3,527 results on '"Analysis"'
Search Results
2. Fractional Fourier Series on the Torus and Applications
- Author
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Chen Wang, Xianming Hou, Qingyan Wu, Pei Dang, and Zunwei Fu
- Subjects
fractional Fourier series ,fractional approximate identity ,fractional Fejér kernel ,fractional Fourier inversion ,convergence ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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3. Stationary Responses of Seven Classes of Fractional Vibrations Driven by Sinusoidal Force
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Ming Li
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fractional vibrations ,fractional phasor ,motion phasor equation ,equivalent vibration parameters ,stationary response ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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4. Two-Dimensional Time-Fractional Nonlinear Drift Reaction–Diffusion Equation Arising in Electrical Field
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Anjuman, Andrew Y. T. Leung, and Subir Das
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nonlinear ,diffusion ,drift reaction ,charge carrier ,fractional ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
Diffusion equations play a crucial role in various scientific and technological domains, including mathematical biology, physics, electrical engineering, and mathematics. This article presents a new formulation of the diffusion equation in the context of electrical engineering. Specifically, the behaviour of the physical quantity of charge carriers (such as concentration) is examined within semiconductor materials. The primary focus of this work is to solve the two-dimensional, time-fractional, nonlinear drift reaction–diffusion equation by applying an appropriate numerical scheme. In recent years, researchers working on nonlinear diffusion equations have proposed several numerical methods, with the shifted airfoil collocation method being one such efficient technique for solving nonlinear partial differential equations. This collocation approach effectively reduces the considered two-dimensional, time-fractional, nonlinear drift reaction–diffusion equation to a system of algebraic equations. The efficiency and effectiveness of the proposed method are validated through an error analysis, comparing the exact solution and the proposed numerical solution for a specific form of the considered mathematical model. The variations in the concentration of charge carriers, driven by the effects of drift and reaction terms, are displayed graphically as the system transitions from a fractional order to an integer order.
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- 2024
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5. Global Mittag-Leffler Attractive Sets, Boundedness, and Finite-Time Stabilization in Novel Chaotic 4D Supply Chain Models with Fractional Order Form
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Muhamad Deni Johansyah, Aceng Sambas, Muhammad Farman, Sundarapandian Vaidyanathan, Song Zheng, Bob Foster, and Monika Hidayanti
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chaos theory ,supply chain dynamics ,control strategies ,finite-time control ,fractional order ,Lyapunov exponents ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
This research explores the complex dynamics of a Novel Four-Dimensional Fractional Supply Chain System (NFDFSCS) that integrates a quadratic interaction term involving the actual demand of customers and the inventory level of distributors. The introduction of the quadratic term results in significantly larger maximal Lyapunov exponents (MLE) compared to the original model, indicating increased system complexity. The existence, uniqueness, and Ulam–Hyers stability of the proposed system are verified. Additionally, we establish the global Mittag-Leffler attractive set (MLAS) and Mittag-Leffler positive invariant set (MLPIS) for the system. Numerical simulations and MATLAB phase portraits demonstrate the chaotic nature of the proposed system. Furthermore, a dynamical analysis achieves verification via the Lyapunov exponents, a bifurcation diagram, a 0–1 test, and a complexity analysis. A new numerical approximation method is proposed to solve non-linear fractional differential equations, utilizing fractional differentiation with a non-singular and non-local kernel. These numerical simulations illustrate the primary findings, showing that both external and internal factors can accelerate the process. Furthermore, a robust control scheme is designed to stabilize the system in finite time, effectively suppressing chaotic behaviors. The theoretical findings are supported by the numerical results, highlighting the effectiveness of the control strategy and its potential application in real-world supply chain management (SCM).
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- 2024
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6. An Efficient Numerical Scheme for a Time-Fractional Black–Scholes Partial Differential Equation Derived from the Fractal Market Hypothesis
- Author
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Samuel M. Nuugulu, Frednard Gideon, and Kailash C. Patidar
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fractal market hypothesis ,option pricing ,fractional Black–Scholes PDEs ,numerical methods ,convergence ,stability analysis ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
Since the early 1970s, the study of Black–Scholes (BS) partial differential equations (PDEs) under the Efficient Market Hypothesis (EMH) has been a subject of active research in financial engineering. It has now become obvious, even to casual observers, that the classical BS models derived under the EMH framework fail to account for a number of realistic price evolutions in real-time market data. An alternative approach to the EMH framework is the Fractal Market Hypothesis (FMH), which proposes better and clearer explanations of market behaviours during unfavourable market conditions. The FMH involves non-local derivatives and integral operators, as well as fractional stochastic processes, which provide better tools for explaining the dynamics of evolving market anomalies, something that classical BS models may fail to explain. In this work, using the FMH, we derive a time-fractional Black–Scholes partial differential equation (tfBS-PDE) and then transform it into a heat equation, which allows for ease of implementing a high-order numerical scheme for solving it. Furthermore, the stability and convergence properties of the numerical scheme are discussed, and overall techniques are applied to pricing European put option problems.
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- 2024
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7. Servo Control of a Current-Controlled Attractive-Force-Type Magnetic Levitation System Using Fractional-Order LQR Control
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Ryo Yoneda, Yuki Moriguchi, Masaharu Kuroda, and Natsuki Kawaguchi
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control ,fractional calculus ,linear quadratic regulator (LQR) control ,linear quadratic integral (LQI) control ,magnetic levitation (Maglev) ,servo control ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
Recent research on fractional-order control laws has introduced the fractional calculus concept into the field of control engineering. As described herein, we apply fractional-order linear quadratic regulator (LQR) control to a current-controlled attractive-force-type magnetic levitation system, which is a strongly nonlinear and unstable system, to investigate its control performance through experimentation. First, to design the controller, a current-controlled attractive-force-type magnetic levitation system expressed as an integer-order system is extended to a fractional-order system expressed using fractional-order derivatives. Then, target value tracking control of a levitated object is achieved by adding states, described by the integrals of the deviation between the output and the target value, to the extended system. Next, a fractional-order LQR controller is designed for the extended system. For state-feedback control, such as fractional-order servo LQR control, which requires the information of all states, a fractional-order state observer is configured to estimate fractional-order states. Simulation results demonstrate that fractional-order servo LQR control can achieve equilibrium point stabilization and enable target value tracking. Finally, to verify the fractional-order servo LQR control effectiveness, experiments using the designed fractional-order servo LQR control law are conducted with comparison to a conventional integer-order servo LQR control.
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- 2024
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8. Analysis of Caputo Sequential Fractional Differential Equations with Generalized Riemann–Liouville Boundary Conditions
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Nallappan Gunasekaran, Murugesan Manigandan, Seralan Vinoth, and Rajarathinam Vadivel
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Caputo fractional derivative ,generalized Riemann–Liouville fractional integral ,existence and uniqueness ,nonlocal conditions ,sequential derivatives ,fixed-point theorem ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
This paper delves into a novel category of nonlocal boundary value problems concerning nonlinear sequential fractional differential equations, coupled with a unique form of generalized Riemann–Liouville fractional differential integral boundary conditions. For single-valued maps, we employ a transformation technique to convert the provided system into an equivalent fixed-point problem, which we then address using standard fixed-point theorems. Following this, we evaluate the stability of these solutions utilizing the Ulam–Hyres stability method. To elucidate the derived findings, we present constructed examples.
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- 2024
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9. Pavement Crack Detection Using Fractal Dimension and Semi-Supervised Learning
- Author
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Wenhao Guo, Leiyang Zhong, Dejin Zhang, and Qingquan Li
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fractal dimension ,crack detection ,semantic similarity ,semi-supervised learning ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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10. A Fractional-Order Creep-Damage Model for Carbonaceous Shale Describing Coupled Damage Caused by Rainfall and Blasting
- Author
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Jing Li, Bin Hu, Jianlong Sheng, and Lei Huang
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fractional order ,creep-damage constitutive model ,carbonaceous shale ,shear creep behavior ,dry–wet cycles ,blasting ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
In order to better understand the shear creep behavior of weak interlayers (carbonaceous shale) under the coupling effect of the rainfall dry–wet cycle and blasting vibration, as well as quantitatively characterize the coupled damage of the rainfall dry–wet cycle and blasting vibration, a series of shear creep tests were carried out. The results show that the combined damage of the rainfall dry–wet cycle and blasting vibration greatly intensifies the creep effect of carbonaceous shale, leading to an increase in deceleration creep time, an increase in steady-state creep rate, and a decrease in long-term strength. The coupling damage of the rainfall dry–wet cycle and blasting vibration in carbonaceous shale was quantitatively characterized. Based on the fractional-order theory, a fractional-order creep-damage constitutive model (DNFVP) was established by introducing the Abel dashpot to describe the coupled damage of the rainfall wet–dry cycle and blasting vibration and the nonlinear creep acceleration characteristics. The three-dimensional creep equation of the model was derived. The effectiveness of the DNFVP model was verified through the inversion of model parameters and fitting of experimental data, providing a basis for in-depth research on the long-term stability of high slopes in mines with weak interlayers.
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- 2024
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11. Voltage Controller Design for Offshore Wind Turbines: A Machine Learning-Based Fractional-Order Model Predictive Method
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Ashkan Safari, Hossein Hassanzadeh Yaghini, Hamed Kharrati, Afshin Rahimi, and Arman Oshnoei
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offshore wind turbines ,AC bus voltage ,state estimation ,intelligent models ,renewable energy systems ,fractional-order modeling ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
Integrating renewable energy sources (RESs), such as offshore wind turbines (OWTs), into the power grid demands advanced control strategies to enhance efficiency and stability. Consequently, a Deep Fractional-order Wind turbine eXpert control system (DeepFWX) model is developed, representing a hybrid proportional/integral (PI) fractional-order (FO) model predictive random forest alternating current (AC) bus voltage controller designed explicitly for OWTs. DeepFWX aims to address the challenges associated with offshore wind energy systems, focusing on achieving the smooth tracking and state estimation of the AC bus voltage. Extensive comparative analyses were performed against other state-of-the-art intelligent models to assess the effectiveness of DeepFWX. Key performance indicators (KPIs) such as MAE, MAPE, RMSE, RMSPE, and R2 were considered. Superior performance across all the evaluated metrics was demonstrated by DeepFWX, as it achieved MAE of [15.03, 0.58], MAPE of [0.09, 0.14], RMSE of [70.39, 5.64], RMSPE of [0.34, 0.85], as well as the R2 of [0.99, 0.99] for the systems states [X1, X2]. The proposed hybrid approach anticipates the capabilities of FO modeling, predictive control, and random forest intelligent algorithms to achieve the precise control of AC bus voltage, thereby enhancing the overall stability and performance of OWTs in the evolving sector of renewable energy integration.
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- 2024
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12. Analyzing Monofractal Short and Very Short Time Series: A Comparison of Detrended Fluctuation Analysis and Convolutional Neural Networks as Classifiers
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Juan L. López and José A. Vásquez-Coronel
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DFA ,Hurst exponent ,synthetic fluctuation ,neural network ,short time series ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
Time series data are a crucial information source for various natural and societal processes. Short time series can exhibit long-range correlations that reveal significant features not easily discernible in longer ones. Such short time series find utility in AI applications for training models to recognize patterns, make predictions, and perform classification tasks. However, traditional methods like DFA fail as classifiers for monofractal short time series, especially when the series are very short. In this study, we evaluate the performance of the traditional DFA method against the CNN-SVM approach of neural networks as classifiers for different monofractal models. We examine their performance as a function of the decreasing length of synthetic samples. The results demonstrate that CNN-SVM achieves superior classification rates compared to DFA. The overall accuracy rate of CNN-SVM ranges between 64% and 98%, whereas DFA’s accuracy rate ranges between 16% and 64%.
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- 2024
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13. Discussion on Weighted Fractional Fourier Transform and Its Extended Definitions
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Tieyu Zhao and Yingying Chi
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weighted fractional Fourier transform ,weighted fractional-order transform ,periodic matrix ,Fourier transform ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
The weighted fractional Fourier transform (WFRFT) has always been considered a development of the discrete fractional Fourier transform (FRFT). This paper points out that the WFRFT is a discrete FRFT of eigenvalue decomposition, which will change the consistent understanding of the WFRFT. Extended definitions based on the WFRFT have been proposed and widely used in information processing. This paper proposes a unified framework for extended definitions, and existing extended definitions can serve as special cases of this unified framework. In further analysis, we find that the existing extended definitions are deficient. With the help of a unified framework, we systematically analyze the reasons for the deficiencies. This has great guiding significance for the application of the WFRFT and its extended definitions.
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- 2024
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14. Mathematical and Physical Analysis of Fractional Estevez–Mansfield–Clarkson Equation
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Haitham Qawaqneh and Yasser Alrashedi
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Estevez–Mansfield–Clarkson equation ,fractional derivative ,stability analysis ,modulation instability ,exact wave solutions ,analytical methods ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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, 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.
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- 2024
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15. On Martínez–Kaabar Fractal–Fractional Volterra Integral Equations of the Second Kind
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Francisco Martínez and Mohammed K. A. Kaabar
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fractal–fractional differentiation ,fractal–fractional integration ,Adomian Decomposition Method ,fractal–fractional integral equations ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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16. Fractional Derivative Model on Physical Fractal Space: Improving Rock Permeability Analysis
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Zelin Liu, Xiaobin Yu, Selin Xie, Hongwei Zhou, and Yajun Yin
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permeability ,pulse transient method ,fractional derivative ,physical fractal space ,gas extraction ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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17. FOMCON Toolbox-Based Direct Approximation of Fractional Order Systems Using Gaze Cues Learning-Based Grey Wolf Optimizer
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Bala Bhaskar Duddeti, Asim Kumar Naskar, Veerpratap Meena, Jitendra Bahadur, Pavan Kumar Meena, and Ibrahim A. Hameed
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reduced order model ,fractional-order systems ,swarm intelligence algorithm ,gaze cues learning-based grey wolf optimizer ,FOMCON toolbox ,metaheuristic ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 (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.
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- 2024
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18. A High-Order Numerical Method Based on a Spatial Compact Exponential Scheme for Solving the Time-Fractional Black–Scholes Model
- Author
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Xinhao Huang and Bo Yu
- Subjects
time fractional Black–Scholes model ,compact exponential scheme ,solvability ,stability ,convergence ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
This paper investigates a high-order numerical method based on a spatial compact exponential scheme for solving the time-fractional Black–Scholes model. Firstly, the original time-fractional Black–Scholes model is converted into an equivalent time-fractional advection–diffusion reaction model by means of a variable transformation technique. Secondly, a novel high-order numerical method is constructed with (2−α) accuracy in time and fourth-order accuracy in space based on a spatial compact exponential scheme, where α is the fractional derivative. The uniqueness of solvability of the derived numerical method is rigorously discussed. Thirdly, the unconditional stability and convergence of the derived numerical method are rigorously analyzed using the Fourier analysis technique. Finally, numerical examples are presented to test the effectiveness of the derived numerical method. The proposed numerical method is also applied to solve the time-fractional Black–Scholes model, whose exact analytical solution is unknown; numerical results are illustrated graphically.
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- 2024
- Full Text
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19. β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations
- Author
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Wei-Shih Du, Michal Fečkan, Marko Kostić, and Daniel Velinov
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β-Banach space ,fractional integral equations ,non-instantaneous impulses ,β–Ulam–Hyers stability ,Schauder’s fixed-point theorem ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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20. Advanced Observation-Based Bipartite Containment Control of Fractional-Order Multi-Agent Systems Considering Hostile Environments, Nonlinear Delayed Dynamics, and Disturbance Compensation
- Author
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Asad Khan, Muhammad Awais Javeed, Saadia Rehman, Azmat Ullah Khan Niazi, and Yubin Zhong
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observation error ,fractional-order multi-agent system ,bipartite containment control ,Lyapunov function ,signed directed network ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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21. High-Order Numerical Approximation for 2D Time-Fractional Advection–Diffusion Equation under Caputo Derivative
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Xindong Zhang, Yan Chen, and Leilei Wei
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barycentric rational interpolation collocation method ,Caputo derivative ,time-fractional advection–diffusion equation ,Gauss–Legendre quadrature rule ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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22. Efficient Numerical Implementation of the Time-Fractional Stochastic Stokes–Darcy Model
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Zharasbek Baishemirov, Abdumauvlen Berdyshev, Dossan Baigereyev, and Kulzhamila Boranbek
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Stokes–Darcy equations ,Caputo fractional derivative ,sparse grid stochastic collocation method ,numerical method ,convergence rate ,ensemble method ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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23. On New Generalized Hermite–Hadamard–Mercer-Type Inequalities for Raina Functions
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Zeynep Çiftci, Merve Coşkun, Çetin Yildiz, Luminiţa-Ioana Cotîrlă, and Daniel Breaz
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Hermite–Hadamard inequality ,Jensen–Mercer inequality ,Hölder inequality ,Raina fractional operator ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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24. Analysis of Fractal Properties of Atmospheric Turbulence and the Practical Applications
- Author
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Zihan Liu, Hongsheng Zhang, Zuntao Fu, Xuhui Cai, and Yu Song
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fractal dimension ,similarity relationship ,Hilbert–Huang transform ,non-turbulent motion ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
- Full Text
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25. Valuation of Currency Option Based on Uncertain Fractional Differential Equation
- Author
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Weiwei Wang, Dan A. Ralescu, and Xiaojuan Xue
- Subjects
fractional-order differential equation ,uncertainty theory ,currency option ,optimistic value ,uncertain hypothesis test ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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26. 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
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Feryal Aladsani and Ahmed Gamal Ibrahim
- Subjects
fractional differential inclusions ,p-proportional ω-weighted κ-Hilfer fractional derivative ,weakly Picard operators ,measure of non-compactness ,Ulam-Hyeres stability ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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, Da,λσ,ρ,p,κ,ω, and then we consider a non-instantaneous impulse differential inclusion containing Da,λσ,ρ,p,κ,ω with order σ∈(1,2) and of kind ρ∈[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 Da,λσ,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.
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- 2024
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27. Logging Evaluation of Irreducible Water Saturation: Fractal Theory and Data-Driven Approach—Case Study of Complex Porous Carbonate Reservoirs in Mishrif Formation
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Jianhong Guo, Zhansong Zhang, Xin Nie, Qing Zhao, and Hengyang Lv
- Subjects
irreducible water saturation ,carbonate rock ,fractal theory ,Thomeer Function ,nuclear magnetic resonance ,logarithm of the mean transverse relaxation time ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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 (T2lm) 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 T2lm 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 T2lm curves, it remains within acceptable limits. Compared to traditional methods, this approach provides enhanced predictive accuracy and broader applicability.
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- 2024
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28. Numerical Analysis and Computation of the Finite Volume Element Method for the Nonlinear Coupled Time-Fractional Schrödinger Equations
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Xinyue Zhao, Yining Yang, Hong Li, Zhichao Fang, and Yang Liu
- Subjects
coupled time-fractional Schrödinger equations ,finite volume element method ,L2 − 1σ formula ,optimal error estimates ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
In this article, our aim is to consider an efficient finite volume element method combined with the L2−1σ 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 L2−1σ 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.
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- 2024
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29. Wind Turbine Blade Fault Diagnosis: Approximate Entropy as a Tool to Detect Erosion and Mass Imbalance
- Author
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Salim Lahmiri
- Subjects
wind turbine blade condition ,fault diagnosis ,correlation dimension ,approximate entropy ,Lyapunov exponent ,crack ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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30. Application of Fractional Calculus in Predicting the Temperature-Dependent Creep Behavior of Concrete
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Jiecheng Chen, Lingwei Gong, and Ruifan Meng
- Subjects
fractional calculus ,creep ,temperature dependence ,concrete ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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31. Remaining Useful Life Prediction for Power Storage Electronic Components Based on Fractional Weibull Process and Shock Poisson Model
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Wanqing Song, Xianhua Yang, Wujin Deng, Piercarlo Cattani, and Francesco Villecco
- Subjects
hybrid energy storage system ,lithium-ion batteries ,supercapacitor ,remaining useful life prediction ,fractional Weibull process ,non-homogeneous compound Poisson process ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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32. Enhanced Thermal and Mass Diffusion in Maxwell Nanofluid: A Fractional Brownian Motion Model
- Author
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Ming Shen, Yihong Liu, Qingan Yin, Hongmei Zhang, and Hui Chen
- Subjects
fractional Brownian motion ,Maxwell nanofluids ,Riemann–Liouville fractional derivative ,improved Buongiorno model ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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33. On Hybrid and Non-Hybrid Discrete Fractional Difference Inclusion Problems for the Elastic Beam Equation
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Faycal Alili, Abdelkader Amara, Khaled Zennir, and Taha Radwan
- Subjects
hybrid elastic beam equation ,fractional difference equation ,fixed point theorems ,existence ,stability analysis ,iterative methods ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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34. 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
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Zahra Sadat Aghayan, Alireza Alfi, Seyed Mehdi Abedi Pahnehkolaei, and António M. Lopes
- Subjects
fractional-order ,descriptor system ,regularity ,neutral delay ,stabilization ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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35. Research on Efficiency and Multifractality of Gold Market under Major Events
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Feifei Wang, Jiaxin Chang, Weizhen Zuo, and Weijie Zhou
- Subjects
gold market ,effectiveness ,multifractality ,major events ,MFDFA ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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) 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.
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- 2024
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36. A Pareto-Optimal-Based Fractional-Order Admittance Control Method for Robot Precision Polishing
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Haotian Wu, Jianzhong Yang, Si Huang, and Xiao Ning
- Subjects
fractional-order admittance control ,fastest tracking differential ,Pareto optimality ,robotic precision polishing ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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%.
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- 2024
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37. Artificial Intelligence in Chromatin Analysis: A Random Forest Model Enhanced by Fractal and Wavelet Features
- Author
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Igor Pantic and Jovana Paunovic Pantic
- Subjects
artificial intelligence ,apoptosis ,chromatin ,fractals ,wavelets ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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38. Difference Approximation for 2D Time-Fractional Integro-Differential Equation with Given Initial and Boundary Conditions
- Author
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Xindong Zhang, Ziyang Luo, Quan Tang, Leilei Wei, and Juan Liu
- Subjects
integro-differential equation ,Riemann–Liouville derivative ,compact finite difference method ,stability ,convergence ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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(τ2+hx4+hy4), where τ is the time step size, hx and hy are the spatial step sizes. Furthermore, several examples were provided to verify the correctness of our theoretical reasoning.
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- 2024
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39. A Delayed Fractional-Order Predator–Prey Model with Three-Stage Structure and Cannibalism for Prey
- Author
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Hui Zhang and Ahmadjan Muhammadhaji
- Subjects
stage structure ,cannibalism ,fractional-order ,local stability ,Hopf bifurcation ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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40. Fractals of Interpolative Kannan Mappings
- Author
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Xiangting Shi, Umar Ishtiaq, Muhammad Din, and Mohammad Akram
- Subjects
metric space ,fractals ,Kannan mapping ,well-posedness ,iterated function system ,iterated multi-function system ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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 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.
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- 2024
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41. Modeling the Dispersion of Waves in a Multilayered Inhomogeneous Membrane with Fractional-Order Infusion
- Author
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Ali M. Mubaraki, Rahmatullah Ibrahim Nuruddeen, Rab Nawaz, and Tayyab Nawaz
- Subjects
Helmholtz equation ,multilayered membrane ,fractional order infusion ,composite media ,vibration ,Bessel equation ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
The dispersion of elastic shear waves in multilayered bodies is a topic of extensive research due to its significance in contemporary science and engineering. Anti-plane shear motion, a two-dimensional mathematical model in solid mechanics, effectively captures shear wave propagation in elastic bodies with relative mathematical simplicity. This study models the vibration of elastic waves in a multilayered inhomogeneous circular membrane using the Helmholtz equation with fractional-order infusion, effectively leveraging the anti-plane shear motion equation to avoid the computational complexity of universal plane motion equations. The method of the separation of variables and the conformable Bessel equation are utilized for the analytical examination of the model’s resulting vibrational displacements, as well as the dispersion relation. Additionally, the influence of various wave phenomena, including the dependencies of the wavenumber on the frequency and the phase speed on the wavenumber, respectively, with the variational effect of the fractional order on wave dispersion is considered. Numerical simulations of prototypical cases validate the formulated model, illustrating its applicability and effectiveness. The study reveals that fractional-order infusion significantly impacts the dispersion of elastic waves in both single- and multilayer membranes. The effects vary depending on the membrane’s structure and the wave propagation regime (long-wave vs. short-wave). These findings underscore the potential of fractional-order parameters in tailoring wave behavior for diverse scientific and engineering applications.
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- 2024
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42. Semi-Regular Continued Fractions with Fast-Growing Partial Quotients
- Author
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Shirali Kadyrov, Aiken Kazin, and Farukh Mashurov
- Subjects
semi-regular continued fractions ,Hausdorff dimension ,box dimension ,dimension theory ,number theory ,partial quotients ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
In number theory, continued fractions are essential tools because they provide distinct representations of real numbers and provide information about their characteristics. Regular continued fractions have been examined in great detail, but less research has been carried out on their semi-regular counterparts, which are produced from the sequences of alternating plus and minus ones. In this study, we investigate the structure and features of semi-regular continuous fractions through the lens of dimension theory. We prove a primary result about the Hausdorff dimension of number sets whose partial quotients increase more quickly than a given pace. Furthermore, we conduct numerical analyses to illustrate the differences between regular and semi-regular continued fractions, shedding light on potential future directions in this field.
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- 2024
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43. Evolution of Pore Structure and Fractal Characteristics in Red Sandstone under Cyclic Impact Loading
- Author
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Huanhuan Qiao, Peng Wang, Zhen Jiang, Yao Liu, Guanglin Tian, and Bokun Zhao
- Subjects
rock ,pore structure ,NMR ,T2 cutoff value ,cyclic impact loading ,fractal dimension ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
Fatigue damage can occur in surface rock engineering due to various factors, including earthquakes, blasting, and impacts. The underlying cause for the variations in physical and mechanical properties of the rock resulting from impact loading is the alteration in the internal pore structure. To investigate the evolution characteristics of the pore structure under impact fatigue damage, red sandstone subjected to cyclic impact compression by split Hopkinson pressure bar (SHPB) was analyzed using nuclear magnetic resonance (NMR) technology. The parameters describing the evolution of pore structure were obtained and quantified using fractal methods. The development of the pore structure in rocks subjected to cyclic impact was quantitatively analyzed, and two fractal evolution models based on pore size and pore connectivity were constructed. The results indicate that with an increasing number of impact loading cycles, the porosity of the red sandstone gradually increases, the T2 cutoff (T2c) value decreases, the most probable gray value of magnetic resonance imaging (MRI) increases, the pores’ connectivity is enhanced, and the fractal dimension decreases gradually. Moreover, the pore distribution space tends to transition from three-dimensional to two-dimensional, suggesting the expansion of dominant pores into clusters, forming microfractures or even macroscopic fissures. The findings provide valuable insights into the impact fatigue characteristics of rocks from a microscopic perspective and contribute to the evaluation of time-varying stability and the assessment of progressive damage in rock engineering.
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- 2024
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44. Adaptive Morphing Activation Function for Neural Networks
- Author
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Oscar Herrera-Alcántara and Salvador Arellano-Balderas
- Subjects
fractional derivative ,activation function ,machine learning ,wavelets ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
A novel morphing activation function is proposed, motivated by the wavelet theory and the use of wavelets as activation functions. Morphing refers to the gradual change of shape to mimic several apparently unrelated activation functions. The shape is controlled by the fractional order derivative, which is a trainable parameter to be optimized in the neural network learning process. Given the morphing activation function, and taking only integer-order derivatives, efficient piecewise polynomial versions of several existing activation functions are obtained. Experiments show that the performance of polynomial versions PolySigmoid, PolySoftplus, PolyGeLU, PolySwish, and PolyMish is similar or better than their counterparts Sigmoid, Softplus, GeLU, Swish, and Mish. Furthermore, it is possible to learn the best shape from the data by optimizing the fractional-order derivative with gradient descent algorithms, leading to the study of a more general formula based on fractional calculus to build and adapt activation functions with properties useful in machine learning.
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- 2024
- Full Text
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45. Analytic Solution of the Time-Fractional Partial Differential Equation Using a Multi-G-Laplace Transform Method
- Author
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Hassan Eltayeb
- Subjects
fractional partial differential equation ,G-double and triple Laplace transform ,decomposition method ,inverse G-double and triple Laplace transform ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
In several recent studies, many researchers have shown the advantage of fractional calculus in the production of particular solutions of a huge number of linear and nonlinear partial differential equations. In this research work, different theorems related to the G-double Laplace transform (DGLT) are proved. The solution of the system of time-fractional partial differential equations is addressed using a new analytical method. This technique is a combination of the multi-G-Laplace transform and decomposition methods (MGLTDM). Moreover, we discuss the convergence of this method. Two examples are provided to check the applicability and efficiency of our technique.
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- 2024
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46. An Application of Multiple Erdélyi–Kober Fractional Integral Operators to Establish New Inequalities Involving a General Class of Functions
- Author
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Asifa Tassaddiq, Rekha Srivastava, Rabab Alharbi, Ruhaila Md Kasmani, and Sania Qureshi
- Subjects
Fox-H function ,generalized inequalities ,multiple Erdélyi–Kober (E–K) fractional integral transforms ,continuous functions ,mathematical operators ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j∈N) positively continuous and decaying functions in the finite interval a≤t≤x, the Fox-H function is involved in establishing new and novel fractional integral inequalities. Since the Fox-H function is the most general special function, the obtained inequalities are therefore sufficiently widespread and significant in comparison to the current literature to yield novel and unique results.
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- 2024
- Full Text
- View/download PDF
47. Calculation of the Relaxation Modulus in the Andrade Model by Using the Laplace Transform
- Author
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Juan Luis González-Santander, Giorgio Spada, Francesco Mainardi, and Alexander Apelblat
- Subjects
Andrade model ,relaxation modulus in linear viscoelasticity ,Mittag-Leffler function ,Laplace transform ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
In the framework of the theory of linear viscoelasticity, we derive an analytical expression of the relaxation modulus in the Andrade model Gαt for the case of rational parameter α=m/n∈(0,1) in terms of Mittag–Leffler functions from its Laplace transform G˜αs. It turns out that the expression obtained can be rewritten in terms of Rabotnov functions. Moreover, for the original parameter α=1/3 in the Andrade model, we obtain an expression in terms of Miller-Ross functions. The asymptotic behaviours of Gαt for t→0+ and t→+∞ are also derived applying the Tauberian theorem. The analytical results obtained have been numerically checked by solving the Volterra integral equation satisfied by Gαt by using a successive approximation approach, as well as computing the inverse Laplace transform of G˜αs by using Talbot’s method.
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- 2024
- Full Text
- View/download PDF
48. Existence of Solutions for Caputo Sequential Fractional Differential Inclusions with Nonlocal Generalized Riemann–Liouville Boundary Conditions
- Author
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Murugesan Manigandan, Saravanan Shanmugam, Mohamed Rhaima, and Elango Sekar
- Subjects
existence of solutions ,boundary conditions ,fractional differential inclusions ,sequential differential inclusions ,fixed-point technique ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
In this study, we explore the existence and uniqueness of solutions for a boundary value problem defined by coupled sequential fractional differential inclusions. This investigation is augmented by the introduction of a novel set of generalized Riemann–Liouville boundary conditions. Utilizing Carathéodory functions and Lipschitz mappings, we establish existence results for these nonlocal boundary conditions. Utilizing fixed-point theorems designed for multi-valued maps, we obtain significant existence results for the problem, considering both convex and non-convex values. The derived results are clearly demonstrated with an illustrative example. Numerical examples are provided to validate the theoretical conclusions, contributing to a deeper understanding of fractional-order boundary value problems.
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- 2024
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49. Morphological Features of Mathematical and Real-World Fractals: A Survey
- Author
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Miguel Patiño-Ortiz, Julián Patiño-Ortiz, Miguel Ángel Martínez-Cruz, Fernando René Esquivel-Patiño, and Alexander S. Balankin
- Subjects
scale invariance ,conformal invariance ,multifractality ,lacunarity ,succolarity ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
The aim of this review paper is to survey the fractal morphology of scale-invariant patterns. We are particularly focusing on the scale and conformal invariance, as well as on the fractal non-uniformity (multifractality), inhomogeneity (lacunarity), and anisotropy (succolarity). We argue that these features can be properly quantified by the following six adimensional numbers: the fractal (e.g., similarity, box-counting, or Assouad) dimension, conformal dimension, degree of multifractal non-uniformity, coefficient of multifractal asymmetry, index of lacunarity, and index of fractal anisotropy. The difference between morphological properties of mathematical and real-world fractals is especially outlined in this review paper.
- Published
- 2024
- Full Text
- View/download PDF
50. Analysis of Truncated M-Fractional Mathematical and Physical (2+1)-Dimensional Nonlinear Kadomtsev–Petviashvili-Modified Equal-Width Model
- Author
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Mohammed Ahmed Alomair and Moin-ud-Din Junjua
- Subjects
nonlinear Kadomtsev–Petviashvili-modified equal-width equation ,fractional derivative ,analytical methods ,exact wave solitons ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
This study focuses on the mathematical and physical analysis of a truncated M-fractional (2+1)-dimensional nonlinear Kadomtsev–Petviashvili-modified equal-width model. The distinct types of the exact wave solitons of an important real-world equation called the truncated M-fractional (2+1)-dimensional nonlinear Kadomtsev–Petviashvili-modified equal-width (KP-mEW) model are achieved. This model is used to explain ocean waves, matter-wave pulses, waves in ferromagnetic media, and long-wavelength water waves. The diverse patterns of waves on the oceans are yielded by the Kadomtsev–Petviashvili-modified equal-width (KP-mEW) equation. We obtain kink-, bright-, and periodic-type soliton solutions by using the expa function and modified extended tanh function methods. The solutions are more valuable than the existing results due to the use of a truncated M-fractional derivative. These solutions may be useful in different areas of science and engineering. The methods applied are simple and useful.
- Published
- 2024
- Full Text
- View/download PDF
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