Fractal and Fractional

14 pages, 317 KiB

Open AccessArticle

On the Approximate Solution of the Cauchy Problem in a Multidimensional Unbounded Domain

by Davron Aslonqulovich Juraev, Ali Shokri and Daniela Marian

Fractal Fract. 2022, 6(7), 403; https://doi.org/10.3390/fractalfract6070403 - 21 Jul 2022

Cited by 3 | Viewed by 1376

In this paper, the Carleman matrix is constructed, and based on it we found explicitly a regularized solution of the Cauchy problem for the matrix factorization of the Helmholtz equation in a multidimensional unbounded domain in [...] Read more.

In this paper, the Carleman matrix is constructed, and based on it we found explicitly a regularized solution of the Cauchy problem for the matrix factorization of the Helmholtz equation in a multidimensional unbounded domain in

R^{m}, (m = 2 k, k \geq 2)

. The corresponding theorems on the stability of the solution of problems are proved. Full article

(This article belongs to the Special Issue Dynamical Systems and Their Applications (DSTA) — in Memory of Prof. Dr. José A. Tenreiro Machado)

16 pages, 8558 KiB

Open AccessArticle

Minkowski’s Loop Fractal Antenna Dedicated to Sixth Generation (6G) Communication

by Maria-Alexandra Paun, Mihai-Virgil Nichita, Vladimir-Alexandru Paun and Viorel-Puiu Paun

Fractal Fract. 2022, 6(7), 402; https://doi.org/10.3390/fractalfract6070402 - 21 Jul 2022

Cited by 9 | Viewed by 2318

Abstract

In this study, we will discuss the engineering construction of a special sixth generation (6G) antenna, based on the fractal called Minkowski’s loop. The antenna has the shape of this known fractal, set at four iterations, to obtain maximum performance. The frequency bands [...] Read more.

In this study, we will discuss the engineering construction of a special sixth generation (6G) antenna, based on the fractal called Minkowski’s loop. The antenna has the shape of this known fractal, set at four iterations, to obtain maximum performance. The frequency bands for which this 6G fractal antenna was designed in the current paper are 170 GHz to 260 GHz (WR-4) and 110 GHz to 170 GHz (WR-6), respectively. The three resonant frequencies, optimally used, are equal to 140 GHz (WR-6) for the first, 182 GHz (WR-4) for the second and 191 GHz (WR-4) for the third. For these frequencies the electromagnetic behaviors of fractal antennas and their graphical representations are highlighted. Full article

(This article belongs to the Special Issue Nonlinear Dynamics in Complex Systems via Fractals and Fractional Calculus)

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20 pages, 895 KiB

Open AccessArticle

Numerical Solution of Time Fractional Black–Scholes Model Based on Legendre Wavelet Neural Network with Extreme Learning Machine

by Xiaoning Zhang, Jianhui Yang and Yuxin Zhao

Fractal Fract. 2022, 6(7), 401; https://doi.org/10.3390/fractalfract6070401 - 21 Jul 2022

Cited by 3 | Viewed by 2074

Abstract

In this paper, the Legendre wavelet neural network with extreme learning machine is proposed for the numerical solution of the time fractional Black–Scholes model. In this way, the operational matrix of the fractional derivative based on the two-dimensional Legendre wavelet is derived and [...] Read more.

In this paper, the Legendre wavelet neural network with extreme learning machine is proposed for the numerical solution of the time fractional Black–Scholes model. In this way, the operational matrix of the fractional derivative based on the two-dimensional Legendre wavelet is derived and employed to solve the European options pricing problem. This scheme converts this problem into the calculation of a set of algebraic equations. The Legendre wavelet neural network is constructed; meanwhile, the extreme learning machine algorithm is adopted to speed up the learning rate and avoid the over-fitting problem. In order to evaluate the performance of this scheme, a comparative study with the implicit differential method is constructed to validate its feasibility and effectiveness. Experimental results illustrate that this scheme offers a satisfactory numerical solution compared to the benchmark method. Full article

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

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18 pages, 2727 KiB

Open AccessArticle

Some Properties of Fractional Cumulative Residual Entropy and Fractional Conditional Cumulative Residual Entropy

by Keqiang Dong, Shushu Li and Dan Li

Fractal Fract. 2022, 6(7), 400; https://doi.org/10.3390/fractalfract6070400 - 21 Jul 2022

Cited by 2 | Viewed by 2206

Abstract

Fractional cumulative residual entropy is a powerful tool for the analysis of complex systems. In this paper, we first provide some properties of fractional cumulative residual entropy (FCRE). Secondly, we generate cumulative residual entropy (CRE) to the case of conditional entropy, named fractional [...] Read more.

Fractional cumulative residual entropy is a powerful tool for the analysis of complex systems. In this paper, we first provide some properties of fractional cumulative residual entropy (FCRE). Secondly, we generate cumulative residual entropy (CRE) to the case of conditional entropy, named fractional conditional cumulative residual entropy (FCCRE), and introduce some properties. Then, we verify the validity of these properties with randomly generated sequences that follow different distributions. Moreover, we give the definition of empirical fractional conditional accumulative residual entropy and prove that it can be used to approximate FCCRE. Finally, the empirical analysis of the aero-engine gas path data is carried out. The results show that FCRE and FCCRE can effectively capture complex information in the gas path system. Full article

(This article belongs to the Special Issue Fractional Behaviors Analysis and Modelling)

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11 pages, 3057 KiB

Open AccessArticle

Exact Solutions of the Nonlinear Modified Benjamin-Bona-Mahony Equation by an Analytical Method

by Trad Alotaibi and Ali Althobaiti

Fractal Fract. 2022, 6(7), 399; https://doi.org/10.3390/fractalfract6070399 - 20 Jul 2022

Cited by 8 | Viewed by 1658

Abstract

The current manuscript investigates the exact solutions of the modified Benjamin-Bona-Mahony (BBM) equation. Due to its efficiency and simplicity, the modified auxiliary equation method is adopted to solve the problem under consideration. As a result, a variety of the exact wave solutions of [...] Read more.

The current manuscript investigates the exact solutions of the modified Benjamin-Bona-Mahony (BBM) equation. Due to its efficiency and simplicity, the modified auxiliary equation method is adopted to solve the problem under consideration. As a result, a variety of the exact wave solutions of the modified BBM equation are obtained. Furthermore, the findings of the current study remain strong since Jacobi function solutions generate hyperbolic function solutions and trigonometric function solutions, as liming cases of interest. Some of the obtained solutions are illustrated graphically using appropriate values for the parameters. Full article

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

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21 pages, 374 KiB

Open AccessArticle

On the Lower and Upper Box Dimensions of the Sum of Two Fractal Functions

by Binyan Yu and Yongshun Liang

Fractal Fract. 2022, 6(7), 398; https://doi.org/10.3390/fractalfract6070398 - 20 Jul 2022

Cited by 12 | Viewed by 1667

Abstract

Let f and g be two continuous functions. In the present paper, we put forward a method to calculate the lower and upper Box dimensions of the graph of

f + g

by classifying all the subsequences tending to zero into different sets. [...] Read more.

Let f and g be two continuous functions. In the present paper, we put forward a method to calculate the lower and upper Box dimensions of the graph of

f + g

by classifying all the subsequences tending to zero into different sets. Using this method, we explore the lower and upper Box dimensions of the graph of

f + g

when the Box dimension of the graph of g is between the lower and upper Box dimensions of the graph of f. In this case, we prove that the upper Box dimension of the graph of

f + g

is just equal to the upper Box dimension of the graph of f. We also prove that the lower Box dimension of the graph of

f + g

could be an arbitrary number belonging to a certain interval. In addition, some other cases when the Box dimension of the graph of g is equal to the lower or upper Box dimensions of the graph of f have also been studied. Full article

13 pages, 5908 KiB

Open AccessArticle

A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme

by Darshana J. Prajapati, Shivam Rawat, Anita Tomar, Mohammad Sajid and R. C. Dimri

Fractal Fract. 2022, 6(7), 397; https://doi.org/10.3390/fractalfract6070397 - 19 Jul 2022

Cited by 8 | Viewed by 2831

Abstract

In this research, we look at the Julia set patterns that are linked to the entire transcendental function

f (z) = a e^{z^{n}} + b z + c

, where

a, b, c \in C

and [...] Read more.

In this research, we look at the Julia set patterns that are linked to the entire transcendental function

f (z) = a e^{z^{n}} + b z + c

, where

a, b, c \in C

and

n \geq 2

, using the Mann iterative scheme, and discuss their dynamical behavior. The sophisticated orbit structure of this function, whose Julia set encompasses the entire complex plane, is described using symbolic dynamics. We also present bifurcation diagrams of Julia sets generated using the proposed iteration and function, which altogether contain four parameters, and discuss the graphical analysis of bifurcation occurring in the family of this function. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

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Graphical abstract

14 pages, 2022 KiB

Open AccessArticle

Volume of Interest-Based Fractal Analysis of Huffaz’s Brain

by Iqbal Jamaludin, Mohd Zulfaezal Che Azemin, Mohd Izzuddin Mohd Tamrin and Abdul Halim Sapuan

Fractal Fract. 2022, 6(7), 396; https://doi.org/10.3390/fractalfract6070396 - 19 Jul 2022

Cited by 3 | Viewed by 1678

Abstract

The robust process in memorising the Quran is expected to cause neuroplasticity changes in the brain. To date, the analysis of neuroplasticity is limited in binary images because greyscale analysis requires the usage of more robust processing techniques. This research work aims to [...] Read more.

The robust process in memorising the Quran is expected to cause neuroplasticity changes in the brain. To date, the analysis of neuroplasticity is limited in binary images because greyscale analysis requires the usage of more robust processing techniques. This research work aims to explore and characterise the complexity of textual memorisation brain structures using fractal analysis between huffaz and non-huffaz applying global box-counting, global Fourier fractal dimension (FFD), and volume of interest (VOI)-based analysis. The study recruited 47 participants from IIUM Kuantan Campus. The huffaz group had their 18 months of systematic memorisation training. The brain images were acquired by using MRI. Global box-counting and FFD analysis were conducted on the brain. Magnetic resonance imaging (MRI) found no significant statistical difference between brains of huffaz and non-huffaz. VOI-based analysis found nine significant areas: two for box-counting analysis (angular gyrus and medial temporal gyrus), six for FFD analysis (BA20, BA30, anterior cingulate, fusiform gyrus, inferior temporal gyrus, and frontal lobe), and only a single area (BA33) showed significant volume differences between huffaz and non-huffaz. The results have highlighted the sensitivity of VOI-based analysis because of its local nature, as compared to the global analysis by box-counting and FFD. Full article

(This article belongs to the Special Issue Methods for Estimation of Fractal Dimension Based on Digital Images)

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16 pages, 371 KiB

Open AccessArticle

A Unified Inertial Iterative Approach for General Quasi Variational Inequality with Application

by Mohammad Akram and Mohammad Dilshad

Fractal Fract. 2022, 6(7), 395; https://doi.org/10.3390/fractalfract6070395 - 18 Jul 2022

Cited by 2 | Viewed by 1932

Abstract

In this paper, we design two inertial iterative methods involving one and two inertial steps for investigating a general quasi-variational inequality in a real Hilbert space. We establish an existence result and a non-trivial example is furnished to substantiate our theoretical findings. We [...] Read more.

In this paper, we design two inertial iterative methods involving one and two inertial steps for investigating a general quasi-variational inequality in a real Hilbert space. We establish an existence result and a non-trivial example is furnished to substantiate our theoretical findings. We discuss the convergence of the inertial iterative algorithms to approximate the solution of a general quasi-variational inequality. Finally, we apply an inertial iterative scheme with two inertial steps to investigate a delay differential equation. The results presented herein can be seen as substantial generalizations of some known results. Full article

(This article belongs to the Special Issue The Solutions of Partial Differential Equations and Recent Applications)

16 pages, 1976 KiB

Open AccessArticle

Forecasting the Volatility of the Stock Index with Deep Learning Using Asymmetric Hurst Exponents

by Poongjin Cho and Minhyuk Lee

Fractal Fract. 2022, 6(7), 394; https://doi.org/10.3390/fractalfract6070394 - 16 Jul 2022

Cited by 9 | Viewed by 4847

Abstract

The prediction of the stock price index is a challenge even with advanced deep-learning technology. As a result, the analysis of volatility, which has been widely studied in traditional finance, has attracted attention among researchers. This paper presents a new forecasting model that [...] Read more.

The prediction of the stock price index is a challenge even with advanced deep-learning technology. As a result, the analysis of volatility, which has been widely studied in traditional finance, has attracted attention among researchers. This paper presents a new forecasting model that combines asymmetric fractality and deep-learning algorithms to predict a one-day-ahead absolute return series, the proxy index of stock price volatility. Asymmetric Hurst exponents are measured to capture the asymmetric long-range dependence behavior of the S&P500 index, and recurrent neural network groups are applied. The results show that the asymmetric Hurst exponents have predictive power for one-day-ahead absolute return and are more effective in volatile market conditions. In addition, we propose a new two-stage forecasting model that predicts volatility according to the magnitude of volatility. This new model shows the best forecasting performance regardless of volatility. Full article

(This article belongs to the Special Issue Fractal and Multifractal Analysis in Financial Markets)

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13 pages, 1284 KiB

Open AccessArticle

Stability Analysis and Computational Interpretation of an Effective Semi Analytical Scheme for Fractional Order Non-Linear Partial Differential Equations

by Javed Iqbal, Khurram Shabbir and Liliana Guran

Fractal Fract. 2022, 6(7), 393; https://doi.org/10.3390/fractalfract6070393 - 16 Jul 2022

Cited by 5 | Viewed by 1893

Abstract

In this study we will check the stability of the semi analytical technique, the Laplace variational iteration (LVI) scheme, which is the combination of a variational iteration technique and the Laplace transform method. Then, we will apply it to solve some non-linear fractional [...] Read more.

In this study we will check the stability of the semi analytical technique, the Laplace variational iteration (LVI) scheme, which is the combination of a variational iteration technique and the Laplace transform method. Then, we will apply it to solve some non-linear fractional order partial differential equations. Since the Laplace transform cannot be applied to non-linear problems, the combination of the variational iteration technique with it will give a better and rapidly convergent sequence. Exact solutions may also exist, but we will show that the coupled technique is much better to approximate the exact solutions. The Caputo–Fabrizio fractional derivative will be used throughout the study. In addition, some possible implications of the results given here are connected with fixed point theory. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

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15 pages, 6979 KiB

Open AccessArticle

The Counterbalance of the Adverse Effect of Abrasion on the Properties of Concrete Incorporating Nano-SiO₂ and Polypropylene Fiber Based on Pore Structure Fractal Characteristics

by Kun Wang, Jinjun Guo, Peng Zhang and Qingxin Meng

Fractal Fract. 2022, 6(7), 392; https://doi.org/10.3390/fractalfract6070392 - 14 Jul 2022

Cited by 10 | Viewed by 1637

Abstract

Abrasion damage is a typical hydraulic structure failure and considerably impacts the durability of buildings. In severe circumstances, it can even prevent hydraulic structures from being used and operated normally. Thus, it is essential to research abrasion-resistant hydraulic systems that are more durable, [...] Read more.

Abrasion damage is a typical hydraulic structure failure and considerably impacts the durability of buildings. In severe circumstances, it can even prevent hydraulic structures from being used and operated normally. Thus, it is essential to research abrasion-resistant hydraulic systems that are more durable, inexpensive, safe, and ecologically friendly, given its unavoidable characteristics. In this context, five dosages of nano-SiO₂ and three dosages of fibers are selected to evaluate and analyze the modification effect of nano-SiO₂ and polypropylene fibers on the abrasion resistance of concrete. The evolution of the concrete properties was characterized based on the abrasion resistance strength. Moreover, the mineralogical composition and microstructure characterization were investigated through X-ray diffraction and scanning probe microscope. Mercury intrusion porosimetry was applied to determine the pore-structure parameters of concrete, such as pore-size distribution and the fractal characteristics. The results indicate that nano-SiO₂ improves the abrasion resistance of concrete by densifying the pore structure and promoting the formation of hydration products. Results reveal that the excessive dosage of fibers agglomerates in the concrete to form an unsubstantial pore structure due to poor dispersibility. The fractal dimension of the pore structure exhibits a close relationship with the abrasion resistance strength of concrete. The implications of these findings inform the design of abrasion and erosion resistance for hydraulic engineering structures. Full article

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15 pages, 4395 KiB

Open AccessArticle

Coupled Seepage Mechanics Model of Coal Containing Methane Based on Pore Structure Fractal Features

by Runsheng Lv, Yuchen Zhu, Xinyu Ma, Xiaoming Ni and Jiangang Ren

Fractal Fract. 2022, 6(7), 391; https://doi.org/10.3390/fractalfract6070391 - 14 Jul 2022

Cited by 3 | Viewed by 1498

Abstract

The paper applies fractal theory to the structure of fractal coal pores and calculates the fractal dimension and integrated fractal dimension for each pore section >100 nm, 100 nm > d > 5.25 nm, and <2 nm. In the experiment, we performed the [...] Read more.

The paper applies fractal theory to the structure of fractal coal pores and calculates the fractal dimension and integrated fractal dimension for each pore section >100 nm, 100 nm > d > 5.25 nm, and <2 nm. In the experiment, we performed the full stress–strain-seepage experiment of methane-bearing coal, revealed the deformation–seepage characteristics of methane-bearing coal under load, and deduced the dynamic prediction mechanical model of methane-bearing coal permeability based on pore heterogeneity, followed by practical verification. The results show that the permeability change in methane-bearing coal is an external manifestation of coal pore deformation, and the two are closely related and affected by changes in the effective stress coefficient. The derived fractal-deformation-coupled methane permeability mechanics model based on coal pore heterogeneity has high accuracy, a general expression for the stress–strain-permeability model based on coal heterogeneity is given, and the fractal Langmuir model is verified to be highly accurate (>0.9) and can be used for coal reservoir permeability prediction. Full article

(This article belongs to the Section Engineering)

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20 pages, 70564 KiB

Open AccessArticle

A Fractional Model of Complex Permittivity of Conductor Media with Relaxation: Theory vs. Experiments

by Armando Ciancio, Vincenzo Ciancio, Alberto d’Onofrio and Bruno Felice Filippo Flora

Fractal Fract. 2022, 6(7), 390; https://doi.org/10.3390/fractalfract6070390 - 14 Jul 2022

Cited by 6 | Viewed by 1727

Abstract

Moving from the study of plasmonic materials with relaxation, in this work we propose a fractional Abraham–Lorentz-like model of the complex permittivity of conductor media. This model extends the Ciancio–Kluitenberg, based on the Mazur–de Groot non-equilibrium thermodynamics theory (NET). The approach based on [...] Read more.

Moving from the study of plasmonic materials with relaxation, in this work we propose a fractional Abraham–Lorentz-like model of the complex permittivity of conductor media. This model extends the Ciancio–Kluitenberg, based on the Mazur–de Groot non-equilibrium thermodynamics theory (NET). The approach based on NET allows us to link the phenomenological function of internal variables and electrodynamics variables for a large range of frequencies. This allows us to closer reproduce experimental data for some key metals, such as Cu, Au and Ag. Particularly, our fitting significantly improves those obtained by Rakic and coworkers and we were able to operate in a larger range of energy values. Moreover, in this work we also provide a definition of a substantial fractional derivative, and we extend the fractional model proposed by Flora et al. Full article

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3 pages, 209 KiB

Open AccessEditorial

Fractional Order Systems and Their Applications

by António M. Lopes and Liping Chen

Fractal Fract. 2022, 6(7), 389; https://doi.org/10.3390/fractalfract6070389 - 13 Jul 2022

Cited by 8 | Viewed by 1685

Abstract

Fractional calculus (FC) generalizes the concepts of derivative and integral to non-integer orders [...] Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

25 pages, 8190 KiB

Open AccessArticle

Circuit Implementation of Variable-Order Scaling Fractal-Ladder Fractor with High Resolution

by Bo Yu, Yifei Pu, Qiuyan He and Xiao Yuan

Fractal Fract. 2022, 6(7), 388; https://doi.org/10.3390/fractalfract6070388 - 12 Jul 2022

Cited by 3 | Viewed by 1864

Abstract

Extensive research has been conducted on the scaling fractal fractor using various structures. The development of high-resolution emulator circuits to achieve a variable-order scaling fractal fractor with high resolution is a major area of interest. We present a scaling fractal-ladder circuit for achieving [...] Read more.

Extensive research has been conducted on the scaling fractal fractor using various structures. The development of high-resolution emulator circuits to achieve a variable-order scaling fractal fractor with high resolution is a major area of interest. We present a scaling fractal-ladder circuit for achieving high-resolution variable-order fractor based on scaling expansion theory using a high-resolution multiplying digital-to-analog converter (HMDAC). Firstly, the circuit configuration of variable-order scaling fractal-ladder fractor (VSFF) is designed. A theoretical demonstration proves that VSFF exhibits the operational characteristics of variable-order fractional calculus. Secondly, a programmable resistor–capacitor series circuit and universal electronic component emulators are developed based on the HMDAC to adjust the resistance and capacitance in the circuit configuration. Lastly, the model, component parameters, approximation performance, and variable-order characteristics are analyzed, and the circuit is physically implemented. The experimental results demonstrate that the circuit exhibits variable-order characteristics, with an operational order ranging from

- 0.7

to

- 0.3

and an operational frequency ranging from

7.72 Hz

to

4.82 kHz

. The peak value of the input signal is

10 V

. This study also proposes a novel method for variable-order fractional calculus based on circuit theory. This study was the first attempt to implement feasible high-resolution continuous variable-order fractional calculus hardware based on VSFF. Full article

(This article belongs to the Special Issue Variable-Order Fractional Problems: Modeling, Analysis, Approximation and Application)

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16 pages, 433 KiB

Open AccessArticle

Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation

by Sandeep Kumar, Rajesh K. Pandey, Kamlesh Kumar, Shyam Kamal and Thach Ngoc Dinh

Fractal Fract. 2022, 6(7), 387; https://doi.org/10.3390/fractalfract6070387 - 11 Jul 2022

Cited by 10 | Viewed by 1742

Abstract

In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the [...] Read more.

In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the accuracy of the proposed method, test examples with different scale and weight functions are considered, and the obtained numerical results are compared with the existing methods in the literature. It is observed that the proposed approach works very well with the generalized fractional derivatives (GFDs), as the presence of scale and weight functions in a generalized fractional derivative (GFD) cause difficulty for its discretization and further analysis. Full article

(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)

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16 pages, 2301 KiB

Open AccessArticle

Second Derivative Block Hybrid Methods for the Numerical Integration of Differential Systems

by Dauda Gulibur Yakubu, Ali Shokri, Geoffrey Micah Kumleng and Daniela Marian

Fractal Fract. 2022, 6(7), 386; https://doi.org/10.3390/fractalfract6070386 - 10 Jul 2022

Cited by 5 | Viewed by 1639

Abstract

The second derivative block hybrid method for the continuous integration of differential systems within the interval of integration was derived. The second derivative block hybrid method maintained the stability properties of the Runge–Kutta methods suitable for solving stiff differential systems. The lack of [...] Read more.

The second derivative block hybrid method for the continuous integration of differential systems within the interval of integration was derived. The second derivative block hybrid method maintained the stability properties of the Runge–Kutta methods suitable for solving stiff differential systems. The lack of such stability properties makes the continuous solution not reliable, especially in solving large stiff differential systems. We derive these methods by using one intermediate off-grid point in between the familiar grid points for continuous solution within the interval of integration. The new family had a high accuracy, non-overlapping piecewise continuous solution with very low error constants and converged under the suitable conditions of stability and consistency. The results of computational experiments are presented to demonstrate the efficiency and usefulness of the methods, which also indicate that the block hybrid methods are competitive with some strong stability stiff integrators. Full article

(This article belongs to the Special Issue Advances in Nonlinear Differential Equations with Applications)

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21 pages, 456 KiB

Open AccessArticle

Exponential Stability of Highly Nonlinear Hybrid Differently Structured Neutral Stochastic Differential Equations with Unbounded Delays

by Boliang Lu, Quanxin Zhu and Ping He

Fractal Fract. 2022, 6(7), 385; https://doi.org/10.3390/fractalfract6070385 - 9 Jul 2022

Cited by 3 | Viewed by 1499

Abstract

This paper mainly studies the exponential stability of the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple unbounded time-dependent delays and different structures. We prove the existence and uniqueness of the exact global solution of the new stochastic system, and then [...] Read more.

This paper mainly studies the exponential stability of the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple unbounded time-dependent delays and different structures. We prove the existence and uniqueness of the exact global solution of the new stochastic system, and then give several criteria of the exponential stability, including the

q_{1}

th moment and almost surely exponential stability. Additionally, some numerical examples are given to illustrate the main results. Such systems are widely applied in physics and other fields. For example, a specific case is pantograph dynamics, in which the delay term is a proportional function. These are widely used to determine the motion of a pantograph head on an electric locomotive collecting current from an overhead trolley wire. Compared with the existing works, our results extend the single constant delay of coefficients to multiple unbounded time-dependent delays, which is more general and applicable. Full article

(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)

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17 pages, 335 KiB

Open AccessArticle

Diamond Alpha Hilbert-Type Inequalities on Time Scales

by Ahmed A. El-Deeb, Dumitru Baleanu, Sameh S. Askar, Clemente Cesarano and Ahmed Abdeldaim

Fractal Fract. 2022, 6(7), 384; https://doi.org/10.3390/fractalfract6070384 - 6 Jul 2022

Cited by 1 | Viewed by 1407

Abstract

In this article, we will prove some new diamond alpha Hilbert-type dynamic inequalities on time scales which are defined as a linear combination of the nabla and delta integrals. These inequalities extend some known dynamic inequalities on time scales, and unify and extend [...] Read more.

In this article, we will prove some new diamond alpha Hilbert-type dynamic inequalities on time scales which are defined as a linear combination of the nabla and delta integrals. These inequalities extend some known dynamic inequalities on time scales, and unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proven by using some algebraic inequalities, diamond alpha Hölder inequality, and diamond alpha Jensen’s inequality on time scales. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

22 pages, 7116 KiB

Open AccessArticle

Effectiveness of Radiation on Magneto-Combined Convective Boundary Layer Flow in Polar Nanofluid around a Spherical Shape

by Mohammed Z. Swalmeh, Feras Shatat, Firas A. Alwawi, Mohd Asrul Hery Ibrahim, Ibrahim Mohammed Sulaiman, Nusayba Yaseen and Mohammad F. M. Naser

Fractal Fract. 2022, 6(7), 383; https://doi.org/10.3390/fractalfract6070383 - 6 Jul 2022

Cited by 13 | Viewed by 1791

Abstract

Many physical aspects emerging from the local structure and micromotions of liquid particles can be studied by utilizing the governing model of micropolar liquid. It has the ability to explain the behavior of a wide range of real fluids, including polymeric solutions, liquid [...] Read more.

Many physical aspects emerging from the local structure and micromotions of liquid particles can be studied by utilizing the governing model of micropolar liquid. It has the ability to explain the behavior of a wide range of real fluids, including polymeric solutions, liquid crystals, lubricants, and animal blood. This earned it a major role in the treatment of many industrial and engineering applications. Radiative heat transmission induced by a combined convection flow of micropolar fluid over a solid sphere, and its enhancement via nanoparticle oxides, are investigated in this study. An applied magnetic field and a constant wall temperature are also considered. The Tiwari–Das model is used to construct the mathematical model. An approximate numerical solution is included using the Keller box method, in which its numerical calculations are performed via MATLAB software, to obtain numerical results and graphic outputs reflecting the effects of critical parameters on the physical quantities associated with heat transfer. The investigation results point out that a weakness in the intensity of the magnetic field, or an increment in the nanoparticle volume fraction, causes an increment in velocity. Raising the radiation parameter promotes energy transport, angular velocity, and velocity. Full article

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

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15 pages, 6754 KiB

Open AccessArticle

Discrete Element Modelling of Fractal Behavior of Particle Size Distribution and Breakage of Ballast under Monotonic Loading

by Cheng Chen, Xin Zhang, Yifei Sun, Lei Zhang, Rui Rui and Zhide Wang

Fractal Fract. 2022, 6(7), 382; https://doi.org/10.3390/fractalfract6070382 - 6 Jul 2022

Cited by 4 | Viewed by 1829

Abstract

Breakage of particles has a great influence on the particle size distribution (PSD) and the associated mechanical behavior of ballast under train loads. A discrete element method (DEM) simulation of triaxial testing under monotonic loading was carried out using FRM (fragment replacement method) [...] Read more.

Breakage of particles has a great influence on the particle size distribution (PSD) and the associated mechanical behavior of ballast under train loads. A discrete element method (DEM) simulation of triaxial testing under monotonic loading was carried out using FRM (fragment replacement method) breakable particles as ballast and a flexible shell model as membrane. The coupled model was validated by comparing the load-deformation responses with those measured in previous experiments and was then used to analyze the contact orientations and the distribution of particle breakage from a micromechanical perspective. The simulation results show that higher confining pressure and larger axial strain may increase the grain breakage (B_g) and the fractal dimension (D) of ballast. It was observed that most breakage was first-generation breakage, and that the proportions of the second- to fifth-generation breakage decreased successively. Moreover, as the axial strain or confining pressure increased, the percentage of small particle fragments increased in correspondence with the PSD curves that remained concave upwards, as the fractal dimension D of PSD increased. In addition, the evolution of D exhibited a linear correlation with grain breakage B_g. Contrarily, a quadratic curve relation between D and volumetric strain was exhibited under different axial strain stages. Therefore, D has the potential to be a key indicator to evaluate the degree of ballast crushing and PSD degradation, which may contribute to better decision making concerning track bed maintenance. Full article

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

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18 pages, 2239 KiB

Open AccessArticle

Optimal H¹-Norm Estimation of Nonconforming FEM for Time-Fractional Diffusion Equation on Anisotropic Meshes

by Yabing Wei, Yanmin Zhao, Shujuan Lü, Fenling Wang and Yayun Fu

Fractal Fract. 2022, 6(7), 381; https://doi.org/10.3390/fractalfract6070381 - 4 Jul 2022

Cited by 2 | Viewed by 1821

Abstract

In this paper, based on the

L 2

-

1_{σ}

scheme and nonconforming

E Q_{1}^{r o t}

finite element method (FEM), a numerical approximation is presented for a class of two-dimensional time-fractional diffusion equations involving variable coefficients. A novel and [...] Read more.

In this paper, based on the

L 2

-

1_{σ}

scheme and nonconforming

E Q_{1}^{r o t}

finite element method (FEM), a numerical approximation is presented for a class of two-dimensional time-fractional diffusion equations involving variable coefficients. A novel and detailed analysis of the equations with an initial singularity is described on anisotropic meshes. The fully discrete scheme is shown to be unconditionally stable, and optimal second-order accuracy for convergence and superconvergence can be achieved in both time and space directions. Finally, the obtained numerical results are compared with the theoretical analysis, which verifies the accuracy of the proposed method. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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21 pages, 4752 KiB

Open AccessArticle

Distributed Adaptive Mittag–Leffler Formation Control for Second-Order Fractional Multi-Agent Systems via Event-Triggered Control Strategy

by Yougang Wang, Jing Zhang and Huaiqin Wu

Fractal Fract. 2022, 6(7), 380; https://doi.org/10.3390/fractalfract6070380 - 4 Jul 2022

Cited by 6 | Viewed by 1693

Abstract

This brief investigates the Mittag–Leffler formation bounded control problem for second-order fractional multi-agent systems (FMASs), where the dynamical nodes of followers are modeled to satisfy quadratic (QUAD) condition. Firstly, under the undirected communication topology, for the considered second-order nonlinear FMASs, a distributed event-triggered [...] Read more.

This brief investigates the Mittag–Leffler formation bounded control problem for second-order fractional multi-agent systems (FMASs), where the dynamical nodes of followers are modeled to satisfy quadratic (QUAD) condition. Firstly, under the undirected communication topology, for the considered second-order nonlinear FMASs, a distributed event-triggered control scheme (ETCS) is designed to realize the global Mittag–Leffler bounded formation control goal. Secondly, by introducing adaptive weights into triggering condition and control protocol, an adaptive event-triggered formation protocol is presented to achieve the global Mittag–Leffler bounded formation. Thirdly, a five-step algorithm is provided to describe protocol execution steps. Finally, two simulation examples are given to verify the effectiveness of the proposed schemes. Full article

(This article belongs to the Section Complexity)

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38 pages, 10447 KiB

Open AccessReview

Fractals: An Eclectic Survey, Part II

by Akhlaq Husain, Manikyala Navaneeth Nanda, Movva Sitaram Chowdary and Mohammad Sajid

Fractal Fract. 2022, 6(7), 379; https://doi.org/10.3390/fractalfract6070379 - 2 Jul 2022

Cited by 23 | Viewed by 5175

Abstract

Fractals are geometric shapes and patterns that can describe the roughness (or irregularity) present in almost every object in nature. Many fractals may repeat their geometry at smaller or larger scales. This paper is the second (and last) part of a series of [...] Read more.

Fractals are geometric shapes and patterns that can describe the roughness (or irregularity) present in almost every object in nature. Many fractals may repeat their geometry at smaller or larger scales. This paper is the second (and last) part of a series of two papers dedicated to an eclectic survey of fractals describing the infinite complexity and amazing beauty of fractals from historical, theoretical, mathematical, aesthetical and technological aspects, including their diverse applications in various fields. In this article, our focus is on engineering, industrial, commercial and futuristic applications of fractals, whereas in the first part, we discussed the basics of fractals, mathematical description, fractal dimension and artistic applications. Among many different applications of fractals, fractal landscape generation (fractal landscapes that can simulate and describe natural terrains and landscapes more precisely by mathematical models of fractal geometry), fractal antennas (fractal-shaped antennas that are designed and used in devices which operate on multiple and wider frequency bands) and fractal image compression (a fractal-based lossy compression method for digital and natural images which uses inherent self-similarity present in an image) are the most creative, engineering-driven, industry-oriented, commercial and emerging applications. We consider each of these applications in detail along with some innovative and future ready applications. Full article

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11 pages, 1202 KiB

Open AccessArticle

Fractal Operators and Fractional Dynamics with 1/2 Order in Biological Systems

by Yajun Yin, Jianqiao Guo, Gang Peng, Xiaobin Yu and Yiya Kong

Fractal Fract. 2022, 6(7), 378; https://doi.org/10.3390/fractalfract6070378 - 2 Jul 2022

Cited by 6 | Viewed by 2332

Abstract

This paper reports the new advances in biological fractal dynamics. The following contents are included: (1) physical (or functional) fractal spaces abstracted from biological materials, biological structures and biological motions; (2) fractal operators on fractal spaces; (3) 1/2-order fractional dynamics controlled by fractal [...] Read more.

This paper reports the new advances in biological fractal dynamics. The following contents are included: (1) physical (or functional) fractal spaces abstracted from biological materials, biological structures and biological motions; (2) fractal operators on fractal spaces; (3) 1/2-order fractional dynamics controlled by fractal operators; and (4) the origin of 1/2-order. Based on the new progress, we can make a judgment that all the two-bifurcation physical functional fractal motions in the living body can be attributed to the fractional dynamics with 1/2-order. Full article

(This article belongs to the Special Issue Fractional Behaviors Analysis and Modelling)

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15 pages, 436 KiB

Open AccessArticle

Numerical Approximation of the Fractional Rayleigh–Stokes Problem Arising in a Generalised Maxwell Fluid

by Le Dinh Long, Bahman Moradi, Omid Nikan, Zakieh Avazzadeh and António M. Lopes

Fractal Fract. 2022, 6(7), 377; https://doi.org/10.3390/fractalfract6070377 - 2 Jul 2022

Cited by 7 | Viewed by 1790

Abstract

This paper presents a numerical technique to approximate the Rayleigh–Stokes model for a generalised Maxwell fluid formulated in the Riemann–Liouville sense. The proposed method consists of two stages. First, the time discretization of the problem is accomplished by using the finite difference. Second, [...] Read more.

This paper presents a numerical technique to approximate the Rayleigh–Stokes model for a generalised Maxwell fluid formulated in the Riemann–Liouville sense. The proposed method consists of two stages. First, the time discretization of the problem is accomplished by using the finite difference. Second, the space discretization is obtained by means of the predictor–corrector method. The unconditional stability result and convergence analysis are analysed theoretically. Numerical examples are provided to verify the feasibility and accuracy of the proposed method. Full article

(This article belongs to the Special Issue Dynamical Systems and Their Applications (DSTA) — in Memory of Prof. Dr. José A. Tenreiro Machado)

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15 pages, 331 KiB

Open AccessArticle

Hermite–Hadamard Type Inequalities Involving (k-p) Fractional Operator for Various Types of Convex Functions

by Vuk Stojiljković, Rajagopalan Ramaswamy, Fahad Alshammari, Ola A. Ashour, Mohammed Lahy Hassan Alghazwani and Stojan Radenović

Fractal Fract. 2022, 6(7), 376; https://doi.org/10.3390/fractalfract6070376 - 2 Jul 2022

Cited by 21 | Viewed by 1754

Abstract

We establish various fractional convex inequalities of the Hermite–Hadamard type with addition to many other inequalities. Various types of such inequalities are obtained, such as

(p, h)

fractional type inequality and many others, as the

(p, h)

[...] Read more.

We establish various fractional convex inequalities of the Hermite–Hadamard type with addition to many other inequalities. Various types of such inequalities are obtained, such as

(p, h)

fractional type inequality and many others, as the

(p, h)

-convexity is the generalization of the other convex inequalities. As a consequence of the

(h, m)

-convexity, the fractional inequality of the

(s, m)

-type is obtained. Many consequences of such fractional inequalities and generalizations are obtained. Full article

(This article belongs to the Special Issue Probabilistic Solutions and Stochastic Representation of Fractional Differential Equations)

13 pages, 386 KiB

Open AccessArticle

The Passivity of Uncertain Fractional-Order Neural Networks with Time-Varying Delays

by Song Xu, Heng Liu and Zhimin Han

Fractal Fract. 2022, 6(7), 375; https://doi.org/10.3390/fractalfract6070375 - 2 Jul 2022

Cited by 3 | Viewed by 1498

Abstract

In this paper, we study the passive problem of uncertain fractional-order neural networks (UFONNs) with time-varying delays. First, we give a sufficient condition for the asymptotic stability of UFONNs with bounded time-varying delays by using the fractional-order Razumikhin theorem. Secondly, according to the [...] Read more.

In this paper, we study the passive problem of uncertain fractional-order neural networks (UFONNs) with time-varying delays. First, we give a sufficient condition for the asymptotic stability of UFONNs with bounded time-varying delays by using the fractional-order Razumikhin theorem. Secondly, according to the above stability criteria and some properties of fractional-order calculus, a delay-dependent condition that can guarantee the passivity of UFONNs with time-varying delays is given in the form of a linear matrix inequality (LMI) that can be reasonably solved in polynomial time using the LMI Control Toolbox. These conditions are not only delay-dependent but also order-dependent, and less conservative than some existing work. Finally, the rationality of the research results is proved by simulation. Full article

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

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19 pages, 7637 KiB

Open AccessArticle

Grid-Connected Inverter Based on a Resonance-Free Fractional-Order LCL Filter

by Xiaogang Wang and Junhui Cai

Fractal Fract. 2022, 6(7), 374; https://doi.org/10.3390/fractalfract6070374 - 1 Jul 2022

Cited by 2 | Viewed by 1959

Abstract

The integer-order LCL (IOLCL) filter has excellent high-frequency harmonic attenuation capability but suffers from resonance, which causes system instability in grid-connected inverter applications. This paper studied a class of resonance-free fractional-order LCL (FOLCL) filters and control problems of single-phase FOLCL-type grid-connected inverters (FOGCI). [...] Read more.

The integer-order LCL (IOLCL) filter has excellent high-frequency harmonic attenuation capability but suffers from resonance, which causes system instability in grid-connected inverter applications. This paper studied a class of resonance-free fractional-order LCL (FOLCL) filters and control problems of single-phase FOLCL-type grid-connected inverters (FOGCI). The Caputo fractional calculus operator was used to describe the fractional-order inductor and capacitor. Compared with the conventional IOLCL filter, by reasonably selecting the orders of the inductor and capacitor, the resonance peak of the FOLCL filter could be effectively avoided. In this way, the FOGCI could operate stably without passive or active dampers, which simplified the design of control system. Furthermore, compared with a single-phase integer-order grid-connected inverter (IOGCI) controlled by an integer-order PI (IOPI) controller, the FOGCI, combined with a fractional-order PI (FOPI) controller, could achieve greater gain and phase margins, which improved the system performance. The correctness of the theoretical analyses was validated through both simulation and hardware-in-the-loop experiments. Full article

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

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Fractal Fract., Volume 6, Issue 7 (July 2022) – 60 articles

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