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Fractal Fract
Volume 7
Issue 4
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Fractal Fract., Volume 7, Issue 4 (April 2023) – 65 articles

Cover Story (view full-size image): In this study, alumina ceramics were obtained from three different sources, with various rare-earth dopants, concentration levels and synthesizing routes; ceramic properties were measured, resulting in a complex, multivariate database. Microstructural features were assessed by using two edge detection techniques and by computing Fractal Dimension (FD). Correlations in the causal chain, materials–composition–processing–microstructure–properties, with multiple choices for each link, were extracted. Principal Component Analysis was used to understand the data on a scientific basis and to extract correlations, along with some other statistical techniques. FD was acknowledged to be an effective quantifier of highly dissimilar and/or noisy microstructures, and the best edge detection technique was selected. View this paper
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18 pages, 9459 KiB  
Article
Microgrid Frequency Regulation Based on a Fractional Order Cascade Controller
by Soroush Oshnoei, Arman Fathollahi, Arman Oshnoei and Mohammad Hassan Khooban
Fractal Fract. 2023, 7(4), 343; https://doi.org/10.3390/fractalfract7040343 - 21 Apr 2023
Cited by 10 | Viewed by 1929
Abstract
Nowadays, the participation of renewable energy sources (RESs) and the integration of these sources with traditional power plants in microgrids (MGs) for providing demand-side power has rapidly grown. Although the presence of RESs in MGs reduces environmental problems, their high participation significantly affects [...] Read more.
Nowadays, the participation of renewable energy sources (RESs) and the integration of these sources with traditional power plants in microgrids (MGs) for providing demand-side power has rapidly grown. Although the presence of RESs in MGs reduces environmental problems, their high participation significantly affects the system’s whole inertia and dynamic stability. This paper focuses on an islanded MG frequency regulation under the high participation of RESs. In this regard, a novel fractional order cascade controller (FOCC) is proposed as the secondary frequency controller. In the proposed FOCC controller structure, a fractional order proportional-integral controller is cascaded with a fractional order tilt-derivative controller. The proposed FOCC controller has a greater degree of freedom and adaptability than integer order controllers and improves the control system’s efficiency. The adjustable coefficients of the proposed controller are tuned via the kidney-inspired algorithm. An energy storage system equipped with virtual inertia is also employed to improve the system inertia. The proposed FOCC controller efficiency is compared with proportional-integral-derivative (PID), tilt-integral-derivative (TID), and fractional order proportional-integral-derivative (FOPID) controllers under different disturbances and operating conditions. The results demonstrate that the presented controller provides better frequency responses compared to the other controllers. Moreover, the sensitivity analysis is performed to show the proposed controller robustness versus the parameters’ changes in the system. Full article
(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering)
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24 pages, 6344 KiB  
Article
Pitch Angle Control of an Airplane Using Fractional Order Direct Model Reference Adaptive Controllers
by Gustavo E. Ceballos Benavides, Manuel A. Duarte-Mermoud, Marcos E. Orchard and Juan Carlos Travieso-Torres
Fractal Fract. 2023, 7(4), 342; https://doi.org/10.3390/fractalfract7040342 - 20 Apr 2023
Cited by 4 | Viewed by 2321
Abstract
This paper deals with the longitudinal movement control of an airplane (pitch angle) using fractional order adaptive controllers (FOACs). It shows the improvements achieved in the plane’s behavior, in terms of the minimization of a given performance index. At the same time, less [...] Read more.
This paper deals with the longitudinal movement control of an airplane (pitch angle) using fractional order adaptive controllers (FOACs). It shows the improvements achieved in the plane’s behavior, in terms of the minimization of a given performance index. At the same time, less control effort is needed to accomplish the control objectives compared with the classic integer order adaptive controllers (IOACs). In this study, fractional order direct model reference adaptive control (FO-DMRAC) is implemented at the simulation level, and exhibits a better performance compared with the classic integer order (IO) version of the DMRAC (IO-DMRAC). It is also shown that the proposed control strategy for FO-DMRAC reduces the resultant system control structure down to a relative degree 2 system, for which the control implementation is simpler and avoids oscillations during the transient period. Moreover, it is interesting to note that this is the first time that an FOAC with fractional adaptive laws is applied to the longitudinal control of an airplane. A suitable model for the longitudinal movement of the airplane related to the pitch angle θ as the output variable with the lifter angle (δe) as the control variable, is first analyzed and discussed to obtain a reliable mathematical model of the plane. All of the other input variables acting on the plane are considered as perturbations. For certain operating conditions defined by the flight conditions, an FO-DMRAC is designed, simulated, and analyzed. Furthermore, a comparison with the implementation of the classical adaptive general direct control (relative degree ≥ 2) is presented. The boundedness and convergence of all of the signals are theoretically proven based on the new Lemma 3, assuring the boundedness of all internal signals ω(t). Full article
(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)
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16 pages, 656 KiB  
Article
Input-Output Finite-Time Stability of Fractional-Order Switched Singular Systems with D-Perturbation
by Qiang Yu and Na Xue
Fractal Fract. 2023, 7(4), 341; https://doi.org/10.3390/fractalfract7040341 - 20 Apr 2023
Cited by 1 | Viewed by 1262
Abstract
The objective of this paper focuses on the stability analysis of the input-output finite-time for a class of fractional-order switched singular systems (FOSSS) with D-perturbation. By using the Φ-dependent average dwell time (ΦDADT) approach together with the multiple Lyapunov functions method, some [...] Read more.
The objective of this paper focuses on the stability analysis of the input-output finite-time for a class of fractional-order switched singular systems (FOSSS) with D-perturbation. By using the Φ-dependent average dwell time (ΦDADT) approach together with the multiple Lyapunov functions method, some sufficient conditions are derived for the considered system to ensure its input-output finite-time stability in terms of linear matrix inequalities. Then, the output feedback controller is designed to ensure the closed-loop system is input-output finite-time stable. Finally, a numerical example illustrates the superiority of the proposed method. Full article
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21 pages, 441 KiB  
Article
Compact Difference Schemes with Temporal Uniform/Non-Uniform Meshes for Time-Fractional Black–Scholes Equation
by Jie Gu, Lijuan Nong, Qian Yi and An Chen
Fractal Fract. 2023, 7(4), 340; https://doi.org/10.3390/fractalfract7040340 - 19 Apr 2023
Cited by 2 | Viewed by 1362
Abstract
In this paper, we are interested in the effective numerical schemes of the time-fractional Black–Scholes equation. We convert the original equation into an equivalent integral-differential equation and then discretize the time-integral term in the equivalent form using the piecewise linear interpolation, while the [...] Read more.
In this paper, we are interested in the effective numerical schemes of the time-fractional Black–Scholes equation. We convert the original equation into an equivalent integral-differential equation and then discretize the time-integral term in the equivalent form using the piecewise linear interpolation, while the compact difference formula is applied in the spatial direction. Thus, we derive a fully discrete compact difference scheme with second-order accuracy in time and fourth-order accuracy in space. Rigorous proofs of the corresponding stability and convergence are given. Furthermore, in order to deal effectively with the non-smooth solution, we extend the obtained results to the case of temporal non-uniform meshes and obtain a temporal non-uniform mesh-based compact difference scheme as well as the numerical theory. Finally, extensive numerical examples are included to demonstrate the effectiveness of the proposed compact difference schemes. Full article
(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)
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15 pages, 2154 KiB  
Article
On the Roots of a Family of Polynomials
by Marilena Jianu
Fractal Fract. 2023, 7(4), 339; https://doi.org/10.3390/fractalfract7040339 - 19 Apr 2023
Cited by 1 | Viewed by 1377
Abstract
The aim of this paper is to give a characterization of the set of roots of a special family of polynomials. This family is relevant in reliability theory since it contains the reliability polynomials of the networks created by series-parallel compositions. We prove [...] Read more.
The aim of this paper is to give a characterization of the set of roots of a special family of polynomials. This family is relevant in reliability theory since it contains the reliability polynomials of the networks created by series-parallel compositions. We prove that the set of roots is bounded, being contained in the two disks of the radius equal to the golden ratio, centered at 0 and at 1. We study the closure of the set of roots and prove that it includes two disks centered at 0 and 1 of a radius slightly greater than 1, as well as the sinusoidal spirals centered at 0 and at 1, respectively. The expression of some limit points is also provided. Full article
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13 pages, 354 KiB  
Brief Report
Unlimited Sampling Theorem Based on Fractional Fourier Transform
by Hui Zhao and Bing-Zhao Li
Fractal Fract. 2023, 7(4), 338; https://doi.org/10.3390/fractalfract7040338 - 18 Apr 2023
Cited by 2 | Viewed by 1569
Abstract
The recovery of bandlimited signals with high dynamic range is a hot issue in sampling research. The unlimited sampling theory expands the recordable range of traditional analog-to-digital converters (ADCs) arbitrarily, and the signal is folded back into a low dynamic range measurement, avoiding [...] Read more.
The recovery of bandlimited signals with high dynamic range is a hot issue in sampling research. The unlimited sampling theory expands the recordable range of traditional analog-to-digital converters (ADCs) arbitrarily, and the signal is folded back into a low dynamic range measurement, avoiding the saturation problem. Since the non-bandlimited signal in the Fourier domain cannot be directly applied to its existing theory, the non-bandlimited signal in the Fourier domain may be bandlimited in the fractional Fourier domain. Therefore, this brief report studies the unlimited sampling problem of high dynamic non-bandlimited signals in the Fourier domain based on the fractional Fourier transform. Firstly, a mathematical signal model for unlimited sampling is proposed. Secondly, based on this mathematical model, the annihilation filtering method is used to estimate the arbitrary folding time. Finally, a novel fractional Fourier domain unlimited sampling theorem is obtained. The theory proves that, based on the folding characteristics of the self-reset ADC, the number of samples is not affected by the modulo threshold, and any folding time can be handled. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)
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14 pages, 332 KiB  
Article
On Coupled System of Langevin Fractional Problems with Different Orders of μ-Caputo Fractional Derivatives
by Lamya Almaghamsi, Ymnah Alruwaily, Kulandhaivel Karthikeyan and El-sayed El-hady
Fractal Fract. 2023, 7(4), 337; https://doi.org/10.3390/fractalfract7040337 - 18 Apr 2023
Cited by 1 | Viewed by 1466
Abstract
In this paper, we study coupled nonlinear Langevin fractional problems with different orders of μ-Caputo fractional derivatives on arbitrary domains with nonlocal integral boundary conditions. In order to ensure the existence and uniqueness of the solutions to the problem at hand, the [...] Read more.
In this paper, we study coupled nonlinear Langevin fractional problems with different orders of μ-Caputo fractional derivatives on arbitrary domains with nonlocal integral boundary conditions. In order to ensure the existence and uniqueness of the solutions to the problem at hand, the tools of the fixed-point theory are applied. An overview of the main results of this study is presented through examples. Full article
(This article belongs to the Special Issue New Insights into Nonlinear Coupled Differential Equations with Its Applications)
15 pages, 1607 KiB  
Article
A Non-Convex Hybrid Overlapping Group Sparsity Model with Hyper-Laplacian Prior for Multiplicative Noise
by Jianguang Zhu, Ying Wei, Juan Wei and Binbin Hao
Fractal Fract. 2023, 7(4), 336; https://doi.org/10.3390/fractalfract7040336 - 17 Apr 2023
Cited by 2 | Viewed by 1484
Abstract
Multiplicative noise removal is a quite challenging problem in image denoising. In recent years, hyper-Laplacian prior information has been successfully introduced in the image denoising problem and significant denoising effects have been achieved. In this paper, we propose a new hybrid regularizer model [...] Read more.
Multiplicative noise removal is a quite challenging problem in image denoising. In recent years, hyper-Laplacian prior information has been successfully introduced in the image denoising problem and significant denoising effects have been achieved. In this paper, we propose a new hybrid regularizer model for removing multiplicative noise. The proposed model consists of the non-convex higher-order total variation and overlapping group sparsity on a hyper-Laplacian prior regularizer. It combines the advantages of the non-convex regularization and the hybrid regularization, which may simultaneously preserve the fine-edge information and reduce the staircase effect at the same time. We develop an effective alternating minimization method for the proposed nonconvex model via an alternating direction method of multipliers framework, where the majorization–minimization algorithm and the iteratively reweighted algorithm are adopted to solve the corresponding subproblems. Numerical experiments show that the proposed model outperforms the most advanced model in terms of visual quality and certain image quality measurements. Full article
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23 pages, 380 KiB  
Article
The Rates of Convergence for Functional Limit Theorems with Stable Subordinators and for CTRW Approximations to Fractional Evolutions
by Vassili N. Kolokoltsov
Fractal Fract. 2023, 7(4), 335; https://doi.org/10.3390/fractalfract7040335 - 17 Apr 2023
Cited by 1 | Viewed by 1498
Abstract
From the initial development of probability theory to the present days, the convergence of various discrete processes to simpler continuous distributions remains at the heart of stochastic analysis. Many efforts have been devoted to functional central limit theorems (also referred to as the [...] Read more.
From the initial development of probability theory to the present days, the convergence of various discrete processes to simpler continuous distributions remains at the heart of stochastic analysis. Many efforts have been devoted to functional central limit theorems (also referred to as the invariance principle), dealing with the convergence of random walks to Brownian motion. Though quite a lot of work has been conducted on the rates of convergence of the weighted sums of independent and identically distributed random variables to stable laws, the present paper is the first to supply the rates of convergence in the functional limit theorem for stable subordinators. On the other hand, there is a lot of activity on the convergence of CTRWs (continuous time random walks) to processes with memory (subordinated Markov process) described by fractional PDEs. Our second main result is the first one yielding rates of convergence in such a setting. Since CTRW approximations may be used for numeric solutions of fractional equations, we obtain, as a direct consequence of our results, the estimates for error terms in such numeric schemes. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)
15 pages, 585 KiB  
Article
A Preconditioned Iterative Method for a Multi-State Time-Fractional Linear Complementary Problem in Option Pricing
by Xu Chen, Xinxin Gong, Siu-Long Lei and Youfa Sun
Fractal Fract. 2023, 7(4), 334; https://doi.org/10.3390/fractalfract7040334 - 17 Apr 2023
Cited by 2 | Viewed by 1382
Abstract
Fractional derivatives and regime-switching models are widely used in various fields of finance because they can describe the nonlocal properties of the solutions and the changes in the market status, respectively. The regime-switching time-fractional diffusion equations that combine both advantages are also used [...] Read more.
Fractional derivatives and regime-switching models are widely used in various fields of finance because they can describe the nonlocal properties of the solutions and the changes in the market status, respectively. The regime-switching time-fractional diffusion equations that combine both advantages are also used in European option pricing; however, to our knowledge, American option pricing based on such models and their numerical methods is yet to be studied. Hence, a fast algorithm for solving the multi-state time-fractional linear complementary problem arising from the regime-switching time-fractional American option pricing models is developed in this paper. To construct the solution strategy, the original problem has been converted into a Hamilton–Jacobi–Bellman equation, and a nonlinear finite difference scheme has been proposed to discretize the problem with stability analysis. A policy-Krylov subspace method is developed to solve the nonlinear scheme. Further, to accelerate the convergence rate of the Krylov method, a tri-diagonal preconditioner is proposed with condition number analysis. Numerical experiments are presented to demonstrate the validity of the proposed nonlinear scheme and the efficiency of the proposed preconditioned policy-Krylov subspace method. Full article
(This article belongs to the Special Issue Application of Fractional Partial Differential Equations in Computational Finance)
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15 pages, 586 KiB  
Article
Fast and Accurate Numerical Algorithm with Performance Assessment for Nonlinear Functional Volterra Equations
by Chinedu Nwaigwe and Sanda Micula
Fractal Fract. 2023, 7(4), 333; https://doi.org/10.3390/fractalfract7040333 - 17 Apr 2023
Cited by 3 | Viewed by 1134
Abstract
An efficient numerical algorithm is developed for solving nonlinear functional Volterra integral equations. The core idea is to define an appropriate operator, then combine the Krasnoselskij iterative scheme with collocation at discrete points and the Newton–Cotes quadrature rule. This results in an explicit [...] Read more.
An efficient numerical algorithm is developed for solving nonlinear functional Volterra integral equations. The core idea is to define an appropriate operator, then combine the Krasnoselskij iterative scheme with collocation at discrete points and the Newton–Cotes quadrature rule. This results in an explicit scheme that does not require solving a nonlinear or linear algebraic system. For the convergence analysis, the discretization error is estimated and proved to converge via a recurrence relation. The discretization error is combined with the Krasnoselskij iteration error to estimate the total approximation error, hence establishing the convergence of the method. Then, numerical experiments are provided, first, to demonstrate the second order convergence of the proposed method, and secondly, to show the better performance of the scheme over the existing nonlinear-based approach. Full article
(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)
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12 pages, 6591 KiB  
Article
A Numerical Solution of Generalized Caputo Fractional Initial Value Problems
by Rania Saadeh, Mohamed A. Abdoon, Ahmad Qazza and Mohammed Berir
Fractal Fract. 2023, 7(4), 332; https://doi.org/10.3390/fractalfract7040332 - 17 Apr 2023
Cited by 30 | Viewed by 2200
Abstract
In this article, the numerical adaptive predictor corrector (Apc-ABM) method is presented to solve generalized Caputo fractional initial value problems. The Apc-ABM method was utilized to establish approximate series solutions. The presented technique is considered to be an extension to the original Adams–Bashforth–Moulton [...] Read more.
In this article, the numerical adaptive predictor corrector (Apc-ABM) method is presented to solve generalized Caputo fractional initial value problems. The Apc-ABM method was utilized to establish approximate series solutions. The presented technique is considered to be an extension to the original Adams–Bashforth–Moulton approach. Numerical simulations and figures are presented and discussed, in order to show the efficiency of the proposed method. In the future, we anticipate that the provided generalized Caputo fractional derivative and the suggested method will be utilized to create and simulate a wide variety of generalized Caputo-type fractional models. We have included examples to demonstrate the accuracy of the present method. Full article
(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)
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8 pages, 279 KiB  
Article
Fractional Itô–Doob Stochastic Differential Equations Driven by Countably Many Brownian Motions
by Abdellatif Ben Makhlouf, Lassaad Mchiri, Hakeem A. Othman and Hafedh M. S. Rguigui
Fractal Fract. 2023, 7(4), 331; https://doi.org/10.3390/fractalfract7040331 - 16 Apr 2023
Viewed by 1251
Abstract
This article is devoted to showing the existence and uniqueness (EU) of a solution with non-Lipschitz coefficients (NLC) of fractional Itô-Doob stochastic differential equations driven by countably many Brownian motions (FIDSDECBMs) of order ϰ∈(0,1) by using the [...] Read more.
This article is devoted to showing the existence and uniqueness (EU) of a solution with non-Lipschitz coefficients (NLC) of fractional Itô-Doob stochastic differential equations driven by countably many Brownian motions (FIDSDECBMs) of order ϰ∈(0,1) by using the Picard iteration technique (PIT) and the semimartingale local time (SLT). Full article
27 pages, 618 KiB  
Article
Enhancing the Mathematical Theory of Nabla Tempered Fractional Calculus: Several Useful Equations
by Yiheng Wei, Linlin Zhao, Xuan Zhao and Jinde Cao
Fractal Fract. 2023, 7(4), 330; https://doi.org/10.3390/fractalfract7040330 - 14 Apr 2023
Cited by 2 | Viewed by 1441
Abstract
Although many applications of fractional calculus have been reported in literature, modeling the physical world using this technique is still a challenge. One of the main difficulties in solving this problem is that the long memory property is necessary, whereas the infinite memory [...] Read more.
Although many applications of fractional calculus have been reported in literature, modeling the physical world using this technique is still a challenge. One of the main difficulties in solving this problem is that the long memory property is necessary, whereas the infinite memory is undesirable. To address this challenge, a new type of nabla fractional calculus with a weight function is formulated, which combines the benefits of nabla fractional calculus and its tempered counterpart, making it highly valuable for modeling practical systems. However, many properties of this calculus are still unclear and need to be discovered. Therefore, this paper gives particular emphasis to the topic, developing some remarkable properties, i.e., the equivalence relation, the nabla Taylor formula, and the nabla Laplace transform of such nabla tempered fractional calculus. All the developed properties greatly enrich the mathematical theory of nabla tempered fractional calculus and provide high value and potential for further applications. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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28 pages, 11403 KiB  
Article
Multiplicative Noise Removal and Contrast Enhancement for SAR Images Based on a Total Fractional-Order Variation Model
by Yamei Zhou, Yao Li, Zhichang Guo, Boying Wu and Dazhi Zhang
Fractal Fract. 2023, 7(4), 329; https://doi.org/10.3390/fractalfract7040329 - 14 Apr 2023
Cited by 5 | Viewed by 2133
Abstract
In this paper, we propose a total fractional-order variation model for multiplicative noise removal and contrast enhancement of real SAR images. Inspired by the high dynamic intensity range of SAR images, the full content of the SAR images is preserved by normalizing the [...] Read more.
In this paper, we propose a total fractional-order variation model for multiplicative noise removal and contrast enhancement of real SAR images. Inspired by the high dynamic intensity range of SAR images, the full content of the SAR images is preserved by normalizing the original data in this model. Then, we propose a degradation model based on the nonlinear transformation to adjust the intensity of image pixel values. With MAP estimator, a corresponding fidelity term is introduced into the model, which is beneficial for contrast enhancement and bias correction in the denoising process. For the regularization term, a gray level indicator is used as a weighted matrix to make the model adaptive. We first apply the scalar auxiliary variable algorithm to solve the proposed model and prove the convergence of the algorithm. By virtue of the discrete Fourier transform (DFT), the model is solved by an iterative scheme in the frequency domain. Experimental results show that the proposed model can enhance the contrast of natural and SAR images while removing multiplicative noise. Full article
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13 pages, 1596 KiB  
Article
Soliton Solutions of Fractional Stochastic Kraenkel–Manna–Merle Equations in Ferromagnetic Materials
by Wael W. Mohammed, M. El-Morshedy, Clemente Cesarano and Farah M. Al-Askar
Fractal Fract. 2023, 7(4), 328; https://doi.org/10.3390/fractalfract7040328 - 14 Apr 2023
Cited by 14 | Viewed by 1646
Abstract
In this study, we take into account the fractional stochastic Kraenkel–Manna–Merle system (FSKMMS). The mapping approach may be used to produce various type of stochastic fractional solutions, such as elliptic, hyperbolic, and trigonometric functions. Solutions to the Kraenkel–Manna–Merle system equation, which explains the [...] Read more.
In this study, we take into account the fractional stochastic Kraenkel–Manna–Merle system (FSKMMS). The mapping approach may be used to produce various type of stochastic fractional solutions, such as elliptic, hyperbolic, and trigonometric functions. Solutions to the Kraenkel–Manna–Merle system equation, which explains the propagation of a magnetic field in a zero-conductivity ferromagnet, may provide insight into a variety of fascinating scientific phenomena. Moreover, we construct a variety of 3D and 2D graphics in MATLAB to illustrate the influence of the stochastic term and the conformable derivative on the exact solutions of the FSKMMS. Full article
(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)
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20 pages, 8929 KiB  
Article
Spherical Box-Counting: Combining 360° Panoramas with Fractal Analysis
by Matthias Kulcke and Wolfgang E. Lorenz
Fractal Fract. 2023, 7(4), 327; https://doi.org/10.3390/fractalfract7040327 - 14 Apr 2023
Cited by 2 | Viewed by 1964
Abstract
In this paper, a new box-counting method to achieve a highly specific topological fingerprinting of architecture in relation to the position of the observer and in the context of its surroundings is proposed. Central to this method is the use of 360-degree spherical [...] Read more.
In this paper, a new box-counting method to achieve a highly specific topological fingerprinting of architecture in relation to the position of the observer and in the context of its surroundings is proposed. Central to this method is the use of 360-degree spherical panoramas as a basis for fractal measurement. Thus, a number of problems of the comparative analysis of the fractal dimension in the field of architecture are explicitly and implicitly addressed, first and foremost being the question of choosing image boundaries while considering adjacent vegetation, urban elements, and other visually present objects for Gestalt analysis of a specific building. Second, the problem of distance and perspective as part of the aesthetic experience based on viewer and object location were taken into account and are addressed. The implications of the use of a spherical perspective as described in this research are also highly relevant for other methods of aesthetic measures in architecture, including those implementing collaborative design processes guided by digital tools and machine learning, among others. Full article
(This article belongs to the Section Geometry)
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22 pages, 1303 KiB  
Article
Numerical Identification of External Boundary Conditions for Time Fractional Parabolic Equations on Disjoint Domains
by Miglena N. Koleva and Lubin G. Vulkov
Fractal Fract. 2023, 7(4), 326; https://doi.org/10.3390/fractalfract7040326 - 13 Apr 2023
Cited by 9 | Viewed by 1602
Abstract
We consider fractional mathematical models of fluid-porous interfaces in channel geometry. This provokes us to deal with numerical identification of the external boundary conditions for 1D and 2D time fractional parabolic problems on disjoint domains. First, we discuss the time discretization, then we [...] Read more.
We consider fractional mathematical models of fluid-porous interfaces in channel geometry. This provokes us to deal with numerical identification of the external boundary conditions for 1D and 2D time fractional parabolic problems on disjoint domains. First, we discuss the time discretization, then we decouple the full inverse problem into two Dirichlet problems at each time level. On this base, we develop decomposition techniques to obtain exact formulas for the unknown boundary conditions at point measurements. A discrete version of the analytical approach is realized on time adaptive mesh for different fractional order of the equations in each of the disjoint domains. A variety of numerical examples are discussed. Full article
(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)
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26 pages, 6479 KiB  
Article
Tool Degradation Prediction Based on Semimartingale Approximation of Linear Fractional Alpha-Stable Motion and Multi-Feature Fusion
by Yuchen Yuan, Jianxue Chen, Jin Rong, Piercarlo Cattani, Aleksey Kudreyko and Francesco Villecco
Fractal Fract. 2023, 7(4), 325; https://doi.org/10.3390/fractalfract7040325 - 12 Apr 2023
Viewed by 1382
Abstract
Tool wear will reduce workpieces’ quality and accuracy. In this paper, the vibration signals of the milling process were analyzed, and it was found that historical fluctuations still have an impact on the existing state. First of all, the linear fractional alpha-stable motion [...] Read more.
Tool wear will reduce workpieces’ quality and accuracy. In this paper, the vibration signals of the milling process were analyzed, and it was found that historical fluctuations still have an impact on the existing state. First of all, the linear fractional alpha-stable motion (LFSM) was investigated, along with a differential iterative model with it as the noise term is constructed according to the fractional-order Ito formula; the general solution of this model is derived by semimartingale approximation. After that, for the chaotic features of the vibration signal, the time-frequency domain characteristics were extracted using principal component analysis (PCA), and the relationship between the variation of the generalized Hurst exponent and tool wear was established using multifractal detrended fluctuation analysis (MDFA). Then, the maximum prediction length was obtained by the maximum Lyapunov exponent (MLE), which allows for analysis of the vibration signal. Finally, tool condition diagnosis was carried out by the evolving connectionist system (ECoS). The results show that the LFSM iterative model with semimartingale approximation combined with PCA and MDFA are effective for the prediction of vibration trends and tool condition. Further, the monitoring of tool condition using ECoS is also effective. Full article
(This article belongs to the Special Issue New Trends in Fractional Stochastic Processes)
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17 pages, 345 KiB  
Article
Onsager’s Energy Conservation of Weak Solutions for a Compressible and Inviscid Fluid
by Xinglong Wu and Qian Zhou
Fractal Fract. 2023, 7(4), 324; https://doi.org/10.3390/fractalfract7040324 - 12 Apr 2023
Cited by 1 | Viewed by 1438
Abstract
In this article, two classes of sufficient conditions of weak solutions are given to guarantee the energy conservation of the compressible Euler equations. Our strategy is to introduce a test function φ(t)vϵ to derive the total energy. The [...] Read more.
In this article, two classes of sufficient conditions of weak solutions are given to guarantee the energy conservation of the compressible Euler equations. Our strategy is to introduce a test function φ(t)vϵ to derive the total energy. The velocity field v needs to be regularized both in time and space. In contrast to the noncompressible Euler equations, the compressible flows we consider here do not have a divergence-free structure. Therefore, it is necessary to make an additional estimate of the pressure p, which takes advantage of an appropriate commutator. In addition, by using the weak convergence, we show that the energy equality is conserved in a point-wise sense. Full article
(This article belongs to the Topic Evolutionary Differential Equations, Dynamic Systems, Computation and Optimization)
12 pages, 794 KiB  
Article
A Fast Computational Scheme for Solving the Temporal-Fractional Black–Scholes Partial Differential Equation
by Rouhollah Ghabaei, Taher Lotfi, Malik Zaka Ullah and Stanford Shateyi
Fractal Fract. 2023, 7(4), 323; https://doi.org/10.3390/fractalfract7040323 - 12 Apr 2023
Cited by 1 | Viewed by 1623
Abstract
In this work, we propose a fast scheme based on higher order discretizations on graded meshes for resolving the temporal-fractional partial differential equation (PDE), which benefits the memory feature of fractional calculus. To avoid excessively increasing the number of discretization points, such as [...] Read more.
In this work, we propose a fast scheme based on higher order discretizations on graded meshes for resolving the temporal-fractional partial differential equation (PDE), which benefits the memory feature of fractional calculus. To avoid excessively increasing the number of discretization points, such as the standard finite difference or meshfree methods, and, at the same time, to increase the efficiency of the solver, we employ discretizations on spatially non-uniform meshes with an attention on the non-smoothness area of the underlying asset. Therefore, the PDE problem is transformed to a linear system of algebraic equations. We perform numerical simulations to observe and check the behavior of the presented scheme in contrast to the existing methods. Full article
(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)
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16 pages, 569 KiB  
Article
Fractional-Order Nonlinear Multi-Agent Systems: A Resilience-Based Approach to Consensus Analysis with Distributed and Input Delays
by Asad Khan, Azmat Ullah Khan Niazi, Waseem Abbasi, Faryal Awan and Anam Khan
Fractal Fract. 2023, 7(4), 322; https://doi.org/10.3390/fractalfract7040322 - 11 Apr 2023
Cited by 3 | Viewed by 1482
Abstract
In this article, a resilient consensus analysis of fractional-order nonlinear leader and follower systems with input and distributed delays is assumed. To make controller design more practical, it is considered that the controller is not implemented as it is, and a disturbance term [...] Read more.
In this article, a resilient consensus analysis of fractional-order nonlinear leader and follower systems with input and distributed delays is assumed. To make controller design more practical, it is considered that the controller is not implemented as it is, and a disturbance term is incorporated into the controller part. A multi-agent system’s topology ahead to a weighted graph which may be directed or undirected is used. The article examines a scenario of leader–follower consensus through the application of algebraic graph theory and the fractional-order Razumikhin method. Numerical simulations are also provided to show the effectiveness of the proposed design for the leader–follower consensus. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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15 pages, 319 KiB  
Article
The Convolution Theorem Involving Windowed Free Metaplectic Transform
by Manjun Cui and Zhichao Zhang
Fractal Fract. 2023, 7(4), 321; https://doi.org/10.3390/fractalfract7040321 - 9 Apr 2023
Cited by 1 | Viewed by 1484
Abstract
The convolution product is widely used in many fields, such as signal processing, numerical analysis and so on; however, the convolution theorem in the domain of the windowed metaplectic transformation (WFMT) has not been studied. The primary goal of this paper is to [...] Read more.
The convolution product is widely used in many fields, such as signal processing, numerical analysis and so on; however, the convolution theorem in the domain of the windowed metaplectic transformation (WFMT) has not been studied. The primary goal of this paper is to give the convolution theorem of WFMT. Firstly, we review the definitions of the FMT and WFMT and give the inversion formula of the WFMT and the relationship between the FMT and WFMT. Then, according to the form of the classical convolution theorem and the convolution operator of the FMT, the convolution theorem in the domain of the WFMT is given. Finally, we prove the existence theorems of the proposed convolution theorem. Full article
12 pages, 446 KiB  
Brief Report
Nonexistence of Finite-Time Stable Equilibria in a Class of Nonlinear Integral Equations
by Aldo Jonathan Muñoz-Vázquez, Oscar Martinez-Fuentes and Guillermo Fernández-Anaya
Fractal Fract. 2023, 7(4), 320; https://doi.org/10.3390/fractalfract7040320 - 8 Apr 2023
Viewed by 1474
Abstract
This brief report studies conditions to ensure the nonexistence of finite-time stable equilibria in a class of systems that are described by means of nonlinear integral equations, whose kernels are part of some Sonine kernel pairs. It is firstly demonstrated that, under certain [...] Read more.
This brief report studies conditions to ensure the nonexistence of finite-time stable equilibria in a class of systems that are described by means of nonlinear integral equations, whose kernels are part of some Sonine kernel pairs. It is firstly demonstrated that, under certain criteria, a real-valued function that converges in finite-time to a constant value, different from the initial condition, and remains there afterwards, cannot have a Sonine derivative that also remains at zero after some finite time. Then, the concept of equilibrium is generalized to the case of equivalent equilibrium, and it is demonstrated that a nonlinear integral equation, whose kernel is part of some Sonine kernel pair, cannot possess equivalent finite-time stable equilibria. Finally, illustrative examples are presented. Full article
(This article belongs to the Special Issue Fractional Order Modeling in Interdisciplinary Applications)
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32 pages, 4095 KiB  
Article
Sub-Diffusion Two-Temperature Model and Accurate Numerical Scheme for Heat Conduction Induced by Ultrashort-Pulsed Laser Heating
by Cuicui Ji and Weizhong Dai
Fractal Fract. 2023, 7(4), 319; https://doi.org/10.3390/fractalfract7040319 - 8 Apr 2023
Viewed by 1372
Abstract
In this study, we propose a new sub-diffusion two-temperature model and its accurate numerical method by introducing the Knudsen number (Kn) and two Caputo fractional derivatives (0<α,β<1) in time into the parabolic [...] Read more.
In this study, we propose a new sub-diffusion two-temperature model and its accurate numerical method by introducing the Knudsen number (Kn) and two Caputo fractional derivatives (0<α,β<1) in time into the parabolic two-temperature model of the diffusive type. We prove that the obtained sub-diffusion two-temperature model is well posed. The numerical scheme is obtained based on the L1 approximation for the Caputo fractional derivatives and the second-order finite difference for the spatial derivatives. Using the discrete energy method, we prove the numerical scheme to be unconditionally stable and convergent with O(τmin{2−α,2−β}+h2), where τ,h are time and space steps, respectively. The accuracy and applicability of the present numerical scheme are tested in two examples. Results show that the numerical solutions are accurate, and the present model and its numerical scheme could be used as a tool by changing the values of the Knudsen number and fractional-order derivatives as well as the parameter in the boundary condition for analyzing the heat conduction in porous media, such as porous thin metal films exposed to ultrashort-pulsed lasers, where the energy transports in phonons and electrons may be ultraslow at different rates. Full article
(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)
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18 pages, 352 KiB  
Article
Migration and Proliferation Dichotomy: A Persistent Random Walk of Cancer Cells
by Hamed Al Shamsi
Fractal Fract. 2023, 7(4), 318; https://doi.org/10.3390/fractalfract7040318 - 7 Apr 2023
Cited by 2 | Viewed by 1655
Abstract
A non-Markovian model of tumor cell invasion with finite velocity is proposed to describe the proliferation and migration dichotomy of cancer cells. The model considers transitions with age-dependent switching rates between three states: moving tumor cells in the positive direction, moving tumor cells [...] Read more.
A non-Markovian model of tumor cell invasion with finite velocity is proposed to describe the proliferation and migration dichotomy of cancer cells. The model considers transitions with age-dependent switching rates between three states: moving tumor cells in the positive direction, moving tumor cells in the negative direction, and resting tumor cells. The first two states correspond to a migratory phenotype, while the third state represents a proliferative phenotype. Proliferation is modeled using a logistic growth equation. The transport of tumor cells is described by a persistent random walk with general residence time distributions. The nonlinear master equations describing the average densities of cancer cells for each of the three states are derived. The present work also includes the analysis of models involving power law distributed random time, highlighting the dominance of the Mittag–Leffler rest state, resulting in subdiffusive behavior. Full article
(This article belongs to the Section Life Science, Biophysics)
16 pages, 777 KiB  
Article
A Seventh Order Family of Jarratt Type Iterative Method for Electrical Power Systems
by Saima Yaseen, Fiza Zafar and Francisco I. Chicharro
Fractal Fract. 2023, 7(4), 317; https://doi.org/10.3390/fractalfract7040317 - 6 Apr 2023
Cited by 4 | Viewed by 1844
Abstract
A load flow study referred to as a power flow study is a numerical analysis of the electricity that flows through any electrical power system. For instance, if a transmission line needs to be taken out of service for maintenance, load flow studies [...] Read more.
A load flow study referred to as a power flow study is a numerical analysis of the electricity that flows through any electrical power system. For instance, if a transmission line needs to be taken out of service for maintenance, load flow studies allow us to assess whether the remaining line can carry the load without exceeding its rated capacity. So, we need to understand about the voltage level and voltage phase angle on each bus under steady-state conditions to keep the bus voltage within a specific range. In this paper, our goal is to present a higher order efficient iterative method to carry out a power flow study to determine the voltages (magnitude and angle) for a specific load, generation and network conditions. We introduce a new seventh-order three-step iterative scheme for obtaining approximate solution of nonlinear systems of equations. We attain the seventh-order convergence by using four function evaluations which makes it worthy of interest. Moreover, we show its applicability to the electrical power system for calculating voltages and phase angles. By calculating the bus angle and voltage level, we conclude that the performance of the power system is assessed in a more efficient manner using the new scheme. In addition, dynamical planes of the methods applied on nonlinear systems of equations show global convergence. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
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15 pages, 814 KiB  
Article
Numerical Investigation of the Three-Dimensional HCIR Partial Differential Equation Utilizing a New Localized RBF-FD Method
by Xiaoxia Ma, Malik Zaka Ullah and Stanford Shateyi
Fractal Fract. 2023, 7(4), 316; https://doi.org/10.3390/fractalfract7040316 - 6 Apr 2023
Viewed by 1258
Abstract
This work is concerned with the computational solution of the time-dependent 3D parabolic Heston–Cox–Ingersoll–Ross (HCIR) PDE, which is of practical importance in mathematical finance. The HCIR dynamic states that the model follows randomness for the underlying asset, the volatility and the rate of [...] Read more.
This work is concerned with the computational solution of the time-dependent 3D parabolic Heston–Cox–Ingersoll–Ross (HCIR) PDE, which is of practical importance in mathematical finance. The HCIR dynamic states that the model follows randomness for the underlying asset, the volatility and the rate of interest. Since the PDE formulation has degeneracy and non-smoothness at some area of its domain, we design a new numerical solver via semi-discretization and the radial basis function–finite difference (RBF-FD) scheme. Our scheme is built on graded meshes so as to employ the lowest possible number of discretized nodes. The stability of our solver is proven analytically. Computational testing is conducted to uphold the analytical findings in practice. Full article
(This article belongs to the Section Numerical and Computational Methods)
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23 pages, 3594 KiB  
Article
Driver Training Based Optimized Fractional Order PI-PDF Controller for Frequency Stabilization of Diverse Hybrid Power System
by Guoqiang Zhang, Amil Daraz, Irfan Ahmed Khan, Abdul Basit, Muhammad Irshad Khan and Mirzat Ullah
Fractal Fract. 2023, 7(4), 315; https://doi.org/10.3390/fractalfract7040315 - 6 Apr 2023
Cited by 27 | Viewed by 2115
Abstract
This work provides an enhanced novel cascaded controller-based frequency stabilization of a two-region interconnected power system incorporating electric vehicles. The proposed controller combines a cascade structure comprising a fractional-order proportional integrator and a proportional derivative with a filter term to handle the frequency [...] Read more.
This work provides an enhanced novel cascaded controller-based frequency stabilization of a two-region interconnected power system incorporating electric vehicles. The proposed controller combines a cascade structure comprising a fractional-order proportional integrator and a proportional derivative with a filter term to handle the frequency regulation challenges of a hybrid power system integrated with renewable energy sources. Driver training-based optimization, an advanced stochastic meta-heuristic method based on human learning, is employed to optimize the gains of the proposed cascaded controller. The performance of the proposed novel controller was compared to that of other control methods. In addition, the results of driver training-based optimization are compared to those of other recent meta-heuristic algorithms, such as the imperialist competitive algorithm and jellyfish swarm optimization. The suggested controller and design technique have been evaluated and validated under a variety of loading circumstances and scenarios, as well as their resistance to power system parameter uncertainties. The results indicate the new controller’s steady operation and frequency regulation capability with an optimal controller coefficient and without the prerequisite for a complex layout procedure. Full article
(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering)
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16 pages, 12490 KiB  
Article
Multistability and Jump in the Harmonically Excited SD Oscillator
by Zhenhua Wang and Huilin Shang
Fractal Fract. 2023, 7(4), 314; https://doi.org/10.3390/fractalfract7040314 - 6 Apr 2023
Cited by 6 | Viewed by 1418
Abstract
Coexisting attractors and the consequent jump in a harmonically excited smooth and discontinuous (SD) oscillator with double potential wells are studied in detail herein. The intra-well periodic solutions in the vicinity of the nontrivial equilibria and the inter-well periodic solutions are generated theoretically. [...] Read more.
Coexisting attractors and the consequent jump in a harmonically excited smooth and discontinuous (SD) oscillator with double potential wells are studied in detail herein. The intra-well periodic solutions in the vicinity of the nontrivial equilibria and the inter-well periodic solutions are generated theoretically. Then, their stability and conditions for local bifurcation are discussed. Furthermore, the point mapping method is utilized to depict the fractal basins of attraction of the attractors intuitively. Complex hidden attractors, such as period-3 responses and chaos, are found. It follows that jumps among multiple attractors can be easily triggered by an increase in the excitation level or a small disturbance of the initial condition. The results offer an opportunity for a more comprehensive understanding and better utilization of the multistability characteristics of the SD oscillator. Full article
(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)
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