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Fractal Fract., Volume 7, Issue 1 (January 2023) – 98 articles

Cover Story (view full-size image): Controlling soft robots is a significant challenge due to the nonlinear elastic nature of the soft materials that conform their structure. This paper studies the identification and control problems of a novel two-degrees-of-freedom, tendon-actuated, soft robotic arm. A decoupled identification approach is presented; later, a fractional order control strategy is proposed and tested experimentally, in comparison with PI solutions. The simulation and experimental results show the goodness of the modeling and control approaches discussed. View this paper
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15 pages, 796 KiB  
Article
Local Convergence of Traub’s Method and Its Extensions
by Muhammed Saeed K, Krishnendu Remesh, Santhosh George, Jidesh Padikkal and Ioannis K. Argyros
Fractal Fract. 2023, 7(1), 98; https://doi.org/10.3390/fractalfract7010098 - 16 Jan 2023
Cited by 3 | Viewed by 1641
Abstract
In this article, we examine the local convergence analysis of an extension of Newton’s method in a Banach space setting. Traub introduced the method (also known as the Arithmetic-Mean Newton’s Method and Weerakoon and Fernando method) with an order of convergence of three. [...] Read more.
In this article, we examine the local convergence analysis of an extension of Newton’s method in a Banach space setting. Traub introduced the method (also known as the Arithmetic-Mean Newton’s Method and Weerakoon and Fernando method) with an order of convergence of three. All the previous works either used higher-order Taylor series expansion or could not derive the desired order of convergence. We studied the local convergence of Traub’s method and two of its modifications and obtained the convergence order for these methods without using Taylor series expansion. The radii of convergence, basins of attraction, comparison of iterations of similar iterative methods, approximate computational order of convergence (ACOC), and a representation of the number of iterations are provided. Full article
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17 pages, 3527 KiB  
Article
Proportional-Integral-Derivative Controller Based-Artificial Rabbits Algorithm for Load Frequency Control in Multi-Area Power Systems
by Ragab El-Sehiemy, Abdullah Shaheen, Ahmed Ginidi and Saad F. Al-Gahtani
Fractal Fract. 2023, 7(1), 97; https://doi.org/10.3390/fractalfract7010097 - 16 Jan 2023
Cited by 31 | Viewed by 2394
Abstract
A major problem in power systems is achieving a match between the load demand and generation demand, where security, dependability, and quality are critical factors that need to be provided to power producers. This paper proposes a proportional–integral–derivative (PID) controller that is optimally [...] Read more.
A major problem in power systems is achieving a match between the load demand and generation demand, where security, dependability, and quality are critical factors that need to be provided to power producers. This paper proposes a proportional–integral–derivative (PID) controller that is optimally designed using a novel artificial rabbits algorithm (ARA) for load frequency control (LFC) in multi-area power systems (MAPSs) of two-area non-reheat thermal systems. The PID controller incorporates a filter with such a derivative coefficient to reduce the effects of the accompanied noise. In this regard, single objective function is assessed based on time-domain simulation to minimize the integral time-multiplied absolute error (ITAE). The proposed ARA adjusts the PID settings to their best potential considering three dissimilar test cases with different sets of disturbances, and the results from the designed PID controller based on the ARA are compared with various published techniques, including particle swarm optimization (PSO), differential evolution (DE), JAYA optimizer, and self-adaptive multi-population elitist (SAMPE) JAYA. The comparisons show that the PID controller’s design, which is based on the ARA, handles the load frequency regulation in MAPSs for the ITAE minimizations with significant effectiveness and success where the statistical analysis confirms its superiority. Considering the load change in area 1, the proposed ARA can acquire significant percentage improvements in the ITAE values of 1.949%, 3.455%, 2.077% and 1.949%, respectively, with regard to PSO, DE, JAYA and SAMPE-JAYA. Considering the load change in area 2, the proposed ARA can acquire significant percentage improvements in the ITAE values of 7.587%, 8.038%, 3.322% and 2.066%, respectively, with regard to PSO, DE, JAYA and SAMPE-JAYA. Considering simultaneous load changes in areas 1 and 2, the proposed ARA can acquire significant improvements in the ITAE values of 60.89%, 38.13%, 55.29% and 17.97%, respectively, with regard to PSO, DE, JAYA and SAMPE-JAYA. Full article
(This article belongs to the Topic Advances in Optimization and Nonlinear Analysis Volume II)
(This article belongs to the Section Engineering)
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11 pages, 899 KiB  
Article
A Family of Transformed Difference Schemes for Nonlinear Time-Fractional Equations
by Hongyu Qin, Xiaoli Chen and Boya Zhou
Fractal Fract. 2023, 7(1), 96; https://doi.org/10.3390/fractalfract7010096 - 14 Jan 2023
Cited by 3 | Viewed by 1489
Abstract
In this paper, we present a class of finite difference methods for numerically solving fractional differential equations. Such numerical schemes are developed based on the change in variable and piecewise interpolations. Error analysis of the numerical schemes is obtained by using a Grönwall-type [...] Read more.
In this paper, we present a class of finite difference methods for numerically solving fractional differential equations. Such numerical schemes are developed based on the change in variable and piecewise interpolations. Error analysis of the numerical schemes is obtained by using a Grönwall-type inequality. Numerical examples are given to confirm the theoretical results. Full article
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26 pages, 7792 KiB  
Article
Non-Linear Analysis of Novel Equivalent Circuits of Single-Diode Solar Cell Models with Voltage-Dependent Resistance
by Mahendiran Vellingiri, Muhyaddin Rawa, Sultan Alghamdi, Abdullah A. Alhussainy, Ahmed S. Althobiti, Martin Calasan, Mihailo Micev, Ziad M. Ali and Shady H. E. Abdel Aleem
Fractal Fract. 2023, 7(1), 95; https://doi.org/10.3390/fractalfract7010095 - 14 Jan 2023
Cited by 4 | Viewed by 1916
Abstract
The most commonly used model of solar cells is the single-diode model, with five unknown parameters. First, this paper proposes three variants of the single-diode model, which imply the voltage dependence of the series resistance, parallel resistance, and both resistors. Second, analytical relationships [...] Read more.
The most commonly used model of solar cells is the single-diode model, with five unknown parameters. First, this paper proposes three variants of the single-diode model, which imply the voltage dependence of the series resistance, parallel resistance, and both resistors. Second, analytical relationships between the current and the voltage expressed were derived using the Lambert W function for each proposed model. Third, the paper presents a hybrid algorithm, Chaotic Snake Optimization (Chaotic SO), combining chaotic sequences with the snake optimization algorithm. The application of the proposed models and algorithm was justified on two well-known solar photovoltaic (PV) cells—RTC France solar cell and Photowatt-PWP201 module. The results showed that the root-mean-square-error (RMSE) values calculated by applying the proposed equivalent circuit with voltage dependence of both resistors are reduced by 20% for the RTC France solar cell and 40% for the Photowatt-PWP201 module compared to the standard single-diode equivalent circuit. Finally, an experimental investigation was conducted into the applicability of the proposed models to a solar laboratory module, and the results obtained proved the relevance and effectiveness of the proposed models. Full article
(This article belongs to the Topic Advances in Optimization and Nonlinear Analysis Volume II)
(This article belongs to the Section Engineering)
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22 pages, 892 KiB  
Article
Generalized Shifted Airfoil Polynomials of the Second Kind to Solve a Class of Singular Electrohydrodynamic Fluid Model of Fractional Order
by Hari M. Srivastava and Mohammad Izadi
Fractal Fract. 2023, 7(1), 94; https://doi.org/10.3390/fractalfract7010094 - 14 Jan 2023
Cited by 13 | Viewed by 1353
Abstract
In this manuscript, we find the numerical solutions of a class of fractional-order differential equations with singularity and strong nonlinearity pertaining to electrohydrodynamic flow in a circular cylindrical conduit. The nonlinearity of the underlying model is removed by the quasilinearization method (QLM) and [...] Read more.
In this manuscript, we find the numerical solutions of a class of fractional-order differential equations with singularity and strong nonlinearity pertaining to electrohydrodynamic flow in a circular cylindrical conduit. The nonlinearity of the underlying model is removed by the quasilinearization method (QLM) and we obtain a family of linearized equations. By making use of the generalized shifted airfoil polynomials of the second kind (SAPSK) together with some appropriate collocation points as the roots of SAPSK, we arrive at an algebraic system of linear equations to be solved in an iterative manner. The error analysis and convergence properties of the SAPSK are established in the L2 and L norms. Through numerical simulations, it is shown that the proposed hybrid QLM-SAPSK approach is not only capable of tackling the inherit singularity at the origin, but also produces effective numerical solutions to the model problem with different nonlinearity parameters and two fractional order derivatives. The accuracy of the present technique is checked via the technique of residual error functions. The QLM-SAPSK technique is simple and efficient for solving the underlying electrohydrodynamic flow model. The computational outcomes are accurate in comparison with those of numerical values reported in the literature. Full article
(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)
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13 pages, 2188 KiB  
Article
Hybrid Approach of Fractional Generalized Pareto Motion and Cosine Similarity Hidden Markov Model for Solar Radiation Forecasting
by Wanqing Song, Wujin Deng, Dongdong Chen, Rong Jin and Aleksey Kudreyko
Fractal Fract. 2023, 7(1), 93; https://doi.org/10.3390/fractalfract7010093 - 13 Jan 2023
Cited by 5 | Viewed by 1684
Abstract
Power from solar energy is not reliable, due to weather-related factors, which diminishes the power system’s reliability. Therefore, this study suggests a way to predict the intensity of solar irradiance using various statistical algorithms and artificial intelligence. In particular, we suggest the use [...] Read more.
Power from solar energy is not reliable, due to weather-related factors, which diminishes the power system’s reliability. Therefore, this study suggests a way to predict the intensity of solar irradiance using various statistical algorithms and artificial intelligence. In particular, we suggest the use of a hybrid predictive model, combining statistical properties and historical data training. In order to evaluate the maximum prediction steps of solar irradiance, the maximum Lyapunov exponent was applied. Then, we used the cosine similarity algorithm in the hidden Markov model for the initial prediction. The combination of the Hurst exponent and tail parameter revealed the self-similarity and long-range dependence of the fractional generalized Pareto motion, which enabled us to consider the iterative predictive model. The initial prediction was substituted into a stochastic differential equation to achieve the final prediction, which prevents error propagation. The effectiveness of the hybrid model was demonstrated in the case study. Full article
(This article belongs to the Special Issue New Trends in Fractional Stochastic Processes)
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42 pages, 629 KiB  
Article
Stability and Controllability Study for Mixed Integral Fractional Delay Dynamic Systems Endowed with Impulsive Effects on Time Scales
by Hasanen A. Hammad and Manuel De la Sen
Fractal Fract. 2023, 7(1), 92; https://doi.org/10.3390/fractalfract7010092 - 13 Jan 2023
Cited by 17 | Viewed by 1694
Abstract
In this article, we investigate a novel class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. Also, fixed-point techniques are applied to study the existence and uniqueness of a solution to the considered systems. Furthermore, sufficient conditions for [...] Read more.
In this article, we investigate a novel class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. Also, fixed-point techniques are applied to study the existence and uniqueness of a solution to the considered systems. Furthermore, sufficient conditions for Ulam–Hyers stability and controllability of the considered systems are established. It turns out that controllability is a very relevant property in dynamic systems and also in differential equations since, if controllability holds, then the solution of a system of differential equations also holds. Finally, an illustrative example of the obtained results is provided. Full article
(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)
18 pages, 998 KiB  
Article
The Dynamic Effects of COVID-19 and the March 2020 Crash on the Multifractality of NASDAQ Insurance Stock Markets
by Xing Li and Fang Su
Fractal Fract. 2023, 7(1), 91; https://doi.org/10.3390/fractalfract7010091 - 13 Jan 2023
Cited by 6 | Viewed by 1605
Abstract
Triggered by COVID-19, one of the most dramatic crashes in the stock market in history occurred in March 2020. The sharp reductions in NASDAQ insurance stock indexes were observed after the occurrence of COVID-19 and in March 2020. In this study, the NASDAQ [...] Read more.
Triggered by COVID-19, one of the most dramatic crashes in the stock market in history occurred in March 2020. The sharp reductions in NASDAQ insurance stock indexes were observed after the occurrence of COVID-19 and in March 2020. In this study, the NASDAQ insurance stock markets (including NASDAQ Insurance Index, Developed Markets Insurance Index, and Emerging Markets Insurance Index) and NASDAQ Composite Index are utilized. The “scissors difference” between the NASDAQ Insurance Index and NASDAQ Composite Index is observed. The dynamic effects of the COVID-19 epidemic and the March 2020 crash on the multifractality of four series are explored. Firstly, the apparent and intrinsic multifractality, the components of multifractality, and the dynamic effects of the COVID-19 epidemic on these indexes are analyzed. Secondly, the multifractal cross-correlation between the NASDAQ Insurance Index and NASDAQ Composite Index is investigated. The dynamic influence of the COVID-19 epidemic on the cross-correlation is examined. The multifractal analysis results reveal that four series both before and after the occurrence of COVID-19 have multifractal characteristics. The stronger multifractal characteristics and the greater multifractal degree are obtained after the occurrence of COVID-19. The intrinsic multifractality of the three indexes ascends largely after the occurrence of COVID-19. The multifractal cross-correlation analysis illustrates that the cross-correlation between two indexes before and after the occurrence of COVID-19 is multifractal. The stronger multifractal cross-correlations and greater multifractal degrees are shown. The contribution of the intrinsic multifractal cross-correlation increased after the occurrence of COVID-19. Full article
(This article belongs to the Special Issue Fractal and Multifractal Analysis in Financial Markets)
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11 pages, 329 KiB  
Article
An ε-Approximate Approach for Solving Variable-Order Fractional Differential Equations
by Yahong Wang, Wenmin Wang, Liangcai Mei, Yingzhen Lin and Hongbo Sun
Fractal Fract. 2023, 7(1), 90; https://doi.org/10.3390/fractalfract7010090 - 13 Jan 2023
Cited by 1 | Viewed by 1341
Abstract
As a mathematical tool, variable-order (VO) fractional calculus (FC) was developed rapidly in the engineering field due to it better describing the anomalous diffusion problems in engineering; thus, the research of the solutions of VO fractional differential equations (FDEs) has become a hot [...] Read more.
As a mathematical tool, variable-order (VO) fractional calculus (FC) was developed rapidly in the engineering field due to it better describing the anomalous diffusion problems in engineering; thus, the research of the solutions of VO fractional differential equations (FDEs) has become a hot topic for the FC community. In this paper, we propose an effective numerical method, named as the ε-approximate approach, based on the least squares theory and the idea of residuals, for the solutions of VO-FDEs and VO fractional integro-differential equations (VO-FIDEs). First, the VO-FDEs and VO-FIDEs are considered to be analyzed in appropriate Sobolev spaces H2n[0,1] and the corresponding orthonormal bases are constructed based on scale functions. Then, the space H2,02[0,1] is chosen which is just suitable for one of the models the authors want to solve to demonstrate the algorithm. Next, the numerical scheme is given, and the stability and convergence are discussed. Finally, four examples with different characteristics are shown, which reflect the accuracy, effectiveness, and wide application of the algorithm. Full article
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22 pages, 6980 KiB  
Article
Modified FOPID Controller for Frequency Regulation of a Hybrid Interconnected System of Conventional and Renewable Energy Sources
by Amil Daraz, Suheel Abdullah Malik, Abdul Basit, Sheraz Aslam and Guoqiang Zhang
Fractal Fract. 2023, 7(1), 89; https://doi.org/10.3390/fractalfract7010089 - 13 Jan 2023
Cited by 40 | Viewed by 2631
Abstract
In this article, a fractional-order proportional-integral-differential (FOPID) controller and its modified structure, called a MFOPID controller, are presented. To guarantee optimal system performance, the gains of the proposed FOPID and MFOPID controllers are well-tuned, employing the Jellyfish Search Optimizer (JSO), a novel and [...] Read more.
In this article, a fractional-order proportional-integral-differential (FOPID) controller and its modified structure, called a MFOPID controller, are presented. To guarantee optimal system performance, the gains of the proposed FOPID and MFOPID controllers are well-tuned, employing the Jellyfish Search Optimizer (JSO), a novel and highly effective bioinspired metaheuristic approach. The proposed controllers are assessed in a hybrid system with two domains, where each domain contains a hybrid of conventional (gas, reheat, and hydro) and renewable generation sources (solar and wind). For a more realistic analysis, the presented system model includes practical limitations with nonlinear characteristics, such as governor dead zone/band (GDZ/GDB), boiler dynamics, generation rate limitation/constraint (GRL/GRC), system uncertainties, communication time delay (CTD), and load changes. The suggested methodology outperforms some newly developed heuristic techniques, including fitness-dependent optimizer (FDO), sine-cosine algorithm (SCA), and firefly algorithm (FA), for the interconnected power system (PS) of two regions with multiple generating units. Furthermore, the proposed MFOPID controller is compared with JSO-tuned PID/FOPID and PI controllers to ascertain its superiority. The results signify that the presented control method and its parametric optimization significantly outperforms the other control strategies with respect to minimum undershoot and peak overshoot, settling times, and ITSE in the system’s dynamic response. The sensitivity analysis outcomes imply that the proposed JSO-MFOPID control method is very reliable and can effectively stabilize the load frequency and interconnection line in a multi-area network with interconnected PS. Full article
(This article belongs to the Special Issue Advances in Fractional-Order PID Controllers)
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26 pages, 443 KiB  
Article
Almost Automorphic Strong Oscillation in Time-Fractional Parabolic Equations
by Tianwei Zhang, Yongkun Li and Jianwen Zhou
Fractal Fract. 2023, 7(1), 88; https://doi.org/10.3390/fractalfract7010088 - 12 Jan 2023
Cited by 2 | Viewed by 1197
Abstract
This paper gives some results on almost automorphic strong solutions to time-fractional partial differential equations by employing a mix o thef Galerkin method, Fourier series, and Picard iteration. As an application, the existence, uniqueness, and global Mittag–Leffler convergence of almost automorphic strong solution [...] Read more.
This paper gives some results on almost automorphic strong solutions to time-fractional partial differential equations by employing a mix o thef Galerkin method, Fourier series, and Picard iteration. As an application, the existence, uniqueness, and global Mittag–Leffler convergence of almost automorphic strong solution are discussed to a concrete time-fractional parabolic equations. To the best of our knowledge, this is the first study on almost automorphic strong solutions on this subject. Full article
(This article belongs to the Section General Mathematics, Analysis)
22 pages, 370 KiB  
Article
Fractional Stochastic Integro-Differential Equations with Nonintantaneous Impulses: Existence, Approximate Controllability and Stochastic Iterative Learning Control
by Kinda Abuasbeh, Nazim I. Mahmudov and Muath Awadalla
Fractal Fract. 2023, 7(1), 87; https://doi.org/10.3390/fractalfract7010087 - 12 Jan 2023
Cited by 1 | Viewed by 1211
Abstract
In this paper, existence/uniqueness of solutions and approximate controllability concept for Caputo type stochastic fractional integro-differential equations (SFIDE) in a Hilbert space with a noninstantaneous impulsive effect are studied. In addition, we study different types of stochastic iterative learning control for SFIDEs with [...] Read more.
In this paper, existence/uniqueness of solutions and approximate controllability concept for Caputo type stochastic fractional integro-differential equations (SFIDE) in a Hilbert space with a noninstantaneous impulsive effect are studied. In addition, we study different types of stochastic iterative learning control for SFIDEs with noninstantaneous impulses in Hilbert spaces. Finally, examples are given to support the obtained results. Full article
(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis II)
18 pages, 426 KiB  
Article
A New Look at the Capacitor Theory
by Manuel Duarte Ortigueira, Valeriy Martynyuk, Volodymyr Kosenkov and Arnaldo Guimarães Batista
Fractal Fract. 2023, 7(1), 86; https://doi.org/10.3390/fractalfract7010086 - 12 Jan 2023
Cited by 12 | Viewed by 2592
Abstract
The mathematical description of the charging process of time-varying capacitors is reviewed and a new formulation is proposed. For it, suitable fractional derivatives are described. The case of fractional capacitors that follow the Curie–von Schweidler law is considered. Through suitable substitutions, a similar [...] Read more.
The mathematical description of the charging process of time-varying capacitors is reviewed and a new formulation is proposed. For it, suitable fractional derivatives are described. The case of fractional capacitors that follow the Curie–von Schweidler law is considered. Through suitable substitutions, a similar scheme for fractional inductors is obtained. Formulae for voltage/current input/output are presented. Backward coherence with classic results is established and generalised to the variable order case. The concept of a tempered fractor is introduced and related to the Davidson–Cole model. Full article
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13 pages, 3767 KiB  
Article
Analyzing Asymmetric Volatility and Multifractal Behavior in Cryptocurrencies Using Capital Asset Pricing Model Filter
by Minhyuk Lee, Younghwan Cho, Seung Eun Ock and Jae Wook Song
Fractal Fract. 2023, 7(1), 85; https://doi.org/10.3390/fractalfract7010085 - 12 Jan 2023
Cited by 2 | Viewed by 2127
Abstract
This research analyzes asymmetric volatility and multifractality in four representative cryptocurrencies using index-based asymmetric multifractal detrended fluctuation analysis. We suggest investigating an idiosyncratic risk premium, which can be obtained by removing the market influence in the cryptocurrency return series. We call the process [...] Read more.
This research analyzes asymmetric volatility and multifractality in four representative cryptocurrencies using index-based asymmetric multifractal detrended fluctuation analysis. We suggest investigating an idiosyncratic risk premium, which can be obtained by removing the market influence in the cryptocurrency return series. We call the process a capital asset pricing model filter. The analyses on the original return series showed no significant sign of asymmetric volatility. However, the filter revealed a distinct asymmetric volatility, distinguishing the uptrend and downtrend fluctuations. Furthermore, the analyses on the idiosyncratic risk premium detected some cases of asymmetry in the degree and source of multifractality, whereas that on the original return series failed to detect the asymmetry. In conclusion, in a highly volatile market, the capital asset pricing model filter can improve an investigation of the asymmetric multifractality in cryptocurrencies. Full article
(This article belongs to the Special Issue Fractal and Multifractal Analysis in Financial Markets)
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18 pages, 1208 KiB  
Article
On the Validation of a Fractional Order Model for Pharmacokinetics Using Clinical Data
by Sinenhlanhla Mtshali and Byron A. Jacobs
Fractal Fract. 2023, 7(1), 84; https://doi.org/10.3390/fractalfract7010084 - 11 Jan 2023
Cited by 3 | Viewed by 1452
Abstract
This study aims to validate the hypothesis that the pharmacokinetics of certain drug regimes are better captured using fractional order differential equations rather than ordinary differential equations. To support this research, two numerical methods, the Grunwald–Letnikov and the L1 approximation, were implemented for [...] Read more.
This study aims to validate the hypothesis that the pharmacokinetics of certain drug regimes are better captured using fractional order differential equations rather than ordinary differential equations. To support this research, two numerical methods, the Grunwald–Letnikov and the L1 approximation, were implemented for the two-compartment model with Michaelis–Menten clearance kinetics for oral and intravenous administration of the drug. The efficacy of the numerical methods is verified through the use of the method of manufactured solutions due to the absence of an analytic solution to the proposed model. The model is derived from a phenomenological process leading to a dimensionally consistent and physically meaningful model. Using clinical data, the model is validated, and it is shown that the optimal model parameters select a fractional order for the clearance dynamic for certain drug regimes. These findings support the hypothesis that fractional differential equations better describe some pharmacokinetics. Full article
(This article belongs to the Section General Mathematics, Analysis)
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19 pages, 6885 KiB  
Article
Conservative Continuous-Stage Stochastic Runge–Kutta Methods for Stochastic Differential Equations
by Xiuyan Li, Zhenyu Wang, Qiang Ma and Xiaohua Ding
Fractal Fract. 2023, 7(1), 83; https://doi.org/10.3390/fractalfract7010083 - 11 Jan 2023
Cited by 1 | Viewed by 1494
Abstract
In this paper, we develop a new class of conservative continuous-stage stochastic Runge–Kutta methods for solving stochastic differential equations with a conserved quantity. The order conditions of the continuous-stage stochastic Runge–Kutta methods are given based on the theory of stochastic B-series and multicolored [...] Read more.
In this paper, we develop a new class of conservative continuous-stage stochastic Runge–Kutta methods for solving stochastic differential equations with a conserved quantity. The order conditions of the continuous-stage stochastic Runge–Kutta methods are given based on the theory of stochastic B-series and multicolored rooted tree. Sufficient conditions for the continuous-stage stochastic Runge–Kutta methods preserving the conserved quantity of stochastic differential equations are derived in terms of the coefficients. Conservative continuous-stage stochastic Runge–Kutta methods of mean square convergence order 1 for general stochastic differential equations, as well as conservative continuous-stage stochastic Runge–Kutta methods of high order for single integrand stochastic differential equations, are constructed. Numerical experiments are performed to verify the conservative property and the accuracy of the proposed methods in the longtime simulation. Full article
(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)
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18 pages, 9203 KiB  
Article
Multistability and Phase Synchronization of Rulkov Neurons Coupled with a Locally Active Discrete Memristor
by Minglin Ma, Yaping Lu, Zhijun Li, Yichuang Sun and Chunhua Wang
Fractal Fract. 2023, 7(1), 82; https://doi.org/10.3390/fractalfract7010082 - 11 Jan 2023
Cited by 49 | Viewed by 2298
Abstract
In order to enrich the dynamic behaviors of discrete neuron models and more effectively mimic biological neural networks, this paper proposes a bistable locally active discrete memristor (LADM) model to mimic synapses. We explored the dynamic behaviors of neural networks by introducing the [...] Read more.
In order to enrich the dynamic behaviors of discrete neuron models and more effectively mimic biological neural networks, this paper proposes a bistable locally active discrete memristor (LADM) model to mimic synapses. We explored the dynamic behaviors of neural networks by introducing the LADM into two identical Rulkov neurons. Based on numerical simulation, the neural network manifested multistability and new firing behaviors under different system parameters and initial values. In addition, the phase synchronization between the neurons was explored. Additionally, it is worth mentioning that the Rulkov neurons showed synchronization transition behavior; that is, anti-phase synchronization changed to in-phase synchronization with the change in the coupling strength. In particular, the anti-phase synchronization of different firing patterns in the neural network was investigated. This can characterize the different firing behaviors of coupled homogeneous neurons in the different functional areas of the brain, which is helpful to understand the formation of functional areas. This paper has a potential research value and lays the foundation for biological neuron experiments and neuron-based engineering applications. Full article
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13 pages, 418 KiB  
Article
New Fractional Integral Inequalities Pertaining to Center-Radius (cr)-Ordered Convex Functions
by Soubhagya Kumar Sahoo, Hleil Alrweili, Savin Treanţă and Zareen A. Khan
Fractal Fract. 2023, 7(1), 81; https://doi.org/10.3390/fractalfract7010081 - 11 Jan 2023
Viewed by 1271
Abstract
In this work, we use the idea of interval-valued convex functions of Center-Radius (cr)-order to give fractional versions of Hermite–Hadamard inequality. The results are supported by some numerical estimations and graphical representations considering some suitable examples. The results are novel in [...] Read more.
In this work, we use the idea of interval-valued convex functions of Center-Radius (cr)-order to give fractional versions of Hermite–Hadamard inequality. The results are supported by some numerical estimations and graphical representations considering some suitable examples. The results are novel in the context of cr-convex interval-valued functions and deal with differintegrals of the p+s2 type. We believe this will be an important contribution to spurring additional research. Full article
(This article belongs to the Topic Advances in Optimization and Nonlinear Analysis Volume II)
(This article belongs to the Section Engineering)
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12 pages, 320 KiB  
Article
Asymptotic Behavior of Delayed Reaction-Diffusion Neural Networks Modeled by Generalized Proportional Caputo Fractional Partial Differential Equations
by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan
Fractal Fract. 2023, 7(1), 80; https://doi.org/10.3390/fractalfract7010080 - 11 Jan 2023
Viewed by 1401
Abstract
In this paper, a delayed reaction-diffusion neural network model of fractional order and with several constant delays is considered. Generalized proportional Caputo fractional derivatives with respect to the time variable are applied, and this type of derivative generalizes several known types in the [...] Read more.
In this paper, a delayed reaction-diffusion neural network model of fractional order and with several constant delays is considered. Generalized proportional Caputo fractional derivatives with respect to the time variable are applied, and this type of derivative generalizes several known types in the literature for fractional derivatives such as the Caputo fractional derivative. Thus, the obtained results additionally generalize some known models in the literature. The long term behavior of the solution of the model when the time is increasing without a bound is studied and sufficient conditions for approaching zero are obtained. Lyapunov functions defined as a sum of squares with their generalized proportional Caputo fractional derivatives are applied and a comparison result for a scalar linear generalized proportional Caputo fractional differential equation with several constant delays is presented. Lyapunov functions and the comparison principle are then combined to establish our main results. Full article
(This article belongs to the Section Mathematical Physics)
14 pages, 399 KiB  
Article
Theoretical Analysis and Simulation of a Fractional-Order Compartmental Model with Time Delay for the Propagation of Leprosy
by Zafar Iqbal, Nauman Ahmed and Jorge E. Macías-Díaz
Fractal Fract. 2023, 7(1), 79; https://doi.org/10.3390/fractalfract7010079 - 10 Jan 2023
Cited by 1 | Viewed by 1631
Abstract
This article investigates the propagation of a deadly human disease, namely leprosy. At the outset, the mathematical model is transformed into a fractional-order model by introducing the Caputo differential operator of arbitrary order. A result is established, which ensures the positivity of the [...] Read more.
This article investigates the propagation of a deadly human disease, namely leprosy. At the outset, the mathematical model is transformed into a fractional-order model by introducing the Caputo differential operator of arbitrary order. A result is established, which ensures the positivity of the fractional-order epidemic model. The stability of the continuous model at different points of equilibria is investigated. The basic reproduction number, R0, is obtained for the leprosy model. It is observed that the leprosy system is locally asymptotically stable at both steady states when R0<1. On the other hand, the fractional-order system is globally asymptotically stable when R0>1. To find the approximate solutions for the continuous epidemic model, a non-standard numerical scheme is constructed. The main features of the non-standard scheme (such as positivity and boundedness of the numerical method) are also confirmed by applying some benchmark results. Simulations and a feasible test example are presented to discern the properties of the numerical method. Our computational results confirm both the analytical and the numerical properties of the finite-difference scheme. Full article
(This article belongs to the Section Mathematical Physics)
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25 pages, 1164 KiB  
Article
Metaheuristic Solution for Stability Analysis of Nonlinear Systems Using an Intelligent Algorithm with Potential Applications
by Faiçal Hamidi, Houssem Jerbi, Hadeel Alharbi, Víctor Leiva, Dumitru Popescu and Wajdi Rajhi
Fractal Fract. 2023, 7(1), 78; https://doi.org/10.3390/fractalfract7010078 - 10 Jan 2023
Cited by 7 | Viewed by 1300
Abstract
In this article, we provide a metaheuristic-based solution for stability analysis of nonlinear systems. We identify the optimal level set in the state space of these systems by combining two optimization phases. This set is in a definite negative region of the time [...] Read more.
In this article, we provide a metaheuristic-based solution for stability analysis of nonlinear systems. We identify the optimal level set in the state space of these systems by combining two optimization phases. This set is in a definite negative region of the time derivative for a polynomial Lyapunov function (LF). Then, we consider a global optimization problem stated in two phases. The first phase is an external optimization to search for a definite positive LF, whose derivative is definite negative under linear matrix inequalities. The candidate LF coefficients are adjusted using a Jaya metaheuristic optimization algorithm. The second phase is an internal optimization to ensure an accurate estimate of the attraction region for each candidate LF that is optimized externally. The key idea of the algorithm is based mainly on a Jaya optimization, which provides an efficient way to characterize accurately the volume and shape of the maximal attraction domains. We conduct numerical experiments to validate the proposed approach. Two potential real-world applications are proposed. Full article
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29 pages, 800 KiB  
Review
A Review on Fractional-Order Modelling and Control of Robotic Manipulators
by Kishore Bingi, B Rajanarayan Prusty and Abhaya Pal Singh
Fractal Fract. 2023, 7(1), 77; https://doi.org/10.3390/fractalfract7010077 - 10 Jan 2023
Cited by 23 | Viewed by 2924
Abstract
Robot manipulators are widely used in many fields and play a vital role in the assembly, maintenance, and servicing of future complex in-orbit infrastructures. They are also helpful in areas where it is undesirable for humans to go, for instance, during undersea exploration, [...] Read more.
Robot manipulators are widely used in many fields and play a vital role in the assembly, maintenance, and servicing of future complex in-orbit infrastructures. They are also helpful in areas where it is undesirable for humans to go, for instance, during undersea exploration, in radioactive surroundings, and other hazardous places. Robotic manipulators are highly coupled and non-linear multivariable mechanical systems designed to perform one of these specific tasks. Further, the time-varying constraints and uncertainties of robotic manipulators will adversely affect the characteristics and response of these systems. Therefore, these systems require effective modelling and robust controllers to handle such complexities, which is challenging for control engineers. To solve this problem, many researchers have used the fractional-order concept in the modelling and control of robotic manipulators; yet it remains a challenge. This review paper presents comprehensive and significant research on state-of-the-art fractional-order modelling and control strategies for robotic manipulators. It also aims to provide a control engineering community for better understanding and up-to-date knowledge of fractional-order modelling, control trends, and future directions. The main table summarises around 95 works closely related to the mentioned issue. Key areas focused on include modelling, fractional-order modelling type, model order, fractional-order control, controller parameters, comparison controllers, tuning techniques, objective function, fractional-order definitions and approximation techniques, simulation tools and validation type. Trends for existing research have been broadly studied and depicted graphically. Further, future perspective and research gaps have also been discussed comprehensively. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Calculus in Robotics)
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29 pages, 2644 KiB  
Article
Escape Criteria for Generating Fractals of Complex Functions Using DK-Iterative Scheme
by Asifa Tassaddiq, Muhammad Tanveer, Muhammad Azhar, Muhammad Arshad and Farha Lakhani
Fractal Fract. 2023, 7(1), 76; https://doi.org/10.3390/fractalfract7010076 - 10 Jan 2023
Cited by 4 | Viewed by 1861
Abstract
Fractals are essential in representing the natural environment due to their important characteristic of self similarity. The dynamical behavior of fractals mostly depends on escape criteria using different iterative techniques. In this article, we establish an escape criteria using DK-iteration as well as [...] Read more.
Fractals are essential in representing the natural environment due to their important characteristic of self similarity. The dynamical behavior of fractals mostly depends on escape criteria using different iterative techniques. In this article, we establish an escape criteria using DK-iteration as well as complex sine function (sin(zm)+c;m2,cC) and complex exponential function (ezm+c;m2,cC). We use this to analyze the dynamical behavior of specific fractals namely Julia set and Mandelbrot set. This is achieved by generalizing the existing algorithms, which led to the visualization of beautiful fractals for m=2,3 and 4. Moreover, the image generation time in seconds using different values of input parameters is also computed. Full article
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18 pages, 4564 KiB  
Article
Construction of New Infinite-Series Exact Solitary Wave Solutions and Its Application to the Korteweg–De Vries Equation
by Guojiang Wu and Yong Guo
Fractal Fract. 2023, 7(1), 75; https://doi.org/10.3390/fractalfract7010075 - 9 Jan 2023
Cited by 4 | Viewed by 1524
Abstract
The Korteweg–de Vries (KDV) equation is one of the most well-known models in nonlinear physics, such as fluid physics, plasma, and ocean engineering. It is very important to obtain the exact solutions of this model in the process of studying these topics. In [...] Read more.
The Korteweg–de Vries (KDV) equation is one of the most well-known models in nonlinear physics, such as fluid physics, plasma, and ocean engineering. It is very important to obtain the exact solutions of this model in the process of studying these topics. In the present paper, using distinct function iteration relations in two ways, namely, squaring infinitely and extracting the square root infinitely, which have not been reported in other documents, we construct abundant types of new infinite-series exact solitary wave solutions using the auxiliary equation method. Most of these solutions have not been reported in previous papers. The numerical analysis of some solutions shows complex solitary wave phenomena. Some solutions can have stable solitary wave structures, while others may have singularities in certain space–time positions. The results show that the analysis model we use is very simple and effective for the construction of new infinite-series solutions and new solitary wave structures of nonlinear models. Full article
(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)
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22 pages, 675 KiB  
Article
Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials
by Waleed Mohamed Abd-Elhameed and Muhammad Mahmoud Alsuyuti
Fractal Fract. 2023, 7(1), 74; https://doi.org/10.3390/fractalfract7010074 - 9 Jan 2023
Cited by 14 | Viewed by 1466
Abstract
The main aim of this paper is to introduce a new class of orthogonal polynomials that generalizes the class of Chebyshev polynomials of the first kind. Some basic properties of the generalized Chebyshev polynomials and their shifted ones are established. Additionally, some new [...] Read more.
The main aim of this paper is to introduce a new class of orthogonal polynomials that generalizes the class of Chebyshev polynomials of the first kind. Some basic properties of the generalized Chebyshev polynomials and their shifted ones are established. Additionally, some new formulas concerned with these generalized polynomials are established. These generalized orthogonal polynomials are employed to treat the multi-term linear fractional differential equations (FDEs) that include some specific problems that arise in many applications. The basic idea behind the derivation of our proposed algorithm is built on utilizing a new power form representation of the shifted generalized Chebyshev polynomials along with the application of the spectral Galerkin method to transform the FDE governed by its initial conditions into a system of linear equations that can be efficiently solved via a suitable numerical solver. Some illustrative examples accompanied by comparisons with some other methods are presented to show that the presented algorithm is useful and effective. Full article
(This article belongs to the Section Mathematical Physics)
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20 pages, 392 KiB  
Article
Abstract Impulsive Volterra Integro-Differential Inclusions
by Wei-Shih Du, Marko Kostić and Daniel Velinov
Fractal Fract. 2023, 7(1), 73; https://doi.org/10.3390/fractalfract7010073 - 9 Jan 2023
Cited by 6 | Viewed by 1122
Abstract
In this work, we provide several applications of (a, k)-regularized C-resolvent families to the abstract impulsive Volterra integro-differential inclusions. The resolvent operator families under our consideration are subgenerated by multivalued linear operators, which can degenerate in the time variable. The [...] Read more.
In this work, we provide several applications of (a, k)-regularized C-resolvent families to the abstract impulsive Volterra integro-differential inclusions. The resolvent operator families under our consideration are subgenerated by multivalued linear operators, which can degenerate in the time variable. The use of regularizing operator C seems to be completely new within the theory of the abstract impulsive Volterra integro-differential equations. Full article
(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)
12 pages, 14347 KiB  
Article
A New Perspective on the Exact Solutions of the Local Fractional Modified Benjamin–Bona–Mahony Equation on Cantor Sets
by Kang-Jia Wang and Feng Shi
Fractal Fract. 2023, 7(1), 72; https://doi.org/10.3390/fractalfract7010072 - 9 Jan 2023
Cited by 20 | Viewed by 1440
Abstract
A new local fractional modified Benjamin–Bona–Mahony equation is proposed within the local fractional derivative in this study for the first time. By defining some elementary functions via the Mittag–Leffler function (MLF) on the Cantor sets (CSs), a set of nonlinear local fractional ordinary [...] Read more.
A new local fractional modified Benjamin–Bona–Mahony equation is proposed within the local fractional derivative in this study for the first time. By defining some elementary functions via the Mittag–Leffler function (MLF) on the Cantor sets (CSs), a set of nonlinear local fractional ordinary differential equations (NLFODEs) is constructed. Then, a fast algorithm namely Yang’s special function method is employed to find the non-differentiable (ND) exact solutions. By this method, we can extract abundant exact solutions in just one step. Finally, the obtained solutions on the CS are outlined in the form of the 3-D plot. The whole calculation process clearly shows that Yang’s special function method is simple and effective, and can be applied to investigate the exact ND solutions of the other local fractional PDEs. Full article
(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)
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23 pages, 3270 KiB  
Article
New Explicit Propagating Solitary Waves Formation and Sensitive Visualization of the Dynamical System
by Rana Muhammad Zulqarnain, Wen-Xiu Ma, Sayed M. Eldin, Khush Bukht Mehdi and Waqas Ali Faridi
Fractal Fract. 2023, 7(1), 71; https://doi.org/10.3390/fractalfract7010071 - 9 Jan 2023
Cited by 9 | Viewed by 1503
Abstract
This work discusses the soliton solutions for the fractional complex Ginzburg–Landau equation in Kerr law media. It is a particularly fascinating model in this context as it is a dissipative variant of the Hamiltonian nonlinear Schrödinger equation with solutions that create localized singularities [...] Read more.
This work discusses the soliton solutions for the fractional complex Ginzburg–Landau equation in Kerr law media. It is a particularly fascinating model in this context as it is a dissipative variant of the Hamiltonian nonlinear Schrödinger equation with solutions that create localized singularities in finite time. The ϕ6-model technique is one of the generalized methodologies exerted on the fractional complex Ginzburg–Landau equation to find the new solitary wave profiles. As a result, solitonic wave patterns develop, including Jacobi elliptic function, periodic, dark, bright, single, dark-bright, exponential, trigonometric, and rational solitonic structures, among others. The assurance of the practicality of the solitary wave results is provided by the constraint condition corresponding to each achieved solution. The graphical 3D and contour depiction of the attained outcomes is shown to define the pulse propagation behaviors while imagining the pertinent data for the involved parameters. The sensitive analysis predicts the dependence of the considered model on initial conditions. It is a reliable and efficient technique used to generate generalized solitonic wave profiles with diverse soliton families. Furthermore, we ensure that all results are innovative and mark remarkable impacts on the prevailing solitary wave theory literature. Full article
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20 pages, 49814 KiB  
Article
An End-to-End Underwater-Image-Enhancement Framework Based on Fractional Integral Retinex and Unsupervised Autoencoder
by Yang Yu and Chenfeng Qin
Fractal Fract. 2023, 7(1), 70; https://doi.org/10.3390/fractalfract7010070 - 9 Jan 2023
Cited by 3 | Viewed by 2718
Abstract
As an essential low-level computer vision task for remotely operated underwater robots and unmanned underwater vehicles to detect and understand the underwater environment, underwater image enhancement is facing challenges of light scattering, absorption, and distortion. Instead of using a specific underwater imaging model [...] Read more.
As an essential low-level computer vision task for remotely operated underwater robots and unmanned underwater vehicles to detect and understand the underwater environment, underwater image enhancement is facing challenges of light scattering, absorption, and distortion. Instead of using a specific underwater imaging model to mitigate the degradation of underwater images, we propose an end-to-end underwater-image-enhancement framework that combines fractional integral-based Retinex and an encoder–decoder network. The proposed variant of Retinex aims to alleviate haze and color distortion in the input image while preserving edges to a large extent by utilizing a modified fractional integral filter. The encoder–decoder network with channel-wise attention modules trained in an unsupervised manner to overcome the lack of paired underwater image datasets is designed to refine the output of the Retinex. Our framework was evaluated under qualitative and quantitative metrics on several public underwater image datasets and yielded satisfactory enhancement results on the evaluation set. Full article
(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)
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14 pages, 10034 KiB  
Article
LFM Signal Parameter Estimation via FTD-FRFT in Impulse Noise
by Xuelian Liu, Xuemei Li, Bo Xiao, Chunyang Wang and Bo Ma
Fractal Fract. 2023, 7(1), 69; https://doi.org/10.3390/fractalfract7010069 - 7 Jan 2023
Cited by 3 | Viewed by 1366
Abstract
LFM signals are widely applied in radar, communication, sonar and many other fields. LFM signals received by these systems contain a lot of noise and outliers. In order to suppress the interference of strong impulse noise on target signals and realize the accurate [...] Read more.
LFM signals are widely applied in radar, communication, sonar and many other fields. LFM signals received by these systems contain a lot of noise and outliers. In order to suppress the interference of strong impulse noise on target signals and realize the accurate estimation of LFM signal parameters, the impulse noise of echo signals need to be filtered. In this paper, to solve the problem of poor performance of LFM signal parameter estimation based on fractional Fourier transform in impulse noise, alpha stable distribution is used to establish the mathematical model of impulse noise. The proposed fastest tracking differentiator with an adaptive tracking factor is used to suppress the strong impulse noise, and fractional Fourier transform is used to estimate the parameter of the LFM signals. The experimental results show that the proposed fastest tracking differentiator with an adaptive tracking factor has a good filtering performance. It can effectively filter the impulse noise in the echo signal and allows the FrFT method to accurate estimate the parameters of the LFM signals in strong impulse noise. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)
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