results show that the reduced approximations constructed using the methodology under consideration have step and Bode responses more in line with the actual FOS. The effectiveness of the advocated strategy is further shown by contrasting several performance metrics with some of the contemporary approaches disseminated in academic journals. Full article

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25 pages, 7836 KiB

Open AccessArticle

Efficient Numerical Implementation of the Time-Fractional Stochastic Stokes–Darcy Model

by Zharasbek Baishemirov, Abdumauvlen Berdyshev, Dossan Baigereyev and Kulzhamila Boranbek

Fractal Fract. 2024, 8(8), 476; https://doi.org/10.3390/fractalfract8080476 - 14 Aug 2024

Viewed by 373

Abstract

This paper presents an efficient numerical method for the fractional-order generalization of the stochastic Stokes–Darcy model, which finds application in various engineering, biomedical and environmental problems involving interaction between free fluid flow and flows in porous media. Unlike the classical model, this model [...] Read more.

This paper presents an efficient numerical method for the fractional-order generalization of the stochastic Stokes–Darcy model, which finds application in various engineering, biomedical and environmental problems involving interaction between free fluid flow and flows in porous media. Unlike the classical model, this model allows taking into account the hereditary properties of the process under uncertainty conditions. The proposed numerical method is based on the combined use of the sparse grid stochastic collocation method, finite element/finite difference discretization, a fast numerical algorithm for computing the Caputo fractional derivative, and a cost-effective ensemble strategy. The hydraulic conductivity tensor is assumed to be uncertain in this problem, which is modeled by the reduced Karhunen–Loève expansion. The stability and convergence of the deterministic numerical method have been rigorously proved and validated by numerical tests. Utilizing the ensemble strategy allowed us to solve the deterministic problem once for all samples of the hydraulic conductivity tensor, rather than solving it separately for each sample. The use of the algorithm for computing the fractional derivatives significantly reduced both computational cost and memory usage. This study also analyzes the influence of fractional derivatives on the fluid flow process within the fractional-order Stokes–Darcy model under uncertainty conditions. Full article

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

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37 pages, 485 KiB

Open AccessArticle

Existence and Stability of Solutions for p-Proportional ω-Weighted κ-Hilfer Fractional Differential Inclusions in the Presence of Non-Instantaneous Impulses in Banach Spaces

by Feryal Aladsani and Ahmed Gamal Ibrahim

Fractal Fract. 2024, 8(8), 475; https://doi.org/10.3390/fractalfract8080475 - 14 Aug 2024

Viewed by 380

Abstract

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the

p

-proportional

ω

-weighted

κ

-Hilfer derivative includes an exponential function, [...] Read more.

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the

p

-proportional

ω

-weighted

κ

-Hilfer derivative includes an exponential function,

D_{a, λ}^{σ, ρ, p, κ, ω}

, and then we consider a non-instantaneous impulse differential inclusion containing

D_{a, λ}^{σ, ρ, p, κ, ω}

with order

σ \in (1, 2)

and of kind

ρ \in [0, 1]

in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the

(p, ω, κ)

-generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the

(p, ω, κ)

-generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator

D_{a, λ}^{σ, ρ, p, κ, ω}

, our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings. Full article

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

16 pages, 8303 KiB

Open AccessArticle

High-Order Numerical Approximation for 2D Time-Fractional Advection–Diffusion Equation under Caputo Derivative

by Xindong Zhang, Yan Chen and Leilei Wei

Fractal Fract. 2024, 8(8), 474; https://doi.org/10.3390/fractalfract8080474 - 13 Aug 2024

Viewed by 525

Abstract

In this paper, we propose a novel approach for solving two-dimensional time-fractional advection–diffusion equations, where the fractional derivative is described in the Caputo sense. The discrete scheme is constructed based on the barycentric rational interpolation collocation method and the Gauss–Legendre quadrature rule. We [...] Read more.

In this paper, we propose a novel approach for solving two-dimensional time-fractional advection–diffusion equations, where the fractional derivative is described in the Caputo sense. The discrete scheme is constructed based on the barycentric rational interpolation collocation method and the Gauss–Legendre quadrature rule. We employ the barycentric rational interpolation collocation method to approximate the unknown function involved in the equation. Through theoretical analysis, we establish the convergence rate of the discrete scheme and show its remarkable accuracy. In addition, we give some numerical examples, to illustrate the proposed method. All the numerical results show the flexible application ability and reliability of the present method. Full article

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

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28 pages, 1196 KiB

Open AccessArticle

Advanced Observation-Based Bipartite Containment Control of Fractional-Order Multi-Agent Systems Considering Hostile Environments, Nonlinear Delayed Dynamics, and Disturbance Compensation

by Asad Khan, Muhammad Awais Javeed, Saadia Rehman, Azmat Ullah Khan Niazi and Yubin Zhong

Fractal Fract. 2024, 8(8), 473; https://doi.org/10.3390/fractalfract8080473 - 13 Aug 2024

Viewed by 498

Abstract

This paper introduces an advanced observer-based control strategy designed for fractional multi-agent systems operating in hostile environments. We take into account the dynamic nature of the agents with nonlinear delayed dynamics and consider external disturbances affecting the system. The manuscript presents an improved [...] Read more.

This paper introduces an advanced observer-based control strategy designed for fractional multi-agent systems operating in hostile environments. We take into account the dynamic nature of the agents with nonlinear delayed dynamics and consider external disturbances affecting the system. The manuscript presents an improved observation-based control approach tailored for fractional-order multi-agent systems functioning in challenging conditions. We also establish various applicable conditions governing the creation of observers and disturbance compensation controllers using the fractional Razmikhin technique, signed graph theory, and matrix transformation. Furthermore, our investigation includes observation-based control on switching networks by employing a typical Lyapunov function approach. Finally, the effectiveness of the proposed strategy is demonstrated through the analysis of two simulation examples. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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16 pages, 299 KiB

Open AccessArticle

On New Generalized Hermite–Hadamard–Mercer-Type Inequalities for Raina Functions

by Zeynep Çiftci, Merve Coşkun, Çetin Yildiz, Luminiţa-Ioana Cotîrlă and Daniel Breaz

Fractal Fract. 2024, 8(8), 472; https://doi.org/10.3390/fractalfract8080472 - 13 Aug 2024

Viewed by 484

Abstract

In this research, we demonstrate novel Hermite–Hadamard–Mercer fractional integral inequalities using a wide class of fractional integral operators (the Raina fractional operator). Moreover, a new lemma of this type is proved, and new identities are obtained using the definition of convex function. In [...] Read more.

In this research, we demonstrate novel Hermite–Hadamard–Mercer fractional integral inequalities using a wide class of fractional integral operators (the Raina fractional operator). Moreover, a new lemma of this type is proved, and new identities are obtained using the definition of convex function. In addition to a detailed derivation of a few special situations, certain known findings are summarized. We also point out that some results in this study, in some special cases, such as setting

α = 0 = φ, γ = 1,

and

w = 0, σ (0) = 1, λ = 1

, are more reasonable than those obtained. Finally, it is believed that the technique presented in this paper will encourage additional study in this field. Full article

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

4 pages, 181 KiB

Open AccessEditorial

Mathematical Inequalities in Fractional Calculus and Applications

by Seth Kermausuor and Eze R. Nwaeze

Fractal Fract. 2024, 8(8), 471; https://doi.org/10.3390/fractalfract8080471 - 13 Aug 2024

Viewed by 498

Abstract

All types of inequalities play a very important role in various aspects of mathematical analysis, such as approximation theory and differential equation theory [...] Full article

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

17 pages, 4253 KiB

Open AccessArticle

Fractional Derivative Model on Physical Fractal Space: Improving Rock Permeability Analysis

by Zelin Liu, Xiaobin Yu, Selin Xie, Hongwei Zhou and Yajun Yin

Fractal Fract. 2024, 8(8), 470; https://doi.org/10.3390/fractalfract8080470 - 12 Aug 2024

Viewed by 614

Abstract

As challenges in gas extraction from coal mines increase, precise measurement of permeability becomes crucial. This study proposes a novel pulse transient method based on a fractional derivative model derived on physical fractal space, incorporating operator algebra and the mechanics–electricity analogy to derive [...] Read more.

As challenges in gas extraction from coal mines increase, precise measurement of permeability becomes crucial. This study proposes a novel pulse transient method based on a fractional derivative model derived on physical fractal space, incorporating operator algebra and the mechanics–electricity analogy to derive a new control equation that more accurately delineates the permeability evolution in coal. To validate the approach, permeability experiments were conducted on coal samples under mining stress conditions. The results showed that the adoption of a physically meaningful fractional-order relaxation equation provides a more accurate description of non-Darcy flow behaviour in rocks than traditional integer-order control equations. Additionally, the method proved effective across different rock types, verifying its broad applicability. By establishing a new theoretical foundation, this approach illustrates how the microscale fractal structure of rocks is fundamentally linked to their macroscale fractional responses, thereby enhancing the understanding of fractional modelling methods in rock mechanics and related domains. Full article

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20 pages, 332 KiB

Open AccessFeature PaperArticle

β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations

by Wei-Shih Du, Michal Fečkan, Marko Kostić and Daniel Velinov

Fractal Fract. 2024, 8(8), 469; https://doi.org/10.3390/fractalfract8080469 - 12 Aug 2024

Viewed by 689

Abstract

In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within [...] Read more.

In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within the framework of

β

-Banach spaces. Moreover, we examine the

β

–Ulam–Hyers stability of the solutions, providing insights into the stability behavior under small perturbations. An illustrative example is presented to demonstrate the practical applicability and effectiveness of the theoretical results obtained. Full article

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

16 pages, 3461 KiB

Open AccessArticle

Pavement Crack Detection Using Fractal Dimension and Semi-Supervised Learning

by Wenhao Guo, Leiyang Zhong, Dejin Zhang and Qingquan Li

Fractal Fract. 2024, 8(8), 468; https://doi.org/10.3390/fractalfract8080468 - 12 Aug 2024

Viewed by 663

Abstract

Pavement cracks are crucial indicators for assessing the structural health of asphalt roads. Existing automated crack detection models depend on large quantities of precisely annotated crack sample data. The irregular morphology of cracks makes manual annotation time-consuming and costly, thereby posing challenges to [...] Read more.

Pavement cracks are crucial indicators for assessing the structural health of asphalt roads. Existing automated crack detection models depend on large quantities of precisely annotated crack sample data. The irregular morphology of cracks makes manual annotation time-consuming and costly, thereby posing challenges to the practical application of these models. This study proposes a pavement crack image detection method integrating fractal dimension analysis and semi-supervised learning. It identifies the self-similarity characteristics within the crack regions by analyzing pavement crack images and using fractal dimensions to preliminarily determine the candidate crack regions. The Crack Similarity Learning Network (CrackSL-Net) is then employed to learn the semantic similarity of crack image regions. Semi-supervised learning facilitates automatic crack detection by combining a small amount of labeled data with a large volume of unlabeled image data. Comparative experiments are conducted on two public pavement crack datasets against the HED, U-Net, and RCF models to comprehensively evaluate the performance of the proposed method. The results indicate that, with a 50% annotation ratio, the proposed method achieves high-precision crack detection, with an intersection over union (IoU) exceeding 0.84, which is close to that of U-Net. Visual analysis of the detection results confirms the method’s effectiveness in identifying cracks in complex environments. Full article

(This article belongs to the Special Issue Fracture Analysis of Materials Based on Fractal Nature)

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19 pages, 10754 KiB

Open AccessArticle

Mathematical and Physical Analysis of Fractional Estevez–Mansfield–Clarkson Equation

by Haitham Qawaqneh and Yasser Alrashedi

Fractal Fract. 2024, 8(8), 467; https://doi.org/10.3390/fractalfract8080467 - 12 Aug 2024

Viewed by 529

Abstract

This paper presents the mathematical and physical analysis, as well as distinct types of exact wave solutions, of an important fluid flow dynamics model called the truncated M-fractional (1+1)-dimensional nonlinear Estevez–Mansfield–Clarkson (EMC) equation. This model is used to explain waves in shallow water, [...] Read more.

This paper presents the mathematical and physical analysis, as well as distinct types of exact wave solutions, of an important fluid flow dynamics model called the truncated M-fractional (1+1)-dimensional nonlinear Estevez–Mansfield–Clarkson (EMC) equation. This model is used to explain waves in shallow water, fluid dynamics, and other areas. We obtain kink, bright, singular, and other types of exact wave solutions using the modified extended direct algebraic method and the improved

(G^{'} / G)

-expansion method. Some solutions do not exist. These solutions may be useful in different areas of science and engineering. The results are represented as three-dimensional, contour, and two-dimensional graphs. Stability analysis is also performed to check the stability of the corresponding model. Furthermore, modulation instability analysis is performed to study the stationary solutions of the corresponding model. The results will be helpful for future studies of the corresponding system. The methods used are easy and useful. Full article

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

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17 pages, 3205 KiB

Open AccessFeature PaperArticle

On Martínez–Kaabar Fractal–Fractional Volterra Integral Equations of the Second Kind

by Francisco Martínez and Mohammed K. A. Kaabar

Fractal Fract. 2024, 8(8), 466; https://doi.org/10.3390/fractalfract8080466 - 7 Aug 2024

Viewed by 646

Abstract

The extension of the theory of generalized fractal–fractional calculus, named in this article as Martínez–Kaabar Fractal–Fractional (MKFF) calculus, is addressed to the field of integral equations. Based on the classic Adomian decomposition method, by incorporating the MKFF

α, γ

-integral operator, we [...] Read more.

The extension of the theory of generalized fractal–fractional calculus, named in this article as Martínez–Kaabar Fractal–Fractional (MKFF) calculus, is addressed to the field of integral equations. Based on the classic Adomian decomposition method, by incorporating the MKFF

α, γ

-integral operator, we establish the so-called extended Adomian decomposition method (EADM). The convergence of this proposed technique is also discussed. Finally, some interesting Volterra Integral equations of non-integer order which possess a fractal effect are solved via our proposed approach. The results in this work provide a novel approach that can be employed in solving various problems in science and engineering, which can overcome the challenges of solving various equations, formulated via other classical fractional operators. Full article

(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function, 2nd Edition)

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Fractal Fract., Volume 8, Issue 8 (August 2024) – 61 articles

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