Fractal and Fractional

16 pages, 3447 KiB

Open AccessArticle

The Multiple Frequency Conversion Sinusoidal Chaotic Neural Network and Its Application

by Zhiqiang Hu, Zhongjin Guo, Gongming Wang, Lei Wang, Xiaodong Zhao and Yongfeng Zhang

Fractal Fract. 2023, 7(9), 697; https://doi.org/10.3390/fractalfract7090697 - 21 Sep 2023

Cited by 2 | Viewed by 997

Aiming at the problem that the global search performance of a transiently chaotic neural network is not ideal, a multiple frequency conversion sinusoidal chaotic neural network (MFCSCNN) model is proposed based on the biological mechanism of the brain, including multiple functional modules and [...] Read more.

Aiming at the problem that the global search performance of a transiently chaotic neural network is not ideal, a multiple frequency conversion sinusoidal chaotic neural network (MFCSCNN) model is proposed based on the biological mechanism of the brain, including multiple functional modules and sinusoidal signals of different frequencies. In this model, multiple FCS functions and Sigmoid functions with different phase angles were used to construct the excitation function of neurons in the form of weighted sum. In this paper, the inverted bifurcation diagram, Lyapunov exponential diagram and parameter range of the model are given. The dynamic characteristics of the model are analyzed and applied to function optimization and combinatorial optimization problems. Experimental results show that the multiple frequency conversion sinusoidal chaotic neural network has better global search performance than the transient chaotic neural network and other related models. Full article

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

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16 pages, 4485 KiB

Open AccessArticle

Fractal Study of the Development Law of Mining Cracks

by Jinsui Wu, Dongyu Xie, Sihai Yi, Shangxian Yin, Dezhi Hu, Yuanyuan Li and Yun Wang

Fractal Fract. 2023, 7(9), 696; https://doi.org/10.3390/fractalfract7090696 - 19 Sep 2023

Cited by 3 | Viewed by 1422

Abstract

Studying mining fracture development is vital for geotechnical and mining engineering and geological disaster prevention. This research assesses crack effects on rock mass stress equilibrium during coal mining, potentially causing geological disasters such as land subsidence and landslides. Using fractal geometry theory, the [...] Read more.

Studying mining fracture development is vital for geotechnical and mining engineering and geological disaster prevention. This research assesses crack effects on rock mass stress equilibrium during coal mining, potentially causing geological disasters such as land subsidence and landslides. Using fractal geometry theory, the present study investigates the development of horizontal and vertical mining cracks, revealing their propagation patterns. The fractal dimension generally increases as the propulsion distance increases; however, fluctuations vary from 250 to 287.5 m, forming a wavering line chart. The proportion of mining fracture area relative to mining space area increases with greater propulsion distance, indicating expanded upward mining space due to separation layers. The horizontal distribution of mining cracks persists, while the vertical distribution decreases, suggesting ground subsidence results from upward transmission. The fastest increase in fractal dimension occurs at 87.5–100 m. At 250 m, it peaks at 1.4136, indicating complex crack structures. During propulsion, the fractal dimension decreases due to upward mining space expansion through overlying rock layer collapse, forming new cracks. The proportion of mining crack area to mining space area increases gradually throughout the mining process. The present study presents a simulation model for crack identification, noting limitations in identifying tiny cracks. Full article

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20 pages, 10371 KiB

Open AccessArticle

Dynamic Compressive Mechanical Property Characteristics and Fractal Dimension Applications of Coal-Bearing Mudstone at Real-Time Temperatures

by Shiru Guo, Lianying Zhang, Hai Pu, Yadong Zheng, Bing Li, Peng Wu, Peitao Qiu, Chao Ma and Yiying Feng

Fractal Fract. 2023, 7(9), 695; https://doi.org/10.3390/fractalfract7090695 - 18 Sep 2023

Cited by 4 | Viewed by 1489

Abstract

Coal-bearing rocks are inevitably exposed to high temperatures and impacts (rapid dynamic load action) during deep-earth resource extraction, necessitating the study of their mechanical properties under such conditions. This paper reports on dynamic compression tests conducted on coal-bearing mudstone specimens at real-time temperatures [...] Read more.

Coal-bearing rocks are inevitably exposed to high temperatures and impacts (rapid dynamic load action) during deep-earth resource extraction, necessitating the study of their mechanical properties under such conditions. This paper reports on dynamic compression tests conducted on coal-bearing mudstone specimens at real-time temperatures (the temperature of the rock remains constant throughout the impact process) ranging from 25 °C to 400 °C using a temperature Hopkinson (T-SHPB) test apparatus developed in-house. The objective is to analyze the relationship between mechanical properties and the fractal dimension of fractured fragments and to explore the mechanical response of coal-bearing mudstone specimens to the combined effects of temperature and impact using macroscopic fracture characteristics. The study found that the peak stress and dynamic elastic modulus initially increased and then decreased with increasing temperature, increasing in the 25–150 °C range and monotonically decreasing in the 150–400 °C range. Based on the distribution coefficients and fractal dimensions of the fractured fragments, it was found that the degree of damage of coal-bearing mudstone shows a trend of an initial decrease and then an increase with increasing temperature. In the temperature range of 25–150 °C, the expansion of clay minerals within the mudstone filled the voids between the skeletal particles, resulting in densification and decreased damage. In the temperature range of 150–400 °C, thermal stresses increased the internal fractures and reduced the overall strength of the mudstone, resulting in increased damage. Negative correlations between fractal dimensions, the modulus of elasticity, and peak stress could be used to predict rock properties in engineering. Full article

(This article belongs to the Special Issue General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics)

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17 pages, 2361 KiB

Open AccessArticle

Nonlinear Filter-Based Adaptive Output-Feedback Control for Uncertain Fractional-Order Nonlinear Systems with Unknown External Disturbance

by Zhiyao Ma and Ke Sun

Fractal Fract. 2023, 7(9), 694; https://doi.org/10.3390/fractalfract7090694 - 18 Sep 2023

Cited by 3 | Viewed by 1425

Abstract

This study is devoted to a nonlinear filter-based adaptive fuzzy output-feedback control scheme for uncertain fractional-order (FO) nonlinear systems with unknown external disturbance. Fuzzy logic systems (FLSs) are applied to estimate unknown nonlinear dynamics, and a new FO fuzzy state observer based on [...] Read more.

This study is devoted to a nonlinear filter-based adaptive fuzzy output-feedback control scheme for uncertain fractional-order (FO) nonlinear systems with unknown external disturbance. Fuzzy logic systems (FLSs) are applied to estimate unknown nonlinear dynamics, and a new FO fuzzy state observer based on a nonlinear disturbance observer is established for simultaneously estimating the unmeasurable states and mixed disturbance. Then, with the aid of auxiliary functions, a novel FO nonlinear filter is given to approximately replace the virtual control functions, together with the corresponding fractional derivative, which not only erases the inherent complexity explosion problem under the framework of backstepping, but also completely compensates for the effects of the boundary errors induced by the constructed filters compared to the previous FO linear filter method. Under certain assumptions, and in line with the FO stability criterion, the stability of the controlled system is ensured. An FO Chua–Hartley simulation study is presented to verify the validity of the proposed method. Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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24 pages, 7239 KiB

Open AccessArticle

Neural Fractional Order PID Controllers Design for 2-Link Rigid Robot Manipulator

by Mohamed Jasim Mohamed, Bashra Kadhim Oleiwi, Layla H. Abood, Ahmad Taher Azar and Ibrahim A. Hameed

Fractal Fract. 2023, 7(9), 693; https://doi.org/10.3390/fractalfract7090693 - 18 Sep 2023

Cited by 5 | Viewed by 1572

Abstract

The robotic manipulator is considered one of the complex systems that include multi-input, multi-output, non-linearity, and highly coupled. The uncertainty in the parameters and external disturbances have a negative influence on the performance of the system. Therefore, the controllers that will be designed [...] Read more.

The robotic manipulator is considered one of the complex systems that include multi-input, multi-output, non-linearity, and highly coupled. The uncertainty in the parameters and external disturbances have a negative influence on the performance of the system. Therefore, the controllers that will be designed for these systems must be able to deal with these complexities and difficulties. The Proportional, Integral, and Derivative (PID) controller is known to be simple and well robust, while the neural network has a solid ability to map complex functions. In this paper, we propose six control structures by combining the benefits of PID controller with integer and fractional order and the benefits of neural networks to produce hybrid controllers for a 2-Link Rigid Robot Manipulator (2-LRRM) handling with the problem of trajectory tracking. The Gorilla Forces Troops Optimization algorithm (GTO) was used to tune the parameters of the proposed controller schemes to minimize the Integral of Time Square Error (ITSE). In addition, the robustness of the performance of the suggested control systems is tested by altering the initial position, external disturbances and parameters and carried out using MATLAB. The best performance of the proposed controllers was the Neural Network Fractional Order Proportional Integral Derivative Controller (NNFOPID). Full article

(This article belongs to the Section Engineering)

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14 pages, 4800 KiB

Open AccessArticle

Robot Manipulator Control Using a Robust Data-Driven Method

by Mehran Rahmani and Sangram Redkar

Fractal Fract. 2023, 7(9), 692; https://doi.org/10.3390/fractalfract7090692 - 18 Sep 2023

Cited by 3 | Viewed by 1737

Abstract

Robotic manipulators with diverse structures find widespread use in both industrial and medical applications. Therefore, designing an appropriate controller is of utmost importance when utilizing such robots. In this research, we present a robust data-driven control method for the regulation of a 2-degree-of-freedom [...] Read more.

Robotic manipulators with diverse structures find widespread use in both industrial and medical applications. Therefore, designing an appropriate controller is of utmost importance when utilizing such robots. In this research, we present a robust data-driven control method for the regulation of a 2-degree-of-freedom (2-DoF) robot manipulator. The nonlinear dynamic model of the 2-DoF robot arm is linearized using Koopman theory. The data mode decomposition (DMD) method is applied to generate the Koopman operator. A fractional sliding mode control (FOSMC) is employed to govern the data-driven linearized dynamic model. We compare the performance of Koopman fractional sliding mode control (KFOSMC) with conventional proportional integral derivative (PID) control and FOSMC prior to linearization by Koopman theory. The results demonstrate that KFOSMC outperforms PID and FOSMC in terms of high tracking performance, low tracking error, and minimal control signals. Full article

(This article belongs to the Special Issue Recent Advances in Adaptive Fractional Sliding Mode Control)

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16 pages, 878 KiB

Open AccessArticle

Study on Abundant Dust-Ion-Acoustic Solitary Wave Solutions of a (3+1)-Dimensional Extended Zakharov–Kuznetsov Dynamical Model in a Magnetized Plasma and Its Linear Stability

by Muhammad Arshad, Aly R. Seadawy, Muhammad Tanveer and Faisal Yasin

Fractal Fract. 2023, 7(9), 691; https://doi.org/10.3390/fractalfract7090691 - 18 Sep 2023

Cited by 12 | Viewed by 1230

Abstract

This article examines how shocks and three-dimensional nonlinear dust-ion-acoustic waves propagate across uniform magnetized electron–positron–ion plasmas. The two-variable

(G^{'} / G, 1 / G)

-expansion and generalized exp(

- ϕ (ξ)

)-expansion techniques are presented to construct [...] Read more.

This article examines how shocks and three-dimensional nonlinear dust-ion-acoustic waves propagate across uniform magnetized electron–positron–ion plasmas. The two-variable

(G^{'} / G, 1 / G)

-expansion and generalized exp(

- ϕ (ξ)

)-expansion techniques are presented to construct the ion-acoustic wave results of a (3+1)-dimensional extended Zakharov–Kuznetsov (eZK) model. As a result, the novel soliton and other wave solutions in a variety of forms, including kink- and anti-kink-type breather waves, dark and bright solitons, kink solitons, and multi-peak solitons, etc., are attained. With the help of software, the solitary wave results (that signify the electrostatic potential field), electric and magnetic fields, and quantum statistical pressures are also constructed. These solutions have numerous applications in various areas of physics and other areas of applied sciences. Graphical representations of some of the obtained results, and the electric and magnetic fields as well as the electrostatic field potential are also presented. These results demonstrate the effectiveness of the presented techniques, which will also be useful in solving many other nonlinear models that arise in mathematical physics and several other applied sciences fields. Full article

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

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24 pages, 2542 KiB

Open AccessArticle

A Marine-Predator-Algorithm-Based Optimum FOPID Controller for Enhancing the Stability and Transient Response of Automatic Voltage Regulators

by Abdullah M. Noman, Sulaiman Z. Almutairi, Mokhtar Aly, Mohammed H. Alqahtani, Ali S. Aljumah and Emad A. Mohamed

Fractal Fract. 2023, 7(9), 690; https://doi.org/10.3390/fractalfract7090690 - 16 Sep 2023

Cited by 9 | Viewed by 1667

Abstract

An improved design optimization method for fractional-order-based proportional integral derivative (FOPID) controllers is proposed in this paper to enhance the stability and transient response of automatic voltage-regulator systems. The FOPID represents a higher degree-of-freedom controller through having five tunable parameters, compared with only [...] Read more.

An improved design optimization method for fractional-order-based proportional integral derivative (FOPID) controllers is proposed in this paper to enhance the stability and transient response of automatic voltage-regulator systems. The FOPID represents a higher degree-of-freedom controller through having five tunable parameters, compared with only three parameters in the integer-order PID controller. In the literature, the performance of the FOPID is highly determined through the design method and its parameter-determination process. Recently, optimum design of the FOPID has found wide employment in several engineering applications through using metaheuristic optimization algorithms. In this paper, an improved method for the FOPID’s parameter optimization is proposed for AVR applications using the marine predator optimization algorithm (MPA). The proposed MPA–FOPID controller is verified through comparing its performance with other featured and newly developed optimization algorithms. The proposed MPA–FOPID comparative analysis has been proven to have better stability, frequency response, robustness, faster response, and disturbance-rejection ability over the other studied methods in this paper. Full article

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22 pages, 446 KiB

Open AccessArticle

Advances in Ostrowski-Mercer Like Inequalities within Fractal Space

by Miguel Vivas-Cortez, Muhammad Uzair Awan, Usama Asif, Muhammad Zakria Javed and Hüseyin Budak

Fractal Fract. 2023, 7(9), 689; https://doi.org/10.3390/fractalfract7090689 - 16 Sep 2023

Cited by 4 | Viewed by 1027

Abstract

The main idea of the current investigation is to explore some new aspects of Ostrowski’s type integral inequalities implementing the generalized Jensen–Mercer inequality established for generalized s-convexity in fractal space. To proceed further with this task, we construct a new generalized integral [...] Read more.

The main idea of the current investigation is to explore some new aspects of Ostrowski’s type integral inequalities implementing the generalized Jensen–Mercer inequality established for generalized s-convexity in fractal space. To proceed further with this task, we construct a new generalized integral equality for first-order local differentiable functions, which will serve as an auxiliary result to restore some new bounds for Ostrowski inequality. We establish our desired results by employing the equality, some renowned generalized integral inequalities like Hölder’s, power mean, Yang-Hölder’s, bounded characteristics of the functions and considering generalized s-convexity characteristics of functions. Also, in support of our main findings, we deliver specific applications to means, and numerical integration and graphical visualization are also presented here. Full article

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

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20 pages, 2744 KiB

Open AccessArticle

Highly Efficient Numerical Algorithm for Nonlinear Space Variable-Order Fractional Reaction–Diffusion Models

by Muhammad Yousuf and Shahzad Sarwar

Fractal Fract. 2023, 7(9), 688; https://doi.org/10.3390/fractalfract7090688 - 15 Sep 2023

Cited by 1 | Viewed by 1036

Abstract

In this paper, we present a new highly efficient numerical algorithm for nonlinear variable-order space fractional reaction–diffusion equations. The algorithm is based on a new method developed by using the Gaussian quadrature pole rational approximation. A splitting technique is used to address the [...] Read more.

In this paper, we present a new highly efficient numerical algorithm for nonlinear variable-order space fractional reaction–diffusion equations. The algorithm is based on a new method developed by using the Gaussian quadrature pole rational approximation. A splitting technique is used to address the issues related to computational efficiency and the stability of the method. Two linear systems need to be solved using the same real-valued discretization matrix. The stability and convergence of the method are discussed analytically and demonstrated through numerical experiments by solving test problems from the literature. The variable-order diffusion effects on the solution profiles are illustrated through graphs. Finally, numerical experiments demonstrate the superiority of the presented method in terms of computational efficiency, accuracy, and reliability. Full article

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19 pages, 365 KiB

Open AccessArticle

Hermite–Hadamard-Type Inequalities via Caputo–Fabrizio Fractional Integral for h-Godunova–Levin and (h₁, h₂)-Convex Functions

by Waqar Afzal, Mujahid Abbas, Waleed Hamali, Ali M. Mahnashi and M. De la Sen

Fractal Fract. 2023, 7(9), 687; https://doi.org/10.3390/fractalfract7090687 - 15 Sep 2023

Cited by 8 | Viewed by 1168

Abstract

This note generalizes several existing results related to Hermite–Hadamard inequality using h-Godunova–Levin and

(h_{1}, h_{2})

-convex functions using a fractional integral operator associated with the Caputo–Fabrizio fractional derivative. This study uses a non-singular kernel and constructs some [...] Read more.

This note generalizes several existing results related to Hermite–Hadamard inequality using h-Godunova–Levin and

(h_{1}, h_{2})

-convex functions using a fractional integral operator associated with the Caputo–Fabrizio fractional derivative. This study uses a non-singular kernel and constructs some new theorems associated with fractional order integrals. Furthermore, we demonstrate that the obtained results are a generalization of the existing ones. To demonstrate the correctness of these results, we developed a few interesting non-trivial examples. Finally, we discuss some applications of our findings associated with special means. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)

14 pages, 9466 KiB

Open AccessArticle

Fractal Quasi-Coulomb Crystals in Ion Trap with Cantor Dust Electrode Configuration

by Semyon Rudyi, Andrei Ivanov and Dmitrii Shcherbinin

Fractal Fract. 2023, 7(9), 686; https://doi.org/10.3390/fractalfract7090686 - 15 Sep 2023

Cited by 3 | Viewed by 1226

Abstract

We propose a new concept of fractal quasi-Coulomb crystals. We have shown that self-similar quasi-Coulomb crystals can be formed in surface electrodynamic traps with the Cantor Dust electrode configuration. Quasi-Coulomb crystal fractal dimension appears to depend on the electrode parameters. We have identified [...] Read more.

We propose a new concept of fractal quasi-Coulomb crystals. We have shown that self-similar quasi-Coulomb crystals can be formed in surface electrodynamic traps with the Cantor Dust electrode configuration. Quasi-Coulomb crystal fractal dimension appears to depend on the electrode parameters. We have identified the conditions for transforming trivial quasi-Coulomb crystals into self-similar crystals and described the features of forming 25 Ca+ self-similar quasi-Coulomb crystals. The local potential well depth and width have been shown to take a discrete value dependent on the distance from the electrode surface. Ions inside the crystals studied possess varied translational secular frequencies. We believe that the extraordinary properties of self-similar quasi-Coulomb crystals may contribute to the new prospects within levitated optomechanics, quantum computing and simulation. Full article

(This article belongs to the Special Issue Fractal Geometries with Applications in Materials)

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10 pages, 295 KiB

Open AccessArticle

Existence of Positive Solutions to Boundary Value Problems with Mixed Riemann–Liouville and Quantum Fractional Derivatives

by Nemat Nyamoradi, Sotiris K. Ntouyas and Jessada Tariboon

Fractal Fract. 2023, 7(9), 685; https://doi.org/10.3390/fractalfract7090685 - 15 Sep 2023

Cited by 1 | Viewed by 1086

Abstract

In this paper, by using the Leggett–Williams fixed-point theorem, we study the existence of positive solutions to fractional differential equations with mixed Riemann–Liouville and quantum fractional derivatives. To prove the effectiveness of our main result, we investigate an interesting example. Full article

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

14 pages, 407 KiB

Open AccessArticle

An Enhanced Numerical Iterative Method for Expanding the Attraction Basins When Computing Matrix Signs of Invertible Matrices

by Lei Shi, Malik Zaka Ullah, Hemant Kumar Nashine, Monairah Alansari and Stanford Shateyi

Fractal Fract. 2023, 7(9), 684; https://doi.org/10.3390/fractalfract7090684 - 14 Sep 2023

Cited by 1 | Viewed by 1123

Abstract

The computation of the sign function of a matrix plays a crucial role in various mathematical applications. It provides a matrix-valued mapping that determines the sign of each eigenvalue of a nonsingular matrix. In this article, we present a novel iterative algorithm designed [...] Read more.

The computation of the sign function of a matrix plays a crucial role in various mathematical applications. It provides a matrix-valued mapping that determines the sign of each eigenvalue of a nonsingular matrix. In this article, we present a novel iterative algorithm designed to efficiently calculate the sign of an invertible matrix, emphasizing the enlargement of attraction basins. The proposed solver exhibits convergence of order four, making it highly efficient for a wide range of matrices. Furthermore, the method demonstrates global convergence properties. We validate the theoretical outcomes through numerical experiments, which confirm the effectiveness and efficiency of our proposed algorithm. Full article

(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition)

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12 pages, 305 KiB

Open AccessArticle

On the Generalization of Ostrowski-Type Integral Inequalities via Fractional Integral Operators with Application to Error Bounds

by Gauhar Rahman, Miguel Vivas-Cortez, Çetin Yildiz, Muhammad Samraiz, Shahid Mubeen and Mansour F. Yassen

Fractal Fract. 2023, 7(9), 683; https://doi.org/10.3390/fractalfract7090683 - 14 Sep 2023

Viewed by 2021

Abstract

The Ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The Ostrowski’s type inequality is frequently used to investigate errors in numerical quadrature rules and computations. In this work, Ostrowski-type inequality is demonstrated using the generalized fractional integral [...] Read more.

The Ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The Ostrowski’s type inequality is frequently used to investigate errors in numerical quadrature rules and computations. In this work, Ostrowski-type inequality is demonstrated using the generalized fractional integral operators. From an application perspective, we present the bounds of the fractional Hadamard inequalities. The results that are being presented involve a number of fractional inequalities that are already known and have been published. Full article

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

19 pages, 3774 KiB

Open AccessArticle

Synchronization of Fractional-Order Delayed Neural Networks Using Dynamic-Free Adaptive Sliding Mode Control

by Majid Roohi, Chongqi Zhang, Mostafa Taheri and Andreas Basse-O’Connor

Fractal Fract. 2023, 7(9), 682; https://doi.org/10.3390/fractalfract7090682 - 13 Sep 2023

Cited by 17 | Viewed by 1653

Abstract

In this work, a dynamic-free adaptive sliding mode control (adaptive-SMC) methodology for the synchronization of a specific class of chaotic delayed fractional-order neural network systems in the presence of input saturation is proposed. By incorporating the frequency distributed model (FDM) and the fractional [...] Read more.

In this work, a dynamic-free adaptive sliding mode control (adaptive-SMC) methodology for the synchronization of a specific class of chaotic delayed fractional-order neural network systems in the presence of input saturation is proposed. By incorporating the frequency distributed model (FDM) and the fractional version of the Lyapunov stability theory, a dynamic-free adaptive SMC methodology is designed to effectively overcome the inherent chaotic behavior exhibited by the delayed FONNSs to achieve synchronization. Notably, the decoupling of the control laws from the nonlinear/linear dynamical components of the system is ensured, taking advantage of the norm-boundedness property of the states in chaotic systems. The effectiveness of the suggested adaptive-SMC method for chaos synchronization in delayed fractional-order Hopfield neural network systems is validated through numerical simulations, demonstrating its robustness and efficiency. The proposed dynamic-free adaptive-SMC approach, incorporating the FDM and fractional Lyapunov stability theorem, offers a promising solution for synchronizing chaotic delayed FONNSs with input saturation, with potential applications in various domains requiring synchronization of such systems. Full article

(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition)

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17 pages, 6522 KiB

Open AccessArticle

Geomechanical Properties of Deep-Sea Pore-Filled Methane Hydrate-Bearing Soils at Critical State Using DEM Analysis

by Jie He, Tao Li and Yi Rui

Fractal Fract. 2023, 7(9), 681; https://doi.org/10.3390/fractalfract7090681 - 12 Sep 2023

Cited by 1 | Viewed by 1183

Abstract

The recognition of the geomechanical properties of methane hydrate-bearing soil (MHBS) is crucial to exploring energy resources. The paper presents the mechanical properties of a pore-filled MHBS at a critical state using the distinct element method (DEM). The pore-filled MHBS was simulated as [...] Read more.

The recognition of the geomechanical properties of methane hydrate-bearing soil (MHBS) is crucial to exploring energy resources. The paper presents the mechanical properties of a pore-filled MHBS at a critical state using the distinct element method (DEM). The pore-filled MHBS was simulated as cemented MH agglomerates to fill the soil pores at varying levels of methane hydration (MH) saturation. A group of triaxial compression (TC) tests were conducted, subjecting MHBS samples to varying effective confining pressures (ECPs). The mechanical behaviors of a pore-filled MHBS were analyzed, as it experienced significant strains leading to a critical state. The findings reveal that the proposed DEM successfully captures the qualitative geomechanical properties of MHBS. As MH saturation increases, the shear strength of MHBS generally rises. Moreover, higher ECPs result in increased shear strength and volumetric contraction. The peak shear strength of MHBS increases with rising MH saturation, while the residual deviator stress remains mainly unchanged at a critical state. There is a good correlation between fabric changes of the MHBS with variations in principal stresses and principal strains. With increasing axial strain, the coordination number (CN) and mechanical coordination number (MCN) increase to peak values as the values of MH saturation and ECPs increase, and reach a stable value at a larger axial strain. Full article

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13 pages, 617 KiB

Open AccessArticle

Option Pricing with Fractional Stochastic Volatilities and Jumps

by Sumei Zhang, Hongquan Yong and Haiyang Xiao

Fractal Fract. 2023, 7(9), 680; https://doi.org/10.3390/fractalfract7090680 - 11 Sep 2023

Cited by 2 | Viewed by 1492

Abstract

Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities [...] Read more.

Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities with mixed-exponential jumps. The characteristic function of the log-return is expressed in terms of the solution of two-dimensional fractional Riccati equations of which closed-form solution does not exist. To obtain the explicit characteristic function, we approximate the pricing model by a semimartingale and convert fractional Riccati equations into a classic PDE. By the multi-dimensional Feynman-Kac theorem and the affine structure of the approximate model, we obtain the solution of the PDE with which the explicit characteristic function and its cumulants are derived. Based on the derived characteristic function and Fourier cosine series expansion, we obtain approximate European options prices. By differential evolution algorithm, we calibrate our approximate model and its two nested models to S&P 500 index options and obtain optimal parameter estimates of these models. Numerical results demonstrate the pricing method is fast and accurate. Empirical results demonstrate our approximate model fits the market best among the three models. Full article

(This article belongs to the Special Issue Application of Fractional Partial Differential Equations in Computational Finance)

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16 pages, 3925 KiB

Open AccessArticle

Sliding-Window TD-FrFT Algorithm for High-Precision Ranging of LFM Signals in the Presence of Impulse Noise

by Bo Xiao, Xuelian Liu, Chunyang Wang, Yuchao Wang and Tingsheng Huang

Fractal Fract. 2023, 7(9), 679; https://doi.org/10.3390/fractalfract7090679 - 11 Sep 2023

Cited by 2 | Viewed by 1170

Abstract

To address the performance degradation of the conventional linear frequency modulation signal ranging method in the presence of impulse noise, this paper proposes a novel technique that integrates a sliding-window tracking differentiator (TD) with the fractional Fourier transform (FrFT) ranging method. First, the [...] Read more.

To address the performance degradation of the conventional linear frequency modulation signal ranging method in the presence of impulse noise, this paper proposes a novel technique that integrates a sliding-window tracking differentiator (TD) with the fractional Fourier transform (FrFT) ranging method. First, the sliding-window TD filtering algorithm is used to suppress the noise in the echo. Subsequently, the filtered signal is subjected to FrFT to calculate the time delay based on the difference in the peak point positions in the fractional domain for realizing target ranging. The simulation results show that the proposed method can effectively suppress impulse noise of different intensities and achieve an accurate and robust ranging of the target. Full article

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

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20 pages, 543 KiB

Open AccessArticle

Finite-Time Synchronization for Stochastic Fractional-Order Memristive BAM Neural Networks with Multiple Delays

by Lili Chen, Minghao Gong, Yanfeng Zhao and Xin Liu

Fractal Fract. 2023, 7(9), 678; https://doi.org/10.3390/fractalfract7090678 - 10 Sep 2023

Cited by 3 | Viewed by 1175

Abstract

This paper studies the finite-time synchronization problem of fractional-order stochastic memristive bidirectional associative memory neural networks (MBAMNNs) with discontinuous jumps. A novel criterion for finite-time synchronization is obtained by utilizing the properties of quadratic fractional-order Gronwall inequality with time delay and the comparison [...] Read more.

This paper studies the finite-time synchronization problem of fractional-order stochastic memristive bidirectional associative memory neural networks (MBAMNNs) with discontinuous jumps. A novel criterion for finite-time synchronization is obtained by utilizing the properties of quadratic fractional-order Gronwall inequality with time delay and the comparison principle. This criterion provides a new approach to analyze the finite-time synchronization problem of neural networks with stochasticity. Finally, numerical simulations are provided to demonstrate the effectiveness and superiority of the obtained results. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)

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19 pages, 4831 KiB

Open AccessArticle

Optimal Fractional-Order Controller for the Voltage Stability of a DC Microgrid Feeding an Electric Vehicle Charging Station

by Sherif A. Zaid, Abualkasim Bakeer, Hani Albalawi, Aadel M. Alatwi, Hossam AbdelMeguid and Ahmed M. Kassem

Fractal Fract. 2023, 7(9), 677; https://doi.org/10.3390/fractalfract7090677 - 9 Sep 2023

Cited by 7 | Viewed by 1522

Abstract

Charging stations are regarded as the cornerstone of electric vehicle (EV) development and utilization. Electric vehicle charging stations (EVCSs) are now energized via standalone microgrids that utilize renewable energy sources and reduce the stress on the utility grid. However, the control and energy [...] Read more.

Charging stations are regarded as the cornerstone of electric vehicle (EV) development and utilization. Electric vehicle charging stations (EVCSs) are now energized via standalone microgrids that utilize renewable energy sources and reduce the stress on the utility grid. However, the control and energy management of EVCSs are challenging tasks because they are nonlinear and time-varying. This study suggests a fractional-order proportional integral (FOPI) controller to improve the performance and energy management of a standalone EVCS microgrid. The microgrid is supplied mainly by photovoltaic (PV) energy and utilizes a battery as an energy storage system (ESS). The FOPI’s settings are best created utilizing the grey wolf optimization (GWO) method to attain the highest performance possible. The grey wolf is run for 100 iterations using 20 wolves. In addition, after 80 iterations for the specified goal function, the GWO algorithm almost discovers the ideal values. For changes in solar insolation, the performance of the proposed FOPI controller is compared with that of a traditional PI controller. The Matlab/Simulink platform models and simulates the EVCS’s microgrid. The results demonstrate that the suggested FOPI controller significantly improves the transient responsiveness of the EVCS performance compared to the standard PI controller. Despite all PV insolation disruptions, the EV battery continues to charge while the ESS battery precisely stores and balances PV energy changes. The results support the suggested FOPI control’s robustness to parameter mismatches. The microgrid’s efficiency fluctuations with the insolation level and state of charge of the EV battery are discussed. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)

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18 pages, 20234 KiB

Open AccessArticle

Fractal Characterization of Multiscale Fracture Network Distribution in Dolomites: Outcrop Analogue of Subsurface Reservoirs

by Ivica Pavičić, Željko Duić, Anja Vrbaški and Ivan Dragičević

Fractal Fract. 2023, 7(9), 676; https://doi.org/10.3390/fractalfract7090676 - 7 Sep 2023

Cited by 3 | Viewed by 1625

Abstract

Fractured aquifers, especially dolomites, are important hydrocarbon reservoirs and sources of thermal and groundwater in many parts of the world, especially in the Alpine-Dinaric-Carpathian region. The most dominant porosity type is fracture porosity, which acts as the preferential fluid pathway in the subsurface, [...] Read more.

Fractured aquifers, especially dolomites, are important hydrocarbon reservoirs and sources of thermal and groundwater in many parts of the world, especially in the Alpine-Dinaric-Carpathian region. The most dominant porosity type is fracture porosity, which acts as the preferential fluid pathway in the subsurface, thus strongly controlling fluid flow. Outcrops provide valuable information for the characterization of fracture networks. Dolomite rock properties and structural and diagenetic processes result in fractured systems that can be considered fractals. The fracture network was analyzed on 14 vertical outcrops in 35 digitized photographs. The values of the fractal dimensions varied slightly by the software and method used, but the trends were consistent, which confirms that all methods are valid. Small values of fractal dimension indicate the dominance of a few small or large fractures, and high values of fractal dimension result from a combination of large numbers of small fractures accompanied by a few large fractures. The mean value of the fractal dimension for analyzed fracture networks was 1.648. The results indicate that the fracture network of the Upper Triassic dolomites can be approximated by fractal distribution and can be considered a natural fractal, and values can be extrapolated to higher and lower scales (1D and 3D). Full article

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

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20 pages, 354 KiB

Open AccessArticle

Coefficient Inequalities of q-Bi-Univalent Mappings Associated with q-Hyperbolic Tangent Function

by Timilehin Gideon Shaba, Serkan Araci, Jong-Suk Ro, Fairouz Tchier, Babatunde Olufemi Adebesin and Saira Zainab

Fractal Fract. 2023, 7(9), 675; https://doi.org/10.3390/fractalfract7090675 - 7 Sep 2023

Cited by 3 | Viewed by 1144

Abstract

The present study introduces a new family of analytic functions by utilizing the q-derivative operator and the q-version of the hyperbolic tangent function. We find certain inequalities, including the coefficient bounds, second Hankel determinants, and Fekete–Szegö inequalities, for this novel family [...] Read more.

The present study introduces a new family of analytic functions by utilizing the q-derivative operator and the q-version of the hyperbolic tangent function. We find certain inequalities, including the coefficient bounds, second Hankel determinants, and Fekete–Szegö inequalities, for this novel family of bi-univalent functions. It is worthy of note that almost all the results are sharp, and their corresponding extremal functions are presented. In addition, some special cases are demonstrated to show the validity of our findings. Full article

21 pages, 6876 KiB

Open AccessArticle

Fractional-Order Total Variation Geiger-Mode Avalanche Photodiode Lidar Range-Image Denoising Algorithm Based on Spatial Kernel Function and Range Kernel Function

by Xuyang Wei, Chunyang Wang, Da Xie, Kai Yuan, Xuelian Liu, Zihao Wang, Xinjian Wang and Tingsheng Huang

Fractal Fract. 2023, 7(9), 674; https://doi.org/10.3390/fractalfract7090674 - 7 Sep 2023

Viewed by 1168

Abstract

A Geiger-mode avalanche photodiode (GM-APD) laser radar range image has much noise when the signal-to-background ratios (SBRs) are low, making it difficult to recover the real target scene. In this paper, based on the GM-APD lidar denoising model of fractional-order total variation (FOTV), [...] Read more.

A Geiger-mode avalanche photodiode (GM-APD) laser radar range image has much noise when the signal-to-background ratios (SBRs) are low, making it difficult to recover the real target scene. In this paper, based on the GM-APD lidar denoising model of fractional-order total variation (FOTV), the spatial relationship and similarity relationship between pixels are obtained by using a spatial kernel function and range kernel function to optimize the fractional differential operator, and a new FOTV GM-APD lidar range-image denoising algorithm is designed. The lost information and range anomalous noise are suppressed while the target details and contour information are preserved. The Monte Carlo simulation and experimental results show that, under the same SBRs and statistical frame number, the proposed algorithm improves the target restoration degree by at least 5.11% and the peak signal-to-noise ratio (PSNR) by at least 24.6%. The proposed approach can accomplish the denoising of GM-APD lidar range images when SBRs are low. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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18 pages, 7937 KiB

Open AccessArticle

A New Hybrid Optimal Auxiliary Function Method for Approximate Solutions of Non-Linear Fractional Partial Differential Equations

by Rashid Ashraf, Rashid Nawaz, Osama Alabdali, Nicholas Fewster-Young, Ali Hasan Ali, Firas Ghanim and Alina Alb Lupaş

Fractal Fract. 2023, 7(9), 673; https://doi.org/10.3390/fractalfract7090673 - 7 Sep 2023

Cited by 3 | Viewed by 1988

Abstract

This study uses the optimal auxiliary function method to approximate solutions for fractional-order non-linear partial differential equations, utilizing Riemann–Liouville’s fractional integral and the Caputo derivative. This approach eliminates the need for assumptions about parameter magnitudes, offering a significant advantage. We validate our approach [...] Read more.

This study uses the optimal auxiliary function method to approximate solutions for fractional-order non-linear partial differential equations, utilizing Riemann–Liouville’s fractional integral and the Caputo derivative. This approach eliminates the need for assumptions about parameter magnitudes, offering a significant advantage. We validate our approach using the time-fractional Cahn–Hilliard, fractional Burgers–Poisson, and Benjamin–Bona–Mahony–Burger equations. Comparative testing shows that our method outperforms new iterative, homotopy perturbation, homotopy analysis, and residual power series methods. These examples highlight our method’s effectiveness in obtaining precise solutions for non-linear fractional differential equations, showcasing its superiority in accuracy and consistency. We underscore its potential for revealing elusive exact solutions by demonstrating success across various examples. Our methodology advances fractional differential equation research and equips practitioners with a tool for solving non-linear equations. A key feature is its ability to avoid parameter assumptions, enhancing its applicability to a broader range of problems and expanding the scope of problems addressable using fractional calculus techniques. Full article

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

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34 pages, 4406 KiB

Open AccessArticle

New Cascaded 1+PII2D/FOPID Load Frequency Controller for Modern Power Grids including Superconducting Magnetic Energy Storage and Renewable Energy

by Fayez F. M. El-Sousy, Mokhtar Aly, Mohammed H. Alqahtani, Ali S. Aljumah, Sulaiman Z. Almutairi and Emad A. Mohamed

Fractal Fract. 2023, 7(9), 672; https://doi.org/10.3390/fractalfract7090672 - 5 Sep 2023

Cited by 5 | Viewed by 1730

Abstract

Having continuous decrease in inertia and being sensitive to load/generation variation are considered crucial challenging problems for modern power grids. The main cause of these problems is the increased penetration capacities of renewables. An unbalanced load with generation power largely affects grids’ frequency [...] Read more.

Having continuous decrease in inertia and being sensitive to load/generation variation are considered crucial challenging problems for modern power grids. The main cause of these problems is the increased penetration capacities of renewables. An unbalanced load with generation power largely affects grids’ frequency and voltage profiles. Load frequency control (LFC) mechanisms are extensively presented to solve these problems. In the literature, LFC methods are still lacking in dealing with system uncertainty, parameter variation, structure changes, and/or disturbance rejection. Therefore, this paper proposes an improved LFC methodology using the hybrid one plus proportional integral double-integral derivative (1+PII2D) cascaded with fractional order proportional-integral-derivative (FOPID), namely, the proposed 1+PII2D/FOPID controller. The contribution of superconducting magnetic energy storage devices (SMES) is considered in the proposed design, also considering hybrid high-voltage DC and AC transmission lines (hybrid HVDC/HVAC). An optimized design of proposed 1+PII2D/FOPID controller is proposed using a new application of the recently presented powerful artificial rabbits optimizers (ARO) algorithm. Various performance comparisons, system changes, parameter uncertainties, and load/generation profiles and changes are considered in the proposed case study. The results proved superior regulation of frequency using proposed 1+PII2D/FOPID control and the ARO optimum parameters. Full article

(This article belongs to the Section Engineering)

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11 pages, 958 KiB

Open AccessArticle

An Accurate Approach to Simulate the Fractional Delay Differential Equations

by Mohamed Adel, Mohamed M. Khader, Salman Algelany and Khaled Aldwoah

Fractal Fract. 2023, 7(9), 671; https://doi.org/10.3390/fractalfract7090671 - 5 Sep 2023

Cited by 3 | Viewed by 1461

Abstract

The fractional Legendre polynomials (FLPs) that we present as an effective method for solving fractional delay differential equations (FDDEs) are used in this work. The Liouville–Caputo sense is used to characterize fractional derivatives. This method uses the spectral collocation technique based on FLPs. [...] Read more.

The fractional Legendre polynomials (FLPs) that we present as an effective method for solving fractional delay differential equations (FDDEs) are used in this work. The Liouville–Caputo sense is used to characterize fractional derivatives. This method uses the spectral collocation technique based on FLPs. The proposed method converts FDDEs into a set of algebraic equations. We lay out a study of the convergence analysis and figure out the upper bound on error for the approximate solution. Examples are provided to demonstrate the precision of the suggested approach. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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16 pages, 508 KiB

Open AccessArticle

Advanced Mathematical Approaches in Psycholinguistic Data Analysis: A Methodological Insight

by Cecilia Castro, Víctor Leiva, Maria do Carmo Lourenço-Gomes and Ana Paula Amorim

Fractal Fract. 2023, 7(9), 670; https://doi.org/10.3390/fractalfract7090670 - 5 Sep 2023

Cited by 1 | Viewed by 1583

Abstract

In the evolving landscape of psycholinguistic research, this study addresses the inherent complexities of data through advanced analytical methodologies, including permutation tests, bootstrap confidence intervals, and fractile or quantile regression. The methodology and philosophy of our approach deeply resonate with fractal and fractional [...] Read more.

In the evolving landscape of psycholinguistic research, this study addresses the inherent complexities of data through advanced analytical methodologies, including permutation tests, bootstrap confidence intervals, and fractile or quantile regression. The methodology and philosophy of our approach deeply resonate with fractal and fractional concepts. Responding to the skewed distributions of data, which are observed in metrics such as reading times, time-to-response, and time-to-submit, our analysis highlights the nuanced interplay between time-to-response and variables like lists, conditions, and plausibility. A particular focus is placed on the implausible sentence response times, showcasing the precision of our chosen methods. The study underscores the profound influence of individual variability, advocating for meticulous analytical rigor in handling intricate and complex datasets. Drawing inspiration from fractal and fractional mathematics, our findings emphasize the broader potential of sophisticated mathematical tools in contemporary research, setting a benchmark for future investigations in psycholinguistics and related disciplines. Full article

(This article belongs to the Special Issue Fractional Models and Statistical Applications)

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13 pages, 1824 KiB

Open AccessArticle

Distribution and Fractal Characteristics of Outdoor Particles in High-Rise Buildings Based on Fractal Theory

by Fuquan Liu, Tao Yu, Wenjun Leng and Xin Zhang

Fractal Fract. 2023, 7(9), 669; https://doi.org/10.3390/fractalfract7090669 - 5 Sep 2023

Cited by 2 | Viewed by 1193

Abstract

It is of great significance to understand the particle distribution characteristics at different heights to effectively control particle pollution. Based on fractal theory, the fractal dimension of outdoor particles in a high-rise building in Xi’an and its relationship with the concentration of particles [...] Read more.

It is of great significance to understand the particle distribution characteristics at different heights to effectively control particle pollution. Based on fractal theory, the fractal dimension of outdoor particles in a high-rise building in Xi’an and its relationship with the concentration of particles with different particle sizes are discussed and analyzed in this paper. The results indicate that the atmosphere in Xi’an is mainly composed of fine particles and that the average proportion of particles ranging from 0 to 1.0 µm is approximately 99.885% of the total particulates. The fractal dimension of particles in the atmosphere at different heights ranges from 5.014 to 5.764, with an average fractal dimension of 5.456. In summer, the fractal dimension of the outdoor particles on the 17th floor was the largest, at 5.764. The fractal dimension in summer is relatively high, being 0.158 higher than that in winter on average. The larger the fractal dimension, the higher the proportion of fine particles. In addition, the fractal dimension can characterize the adsorption of toxic and harmful gases by particles well. It provides parameter support for understanding particle distribution and the effective control of atmospheric particles at different heights and application values. Full article

(This article belongs to the Special Issue Applications of Fractal Approaches in Seismology and Atmospheric Science)

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22 pages, 3001 KiB

Open AccessArticle

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

by Abdelfattah Mustafa, Ola Ragb, Mohamed Salah, Reda S. Salama and Mokhtar Mohamed

Fractal Fract. 2023, 7(9), 668; https://doi.org/10.3390/fractalfract7090668 - 4 Sep 2023

Cited by 1 | Viewed by 1114

Abstract

The aim of this study is to utilize a differential quadrature method with various kernels, such as Lagrange interpolation and discrete singular convolution, to tackle problems related to the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The governing equation for [...] Read more.

The aim of this study is to utilize a differential quadrature method with various kernels, such as Lagrange interpolation and discrete singular convolution, to tackle problems related to the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The governing equation for convection and diffusion depends on both spatial and transient factors. By using the block marching technique, we transform these equations into an algebraic system using differential quadrature methods and the Caputo-type fractional operator. Next, we develop a MATLAB program that generates code capable of solving the fractional convection–diffusion equation in (1+2) dimensions for each shape function. Our goal is to ensure that our methods are reliable, accurate, efficient, and capable of convergence. To achieve this, we conduct two experiments, comparing the numerical and graphical results with both analytical and numerical solutions. Additionally, we evaluate the accuracy of our findings using the L_∞ error. Our tests show that the differential quadrature method, which relies mainly on the discrete singular convolution shape function, is a highly effective numerical approach for fractional convective diffusion problems. It offers superior accuracy, faster convergence, and greater reliability than other techniques. Furthermore, we study the impact of fractional order derivatives, velocity, and positive diffusion parameters on the results. Full article

(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)

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Fractal Fract., Volume 7, Issue 9 (September 2023) – 59 articles

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