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Fractal Fract., Volume 8, Issue 2 (February 2024) – 50 articles

Cover Story (view full-size image): The Wright function is a fascinating mathematical entity of its own merit. It was first introduced in 1935, but it gained more attention from mathematicians when they found out how it relates to the theory of space- and time-fractional diffusion equations. With two continuous parameters, it is possible to form complex connections with many other elementary and special functions. This work shows all the cases where the Wright function can be expressed as sums of generalized hypergeometric functions and polynomials, which then reveal the links with the theory of the Airy, Bessel, Gaussian, exponential, and error functions, etc. A concrete application of these results is expected in computer algebra for validating numerical methods for computing the Wright function and computing inverse Laplace transforms. View this paper
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10 pages, 308 KiB  
Article
Topological Properties of Polymeric Networks Modelled by Generalized Sierpiński Graphs
by Alaa Altassan and Muhammad Imran
Fractal Fract. 2024, 8(2), 123; https://doi.org/10.3390/fractalfract8020123 - 19 Feb 2024
Viewed by 903
Abstract
In this article, we compute the irregularity measures of generalized Sierpiński graphs and obtain some bounds on these irregularities. Moreover, we discuss some bounds on connectivity indices for generalized Sierpiński graphs of any arbitrary graph H along with classification of the extremal graphs [...] Read more.
In this article, we compute the irregularity measures of generalized Sierpiński graphs and obtain some bounds on these irregularities. Moreover, we discuss some bounds on connectivity indices for generalized Sierpiński graphs of any arbitrary graph H along with classification of the extremal graphs used to attain them. Full article
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13 pages, 3975 KiB  
Article
System Identification and Fractional-Order Proportional–Integral–Derivative Control of a Distributed Piping System
by Xiaomeng Zhang, Shuo Zhang, Furui Xiong, Lu Liu, Lichuan Zhang, Xuan Han, Heng Wang, Yanzhu Zhang and Ranzhen Ren
Fractal Fract. 2024, 8(2), 122; https://doi.org/10.3390/fractalfract8020122 - 19 Feb 2024
Viewed by 963
Abstract
The vibration of piping systems is one of the most important causes of accelerated equipment wear and reduced work efficiency and safety. In this study, an active vibration control method based on a fractional-order proportional–integral–derivative (PID) controller was proposed to suppress pipeline vibration [...] Read more.
The vibration of piping systems is one of the most important causes of accelerated equipment wear and reduced work efficiency and safety. In this study, an active vibration control method based on a fractional-order proportional–integral–derivative (PID) controller was proposed to suppress pipeline vibration and reduce pipeline damage. First, a mathematical model of the distributed piping system was established using the finite element analysis method, and the characteristics of the distributed piping system were studied effectively. Further, the time-frequency domain parameter identification method was used to realise the system identification of the cross-point vibration transfer function between the brake and sensor, and the particle swarm optimisation algorithm was utilised to further optimise the transfer function parameters to improve the system identification accuracy. Therefore, a fractional-order PID controller was designed using the D-decomposition method, and the optimal controller parameters were obtained. The experimental and numerical simulation results show that the improved system identification algorithm can significantly improve modelling accuracy. In addition, the designed fractional-order PID controller can effectively reduce the system’s overshoot, oscillation time, and adjustment time, thereby reducing the vibration response of piping systems. Full article
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10 pages, 1773 KiB  
Article
The El Niño Southern Oscillation Recharge Oscillator with the Stochastic Forcing of Long-Term Memory
by Xiaofeng Li and Yaokun Li
Fractal Fract. 2024, 8(2), 121; https://doi.org/10.3390/fractalfract8020121 - 18 Feb 2024
Viewed by 830
Abstract
The influence of the fast-varying variables that have a long-term memory on the El Niño Southern Oscillation (ENSO) is investigated by adding a fractional Ornstein–Uhlenbeck (FOU) process stochastic noise on the simple recharge oscillator (RO) model. The FOU process noise converges to zero [...] Read more.
The influence of the fast-varying variables that have a long-term memory on the El Niño Southern Oscillation (ENSO) is investigated by adding a fractional Ornstein–Uhlenbeck (FOU) process stochastic noise on the simple recharge oscillator (RO) model. The FOU process noise converges to zero very slowly with a negative power law. The corresponding non-zero ensemble mean during the integration period can exert a pronounced influence on the ensemble-mean dynamics of the RO model. The state-dependent noise, also called the multiplicative noise, can present its influence by reducing the relaxation coefficient and by introducing periodic external forcing. The decreasing relaxation coefficient can enhance the oscillation amplitude and shorten the oscillation period. The forced frequency is close to the natural frequency. The two mechanisms together can further amplify the amplitude and shorten the period, compared with the state-independent noise or additive noise, which only exhibits its influence by introducing non-periodic external forcing. These two mechanisms explicitly elucidate the influence of the stochastic forcing on the ensemble-mean dynamics of the RO model. It provides comprehensive knowledge to better understand the interaction between the fast-varying stochastic forcing and the slow-varying deterministic system and deserves further investigation. Full article
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21 pages, 6171 KiB  
Article
Magnetically Suspended Control Sensitive Gyroscope Rotor High-Precision Deflection Decoupling Method using Quantum Neural Network and Fractional-Order Terminal Sliding Mode Control
by Yuan Ren, Lei Li, Weijie Wang, Lifen Wang and Weikun Pang
Fractal Fract. 2024, 8(2), 120; https://doi.org/10.3390/fractalfract8020120 - 17 Feb 2024
Viewed by 1093
Abstract
To achieve high-precision deflection control of a Magnetically Suspended Control and Sensitive Gyroscope rotor under high dynamic conditions, a deflection decoupling method using Quantum Radial Basis Function Neural Network and fractional-order terminal sliding mode control is proposed. The convergence speed and time complexity [...] Read more.
To achieve high-precision deflection control of a Magnetically Suspended Control and Sensitive Gyroscope rotor under high dynamic conditions, a deflection decoupling method using Quantum Radial Basis Function Neural Network and fractional-order terminal sliding mode control is proposed. The convergence speed and time complexity of the neural network controller limit the control accuracy and stability of rotor deflection under high-bandwidth conditions. To solve the problem, a quantum-computing-based structure optimization method for the Radial Basis Function Neural Network is proposed for the first time, where the input and the center of hidden layer basis function of the neural network are quantum-coded, and quantum rotation gates are designed to replace the Gaussian function. The parallel characteristic of quantum computing is utilized to reduce the time complexity and improve the convergence speed of the neural network. On top of that, in order to further address the issue of input jitter, a fractional-order terminal sliding mode controller based on the Quantum Radial Basis Function Neural Network is designed, the fractional-order differential sliding mode surface and the fractional-order convergence law are proposed to reduce the input jitter and achieve finite-time convergence of the controller, and the Quantum Radial Basis Function Neural Network is used to approximate the residual coupling and external disturbances of the system, resulting in improving the rotor deflection control accuracy. The semi-physical simulation experiments demonstrate the effectiveness and superiority of the proposed method. Full article
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16 pages, 5507 KiB  
Article
Discrete Element Method Analyses of Bond Degradation Evolutions for Cemented Soils
by Jie He, Tao Li and Yi Rui
Fractal Fract. 2024, 8(2), 119; https://doi.org/10.3390/fractalfract8020119 - 17 Feb 2024
Viewed by 867
Abstract
The degradation of soil bonding, which can be described by the evolution of bond degradation variables, is essential in the constitutive modeling of cemented soils. A degradation variable with a value of 0/1.0 indicates that the applied stress is completely sustained by bonded [...] Read more.
The degradation of soil bonding, which can be described by the evolution of bond degradation variables, is essential in the constitutive modeling of cemented soils. A degradation variable with a value of 0/1.0 indicates that the applied stress is completely sustained by bonded particles/unbounded grains. The discrete element method (DEM) was used for cemented soils to analyze the bond degradation evolution and to evaluate the degradation variables at the contact scale. Numerical cemented soil samples with different bonding strengths were first prepared using an advanced contact model (CM). Constant stress ratio compression, one-dimensional compression, conventional triaxial tests (CTTs), and true triaxial tests (TTTs) were then implemented for the numerical samples. After that, the numerical results were adopted to investigate the evolution of the bond degradation variables BN and B0. In the triaxial tests, B0 evolves to be near to or larger than BN due to shearing, which indicates that shearing increases the bearing rate of bond contacts. Finally, an approximate stress-path-independent bond degradation variable Bσ was developed. The evolution of Bσ with the equivalent plastic strain can be effectively described by an exponential function and a hyperbolic function. Full article
(This article belongs to the Special Issue Fractals in Geology and Geochemistry)
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15 pages, 5089 KiB  
Article
An Improved Rock Resistivity Model Based on Multi-Fractal Characterization Method for Sandstone Micro-Pore Structure Using Capillary Pressure
by Weibiao Xie, Qiuli Yin, Jingbo Zeng, Fan Yang, Pan Zhang and Binpeng Yan
Fractal Fract. 2024, 8(2), 118; https://doi.org/10.3390/fractalfract8020118 - 16 Feb 2024
Viewed by 837
Abstract
Micro-pore structures are an essential factor for the electrical properties of porous rock. Theoretical electrical conductivity models considering pore structure can highly improve the accuracy of reservoir estimation. In this study, a pore structure characterization method based on a multi-fractal theory using capillary [...] Read more.
Micro-pore structures are an essential factor for the electrical properties of porous rock. Theoretical electrical conductivity models considering pore structure can highly improve the accuracy of reservoir estimation. In this study, a pore structure characterization method based on a multi-fractal theory using capillary pressure is developed. Next, a theoretical electrical conductivity equation is derived based on the new pore structure characterization method. Furthermore, a distinct interrelationship between fractal dimensions of capillary pressure curves (Dv) and of resistivity index curves (Dt and Dr) is obtained. The experimental data of 7 sandstone samples verify that the fitting result by the new pore structure characterization method is highly identical to the experimental capillary pressure curves, and the accuracy of the improved rock resistivity model is higher than the Archie model. In addition, capillary pressure curves can be directly converted to resistivity index curves according to the relationship model between fractal dimensions of capillary pressure curves (Dv) and resistivity index curves (Dt and Dr). This study provides new ideas to improve the accuracy of pore structure characterization and oil saturation calculation; it has good application prospects and guiding significance in reservoir evaluation and rock physical characteristics research. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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23 pages, 2519 KiB  
Article
Multifractal Detrended Cross-Correlations between Green Bonds and Commodity Markets: An Exploration of the Complex Connections between Green Finance and Commodities from the Econophysics Perspective
by Turker Acikgoz, Soner Gokten and Abdullah Bugra Soylu
Fractal Fract. 2024, 8(2), 117; https://doi.org/10.3390/fractalfract8020117 - 15 Feb 2024
Viewed by 1067
Abstract
Green bonds represent a compelling financial innovation that presents a financial perspective solution to address climate change and promote sustainable development. On the other hand, the recent process of financialisation of commodities disrupts the dynamics of the commodity market, increasing its correlation with [...] Read more.
Green bonds represent a compelling financial innovation that presents a financial perspective solution to address climate change and promote sustainable development. On the other hand, the recent process of financialisation of commodities disrupts the dynamics of the commodity market, increasing its correlation with financial markets and raising the risks associated with commodities. In this context, understanding the dynamics of the interconnectivity between green bonds and commodity markets is crucial for risk management and portfolio diversification. This study aims to reveal the multifractal cross-correlations between green bonds and commodities by employing methods from statistical physics. We apply multifractal detrended cross-correlation analysis (MFDCCA) to both return and volatility series, demonstrating that green bonds and commodities exhibit multifractal characteristics. The analysis reveals long-range power-law cross-correlations between these two markets. Specifically, volatility cross-correlations persist across various fluctuations, while return series display persistence in small fluctuations and antipersistence in large fluctuations. These findings carry significant practical implications for hedging and risk diversification purposes. Full article
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24 pages, 12369 KiB  
Article
Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration
by Rekha Srivastava, Asifa Tassaddiq and Ruhaila Md Kasmani
Fractal Fract. 2024, 8(2), 116; https://doi.org/10.3390/fractalfract8020116 - 15 Feb 2024
Cited by 1 | Viewed by 926
Abstract
Fractals are a common characteristic of many artificial and natural networks having topological patterns of a self-similar nature. For example, the Mandelbrot set has been investigated and extended in several ways since it was first introduced, whereas some authors characterized it using various [...] Read more.
Fractals are a common characteristic of many artificial and natural networks having topological patterns of a self-similar nature. For example, the Mandelbrot set has been investigated and extended in several ways since it was first introduced, whereas some authors characterized it using various complex functions or polynomials, others generalized it using iterations from fixed-point theory. In this paper, we generate Mandelbrot sets using the hybrid Picard S-iterations. Therefore, an escape criterion involving complex functions is proved and used to provide numerical and graphical examples. We produce a wide range of intriguing fractal patterns with the suggested method, and we compare our findings with the classical S-iteration. It became evident that the newly proposed iteration method produces novel images that are more spontaneous and fascinating than those produced by the S-iteration. Therefore, the generated sets behave differently based on the parameters involved in different iteration schemes. Full article
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21 pages, 1940 KiB  
Article
Oscillation Criteria for Nonlinear Third-Order Delay Dynamic Equations on Time Scales Involving a Super-Linear Neutral Term
by Qinghua Feng and Bin Zheng
Fractal Fract. 2024, 8(2), 115; https://doi.org/10.3390/fractalfract8020115 - 14 Feb 2024
Viewed by 848
Abstract
In the sense of an arbitrary time scale, some new sufficient conditions on oscillation are presented in this paper for a class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term. The established oscillation results include [...] Read more.
In the sense of an arbitrary time scale, some new sufficient conditions on oscillation are presented in this paper for a class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term. The established oscillation results include known Kamenev and Philos-type oscillation criteria and are new oscillation results so far in the literature. Some inequalities, the Riccati transformation, the integral technique, and the theory of time scale are used in the establishment of these oscillation criteria. The proposed results unify continuous and discrete analysis, and the process of deduction is further extended to another class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term and a damping term. As applications for the established oscillation criteria, some examples are given. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
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21 pages, 761 KiB  
Article
Chaos, Fractionality, Nonlinear Contagion, and Causality Dynamics of the Metaverse, Energy Consumption, and Environmental Pollution: Markov-Switching Generalized Autoregressive Conditional Heteroskedasticity Copula and Causality Methods
by Melike Bildirici, Özgür Ömer Ersin and Blend Ibrahim
Fractal Fract. 2024, 8(2), 114; https://doi.org/10.3390/fractalfract8020114 - 14 Feb 2024
Viewed by 1012
Abstract
Metaverse (MV) technology introduces new tools for users each day. MV companies have a significant share in the total stock markets today, and their size is increasing. However, MV technologies are questioned as to whether they contribute to environmental pollution with their increasing [...] Read more.
Metaverse (MV) technology introduces new tools for users each day. MV companies have a significant share in the total stock markets today, and their size is increasing. However, MV technologies are questioned as to whether they contribute to environmental pollution with their increasing energy consumption (EC). This study explores complex nonlinear contagion with tail dependence and causality between MV stocks, EC, and environmental pollution proxied with carbon dioxide emissions (CO2) with a decade-long daily dataset covering 18 May 2012–16 March 2023. The Mandelbrot–Wallis and Lo’s rescaled range (R/S) tests confirm long-term dependence and fractionality, and the largest Lyapunov exponents, Shannon and Havrda, Charvât, and Tsallis (HCT) entropy tests followed by the Kolmogorov–Sinai (KS) complexity measure confirm chaos, entropy, and complexity. The Brock, Dechert, and Scheinkman (BDS) test of independence test confirms nonlinearity, and White‘s test of heteroskedasticity of nonlinear forms and Engle’s autoregressive conditional heteroskedasticity test confirm heteroskedasticity, in addition to fractionality and chaos. In modeling, the marginal distributions are modeled with Markov-Switching Generalized Autoregressive Conditional Heteroskedasticity Copula (MS-GARCH–Copula) processes with two regimes for low and high volatility and asymmetric tail dependence between MV, EC, and CO2 in all regimes. The findings indicate relatively higher contagion with larger copula parameters in high-volatility regimes. Nonlinear causality is modeled under regime-switching heteroskedasticity, and the results indicate unidirectional causality from MV to EC, from MV to CO2, and from EC to CO2, in addition to bidirectional causality among MV and EC, which amplifies the effects on air pollution. The findings of this paper offer vital insights into the MV, EC, and CO2 nexus under chaos, fractionality, and nonlinearity. Important policy recommendations are generated. Full article
(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)
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20 pages, 330 KiB  
Article
Existence of Ground State Solutions for a Class of Non-Autonomous Fractional Kirchhoff Equations
by Guangze Gu, Changyang Mu and Zhipeng Yang
Fractal Fract. 2024, 8(2), 113; https://doi.org/10.3390/fractalfract8020113 - 14 Feb 2024
Viewed by 873
Abstract
We take a look at the fractional Kirchhoff problem in this paper. Using a variational approach, we show that there exists a ground state solution for this problem. Furthermore, using the approach developed by Szulkin and Weth, we also find that positive ground [...] Read more.
We take a look at the fractional Kirchhoff problem in this paper. Using a variational approach, we show that there exists a ground state solution for this problem. Furthermore, using the approach developed by Szulkin and Weth, we also find that positive ground state solutions exist for the fractional Kirchhoff equation with p=4. Full article
(This article belongs to the Special Issue Variational Problems and Fractional Differential Equations)
14 pages, 301 KiB  
Article
A Temporal Second-Order Difference Scheme for Variable-Order-Time Fractional-Sub-Diffusion Equations of the Fourth Order
by Xin Zhang, Yu Bo and Yuanfeng Jin
Fractal Fract. 2024, 8(2), 112; https://doi.org/10.3390/fractalfract8020112 - 13 Feb 2024
Viewed by 871
Abstract
In this article, we develop a compact finite difference scheme for a variable-order-time fractional-sub-diffusion equation of a fourth-order derivative term via order reduction. The proposed scheme exhibits fourth-order convergence in space and second-order convergence in time. Additionally, we provide a detailed proof for [...] Read more.
In this article, we develop a compact finite difference scheme for a variable-order-time fractional-sub-diffusion equation of a fourth-order derivative term via order reduction. The proposed scheme exhibits fourth-order convergence in space and second-order convergence in time. Additionally, we provide a detailed proof for the existence and uniqueness, as well as the stability of scheme, along with a priori error estimates. Finally, we validate our theoretical results through various numerical computations. Full article
20 pages, 394 KiB  
Article
A Unified Approach to Solvability and Stability of Multipoint BVPs for Langevin and Sturm–Liouville Equations with CH–Fractional Derivatives and Impulses via Coincidence Theory
by Kaihong Zhao, Juqing Liu and Xiaojun Lv
Fractal Fract. 2024, 8(2), 111; https://doi.org/10.3390/fractalfract8020111 - 13 Feb 2024
Cited by 3 | Viewed by 1000
Abstract
The Langevin equation is a model for describing Brownian motion, while the Sturm–Liouville equation is an important mechanical model. This paper focuses on the solvability and stability of nonlinear impulsive Langevin and Sturm–Liouville equations with Caputo–Hadamard (CH) fractional derivatives and multipoint boundary value [...] Read more.
The Langevin equation is a model for describing Brownian motion, while the Sturm–Liouville equation is an important mechanical model. This paper focuses on the solvability and stability of nonlinear impulsive Langevin and Sturm–Liouville equations with Caputo–Hadamard (CH) fractional derivatives and multipoint boundary value conditions. To unify the two types of equations, we investigate a general nonlinear impulsive coupled implicit system. By cleverly constructing relevant operators involving impulsive terms, we establish the coincidence degree theory and obtain the solvability. We explore the stability of solutions using nonlinear analysis and inequality techniques. As the most direct application, we naturally obtained the solvability and stability of the Langevin and Sturm–Liouville equations mentioned above. Finally, an example is provided to demonstrate the validity and availability of our major findings. Full article
21 pages, 1306 KiB  
Article
Error-Based Switched Fractional Order Model Reference Adaptive Control for MIMO Linear Time Invariant Systems
by Norelys Aguila-Camacho and Javier A. Gallegos
Fractal Fract. 2024, 8(2), 109; https://doi.org/10.3390/fractalfract8020109 - 13 Feb 2024
Cited by 1 | Viewed by 917
Abstract
This paper presents the design and analysis of Switched Fractional Order Model Reference Adaptive Controllers (SFOMRAC) for Multiple Input Multiple Output (MIMO) linear systems with unknown parameters. The proposed controller uses adaptive laws whose derivation order switches between a fractional order and the [...] Read more.
This paper presents the design and analysis of Switched Fractional Order Model Reference Adaptive Controllers (SFOMRAC) for Multiple Input Multiple Output (MIMO) linear systems with unknown parameters. The proposed controller uses adaptive laws whose derivation order switches between a fractional order and the integer order, according to a certain level of control error. The switching aims to use fractional orders when the control error is larger to improve transient response and system performance during large disturbed states, and to obtain smoother control signals, leading to a better control energy usage. Then, it switches to the integer order when the control error is smaller to improve steady state. Boundedness of all the signals in the scheme is analytically proved, as well as convergence of the control error to zero. Moreover, these properties are extended to the case when system states are affected by a bounded non-parametric disturbance. Simulation studies are carried out using different representative plants to be controlled, showing that fractional orders and switching error levels can be found in most of the cases, such as when SFOMRAC achieves a better balance among control energy and system performance than the non-switched equivalent strategies. Full article
(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems, 2nd Edition)
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10 pages, 972 KiB  
Article
Ultrafast Diffusion Modeling via the Riemann–Liouville Nonlocal Structural Derivative and Its Application in Porous Media
by Wei Xu, Hui Liu, Lijuan Chen and Yongtao Zhou
Fractal Fract. 2024, 8(2), 110; https://doi.org/10.3390/fractalfract8020110 - 12 Feb 2024
Viewed by 905
Abstract
Ultrafast diffusion disperses faster than super-diffusion, and this has been proven by several theoretical and experimental investigations. The mean square displacement of ultrafast diffusion grows exponentially, which provides a significant challenge for modeling. Due to the inhomogeneity, nonlinear interactions, and high porosity of [...] Read more.
Ultrafast diffusion disperses faster than super-diffusion, and this has been proven by several theoretical and experimental investigations. The mean square displacement of ultrafast diffusion grows exponentially, which provides a significant challenge for modeling. Due to the inhomogeneity, nonlinear interactions, and high porosity of cement materials, the motion of particles on their surfaces satisfies the conditions for ultrafast diffusion. The investigation of the diffusion behavior in cementitious materials is crucial for predicting the mechanical properties of cement. In this study, we first attempted to investigate the dynamic of ultrafast diffusion in cementitious materials underlying the Riemann–Liouville nonlocal structural derivative. We constructed a Riemann–Liouville nonlocal structural derivative ultrafast diffusion model with an exponential function and then extended the modeling strategy using the Mittag–Leffler function. The mean square displacement is analogous to the integral of the corresponding structural derivative, providing a reference standard for the selection of structural functions in practical applications. Based on experimental data on cement mortar, the accuracy of the Riemann–Liouville nonlocal structural derivative ultrafast diffusion model was verified. Compared to the power law diffusion and the exponential law diffusion, the mean square displacement with respect to the Mittag–Leffler law is closely tied to the actual data. The modeling approach based on the Riemann–Liouville nonlocal structural derivative provides an efficient tool for depicting ultrafast diffusion in porous media. Full article
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21 pages, 5481 KiB  
Article
Analysis and Warning Prediction of Tunnel Deformation Based on Multifractal Theory
by Chengtao Yang, Rendong Huang, Dunwen Liu, Weichao Qiu, Ruiping Zhang and Yu Tang
Fractal Fract. 2024, 8(2), 108; https://doi.org/10.3390/fractalfract8020108 - 12 Feb 2024
Viewed by 1022
Abstract
To better analyze the fluctuation characteristics and development law of tunnel deformation data, multifractal theory is applied to tunnel deformation analysis. That is, the multifractal detrended fluctuation analysis (MF-DFA) model is first utilized to carry out the multifractal characterization of tunnel deformation data. [...] Read more.
To better analyze the fluctuation characteristics and development law of tunnel deformation data, multifractal theory is applied to tunnel deformation analysis. That is, the multifractal detrended fluctuation analysis (MF-DFA) model is first utilized to carry out the multifractal characterization of tunnel deformation data. Further, Mann–Kendall (M–K) analysis is utilized to construct the dual criterion (∆α indicator criterion and ∆f(α) indicator criterion) for the tunnel deformation early warning study. In addition, the particle swarm optimization long-short-term memory (PSO-LSTM) prediction model is used for predicting tunnel settlement. The results show that, in reference to the tunnel warning level criteria and based on the Z-value results of the indicator criterion, the warning level of all four sections is class II. At the same time, through the analysis of tunnel settlement predictions, the PSO-LSTM model has a better prediction effect and stability for tunnel settlement. The predicted results show a slow increase in tunnel settlement over the next 5 days. Finally, the tunnel warning level and the predicted results of tunnel settlement are analyzed in a comprehensive manner. The deformation will increase slowly in the future. Therefore, monitoring and measurement should be strengthened, and disaster preparedness plans should be prepared. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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21 pages, 674 KiB  
Article
Properties and Applications of Symmetric Quantum Calculus
by Miguel Vivas-Cortez, Muhammad Zakria Javed, Muhammad Uzair Awan, Silvestru Sever Dragomir and Ahmed M. Zidan
Fractal Fract. 2024, 8(2), 107; https://doi.org/10.3390/fractalfract8020107 - 12 Feb 2024
Viewed by 1444
Abstract
Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and [...] Read more.
Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and studied various properties of both operators as well. Using these concepts, we deliver new variants of Young’s inequality, Hölder’s inequality, Minkowski’s inequality, Hermite–Hadamard’s inequality, Ostrowski’s inequality, and Gruss–Chebysev inequality. We report the Hermite–Hadamard’s inequalities by taking into account the differentiability of convex mappings. These fundamental results are pivotal to studying the various other problems in the field of inequalities. The validation of results is also supported with some visuals. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
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13 pages, 460 KiB  
Article
Error Analysis of the Nonuniform Alikhanov Scheme for the Fourth-Order Fractional Diffusion-Wave Equation
by Zihao An and Chaobao Huang
Fractal Fract. 2024, 8(2), 106; https://doi.org/10.3390/fractalfract8020106 - 10 Feb 2024
Viewed by 868
Abstract
This paper considers the numerical approximation to the fourth-order fractional diffusion-wave equation. Using a separation of variables, we can construct the exact solution for such a problem and then analyze its regularity. The obtained regularity result indicates that the solution behaves as a [...] Read more.
This paper considers the numerical approximation to the fourth-order fractional diffusion-wave equation. Using a separation of variables, we can construct the exact solution for such a problem and then analyze its regularity. The obtained regularity result indicates that the solution behaves as a weak singularity at the initial time. Using the order reduction method, the fourth-order fractional diffusion-wave equation can be rewritten as a coupled system of low order, which is approximated by the nonuniform Alikhanov scheme in time and the finite difference method in space. Furthermore, the H2-norm stability result is obtained. With the help of this result and a priori bounds of the solution, an α-robust error estimate with optimal convergence order is derived. In order to further verify the accuracy of our theoretical analysis, some numerical results are provided. Full article
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14 pages, 626 KiB  
Article
A Hybrided Method for Temporal Variable-Order Fractional Partial Differential Equations with Fractional Laplace Operator
by Chengyi Wang and Shichao Yi
Fractal Fract. 2024, 8(2), 105; https://doi.org/10.3390/fractalfract8020105 - 08 Feb 2024
Viewed by 906
Abstract
In this paper, we present a more general approach based on a Picard integral scheme for nonlinear partial differential equations with a variable time-fractional derivative of order α(x,t)(1,2) and space-fractional order [...] Read more.
In this paper, we present a more general approach based on a Picard integral scheme for nonlinear partial differential equations with a variable time-fractional derivative of order α(x,t)(1,2) and space-fractional order s(0,1), where v=u(t) is introduced as the new unknown function and u is recovered using the quadrature. In order to get rid of the constraints of traditional plans considering the half-time situation, integration by parts and the regularity process are introduced on the variable v. The convergence order can reach O(τ2+h2), different from the common L1,2α schemes with convergence rate O(τ2,3α(x,t)) under the infinite norm. In each integer time step, the stability, solvability and convergence of this scheme are proved. Several error results and convergence rates are calculated using numerical simulations to evidence the theoretical values of the proposed method. Full article
(This article belongs to the Section Numerical and Computational Methods)
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15 pages, 3717 KiB  
Article
Real-Time Synchronisation of Multiple Fractional-Order Chaotic Systems: An Application Study in Secure Communication
by Bachir Nail, Mahedi Abdelghani Atoussi, Slami Saadi, Imad Eddine Tibermacine and Christian Napoli
Fractal Fract. 2024, 8(2), 104; https://doi.org/10.3390/fractalfract8020104 - 08 Feb 2024
Viewed by 1523
Abstract
In this paper, we use two Fractional-Order Chaotic Systems (FOCS)—one for the sender and one for the receiver—to determine the optimal synchronisation for a secure communication technique. With the help of the Step-By-Step Sliding-Mode Observer (SBS-SMO), this synchronisation is accomplished. An innovative optimisation [...] Read more.
In this paper, we use two Fractional-Order Chaotic Systems (FOCS)—one for the sender and one for the receiver—to determine the optimal synchronisation for a secure communication technique. With the help of the Step-By-Step Sliding-Mode Observer (SBS-SMO), this synchronisation is accomplished. An innovative optimisation method, known as the artificial Harris hawks optimisation (HHO), was employed to enhance the observer’s performance. This method eliminates the random parameter selection process and instead selects the optimal values for the observer. In a short amount of time, the secret message that could have been in the receiver portion (signal, voice, etc.) was successfully recovered using the proposed scheme. The experimental validation of the numerical results was carried out with the assistance of an Arduino microcontroller and several electronic components. In addition, the findings of the experiments were compared with the theoretical calculations, revealing a satisfactory level of agreement. Full article
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32 pages, 13195 KiB  
Article
The Application of Fractional Derivative Viscoelastic Models in the Finite Element Method: Taking Several Common Models as Examples
by Guozhi Zheng, Naitian Zhang and Songtao Lv
Fractal Fract. 2024, 8(2), 103; https://doi.org/10.3390/fractalfract8020103 - 07 Feb 2024
Viewed by 1092
Abstract
This paper aims to incorporate the fractional derivative viscoelastic model into a finite element analysis. Firstly, based on the constitutive equation of the fractional derivative three-parameter solid model (FTS), the constitutive equation is discretized by using the Grünwald–Letnikov definition of the fractional derivative, [...] Read more.
This paper aims to incorporate the fractional derivative viscoelastic model into a finite element analysis. Firstly, based on the constitutive equation of the fractional derivative three-parameter solid model (FTS), the constitutive equation is discretized by using the Grünwald–Letnikov definition of the fractional derivative, and the stress increment and strain increment relationship and Jacobian matrix are obtained by using the difference method. Subsequently, we degrade the model to establish stress increment and strain increment relationships and Jacobian matrices for the fractional derivative Kelvin model (FK) and fractional derivative Maxwell model (FM). Finally, we further degrade the fractional derivative viscoelastic model to derive stress increment and strain increment relationships and Jacobian matrices for a three-component solid model and Kelvin and Maxwell models. Based on these developments, a UMAT subroutine is implemented in ABAQUS 6.14 finite element software. Three different loading modes, including static load, dynamic load, and mobile load, are analyzed and calculated. The calculations primarily involve a convergence analysis, verification of numerical solutions, and comparative analysis of responses among different viscoelastic models. Full article
(This article belongs to the Section Engineering)
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12 pages, 2733 KiB  
Article
Space–Time Variations in the Long-Range Dependence of Sea Surface Chlorophyll in the East China Sea and the South China Sea
by Junyu He and Ming Li
Fractal Fract. 2024, 8(2), 102; https://doi.org/10.3390/fractalfract8020102 - 07 Feb 2024
Viewed by 1036
Abstract
Gaining insights into the space–time variations in the long-range dependence of sea surface chlorophyll is crucial for the early detection of environmental issues in oceans. To this end, 12 locations were selected along the Yangtze River and Pearl River estuaries, varying in distances [...] Read more.
Gaining insights into the space–time variations in the long-range dependence of sea surface chlorophyll is crucial for the early detection of environmental issues in oceans. To this end, 12 locations were selected along the Yangtze River and Pearl River estuaries, varying in distances from the Chinese coastline. Daily satellite-observed sea surface chlorophyll concentration data at these 12 locations were collected from the Copernicus Marine Service website, spanning from December 1997 to November 2023. The main objective of the current study is to introduce a multi-fractional generalized Cauchy model for calculating the values of Hurst exponents and quantitatively assessing the long-range dependence strength of sea surface chlorophyll at different spatial locations and time instants during the study period. Furthermore, ANOVA was utilized to detect the differences of calculated Hurst exponent values among the locations during various months and seasons. From a spatial perspective, the findings reveal a significantly stronger long-range dependence of sea surface chlorophyll in offshore regions compared to nearshore areas, with Hurst exponent values > 0.5 versus <0.5. It is noteworthy that the values of Hurst exponents at each location exhibit significant differences during various seasons, from a temporal perspective. Specifically, the long-range dependence of sea surface chlorophyll in summer in the nearshore region is weaker than in other seasons, whereas that in the offshore region is stronger than in other seasons. The study concludes that long-range dependence is inversely related to the distance from the coastline, and anthropogenic activity plays a dominant role in shaping the long-range dependence of sea surface chlorophyll in the coastal regions of China. Full article
(This article belongs to the Special Issue Fractional Processes and Systems in Computer Science and Engineering)
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27 pages, 480 KiB  
Review
Review of the Fractional Black-Scholes Equations and Their Solution Techniques
by Hongmei Zhang, Mengchen Zhang, Fawang Liu and Ming Shen
Fractal Fract. 2024, 8(2), 101; https://doi.org/10.3390/fractalfract8020101 - 07 Feb 2024
Viewed by 1165
Abstract
The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, [...] Read more.
The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs. Full article
17 pages, 3893 KiB  
Article
Fractional-Order Control Method Based on Twin-Delayed Deep Deterministic Policy Gradient Algorithm
by Guangxin Jiao, Zhengcai An, Shuyi Shao and Dong Sun
Fractal Fract. 2024, 8(2), 99; https://doi.org/10.3390/fractalfract8020099 - 06 Feb 2024
Viewed by 1005
Abstract
In this paper, a fractional-order control method based on the twin-delayed deep deterministic policy gradient (TD3) algorithm in reinforcement learning is proposed. A fractional-order disturbance observer is designed to estimate the disturbances, and the radial basis function network is selected to approximate system [...] Read more.
In this paper, a fractional-order control method based on the twin-delayed deep deterministic policy gradient (TD3) algorithm in reinforcement learning is proposed. A fractional-order disturbance observer is designed to estimate the disturbances, and the radial basis function network is selected to approximate system uncertainties in the system. Then, a fractional-order sliding-mode controller is constructed to control the system, and the parameters of the controller are tuned using the TD3 algorithm, which can optimize the control effect. The results show that the fractional-order control method based on the TD3 algorithm can not only improve the closed-loop system performance under different operating conditions but also enhance the signal tracking capability. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)
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27 pages, 20554 KiB  
Article
Novel Meta-Fractal Wearable Sensors and Antennas for Medical, Communication, 5G, and IoT Applications
by Albert Sabban
Fractal Fract. 2024, 8(2), 100; https://doi.org/10.3390/fractalfract8020100 - 06 Feb 2024
Cited by 1 | Viewed by 1191
Abstract
Future communication, 5G, medical, and IoT systems need compact, green, efficient wideband sensors, and antennas. Novel linear and dual-polarized antennas for 5G, 6G, medical devices, Internet of Things (IoT) systems, and healthcare monitoring sensors are presented in this paper. One of the major [...] Read more.
Future communication, 5G, medical, and IoT systems need compact, green, efficient wideband sensors, and antennas. Novel linear and dual-polarized antennas for 5G, 6G, medical devices, Internet of Things (IoT) systems, and healthcare monitoring sensors are presented in this paper. One of the major goals in the evaluation of medical, 5G, and smart wireless communication devices is the development of efficient, compact, low-cost antennas and sensors. Moreover, passive and active sensors may be self-powered by connecting an energy-harvesting unit to the antenna to collect electromagnetic radiation and charge the wearable sensor battery. Wearable sensors and antennas can be employed in smart grid applications that provide communication between neighbors, localized management, bidirectional power transfer, and effective demand response. A low-cost wearable antenna may be developed by etching the printed feed and matching the network on the same substrate in the printed antenna. Active modules may be placed on the same dielectric board. The antenna design parameters and a comparison between the computation and measured electrical performance of the antennas are presented in this paper. The electrical characteristics of the new compact antennas in the vicinity of the patient’s body were simulated by using electromagnetic simulation techniques. Fractal and metamaterial efficient antennas and sensors were evaluated to maximize the electrical characteristics of smart communication and medical devices. The dual- and circularly polarized antennas developed in this paper are crucial to the evaluation of wideband and multiband compact 5G, 6G, and IoT advanced systems. The new efficient sensors and antennas maximize the system’s dynamic range and electrical characteristics. The new efficient wearable antennas and sensors are compact, wideband, and low-cost. The operating resonant frequency of the metamaterial antennas with circular split-ring resonators (CSRRs) may be 5% to 9% lower than the resonant frequency of the sensor without CSRRs. The directivity and gain of the metamaterial fractal antennas with CSRRs may be up to 3 dB higher than the antennas without CSRRs. The directivity and gain of the metamaterial fractal passive sensors with CSRRs may be up to 8.5 dBi. This study presents new wideband active meta-fractal antennas and sensors. The bandwidth of the new sensors is around 9% to 20%. At 2.83 GHz, the receiving active sensor gain is 13.5 dB and drops to 8 dB at 3.2 GHz. The receiving module noise figure with TAV541 LNA is around 1dB. Full article
(This article belongs to the Special Issue Advances in Fractal Antennas: Design, Modeling and Applications)
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22 pages, 2920 KiB  
Article
Understanding the Complex Adaptive Characteristics of Cross-Regional Emergency Collaboration in China: A Stochastic Evolutionary Game Approach
by Jida Liu and Changqi Dong
Fractal Fract. 2024, 8(2), 98; https://doi.org/10.3390/fractalfract8020098 - 05 Feb 2024
Viewed by 923
Abstract
Regional integration and pairing assistance are two forms of cross-regional emergency collaboration practice carried out by the Chinese government. Based on the Chinese government’s emergency management practice, evolutionary game models of cross-regional emergency collaboration were constructed. Further, the traditional evolutionary game model was [...] Read more.
Regional integration and pairing assistance are two forms of cross-regional emergency collaboration practice carried out by the Chinese government. Based on the Chinese government’s emergency management practice, evolutionary game models of cross-regional emergency collaboration were constructed. Further, the traditional evolutionary game model was improved by introducing the stochastic process, and Gaussian white noise was introduced as a random disturbance. The stochastic evolutionary game model was constructed, and the existence and stability of the equilibrium solutions of the two kinds of stochastic evolutionary game systems for cross-regional emergency collaboration were verified based on the stability discrimination theorem of stochastic differential equations. We used numerical simulations to simulate the evolution trajectories of the regional integration and the pairing assistance stochastic evolutionary game system. In the regional integration game system, when the efficiency of emergency collaboration, the emergency capital stock, and the externality coefficients are higher, positive emergency strategies are more likely to become the stable state of the game subjects’ strategy selection. In the pairing assistance game system, the efficiency of emergency collaboration, the rewards and benefits from the central government, and the matching degree between governments all had positive effects on the formation of the positive emergency strategies of the game subjects. In addition, the pairing assistance mechanism for sustainable development requires external support from the central government. Full article
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19 pages, 655 KiB  
Article
Fixed-Time Synchronization for Fractional-Order Cellular Inertial Fuzzy Neural Networks with Mixed Time-Varying Delays
by Yeguo Sun, Yihong Liu and Lei Liu
Fractal Fract. 2024, 8(2), 97; https://doi.org/10.3390/fractalfract8020097 - 04 Feb 2024
Viewed by 926
Abstract
Due to the widespread application of neural networks (NNs), and considering the respective advantages of fractional calculus (FC), inertial neural networks (INNs), cellular neural networks (CNNs), and fuzzy neural networks (FNNs), this paper investigates the fixed-time synchronization (FDTS) issues for a particular category [...] Read more.
Due to the widespread application of neural networks (NNs), and considering the respective advantages of fractional calculus (FC), inertial neural networks (INNs), cellular neural networks (CNNs), and fuzzy neural networks (FNNs), this paper investigates the fixed-time synchronization (FDTS) issues for a particular category of fractional-order cellular-inertial fuzzy neural networks (FCIFNNs) that involve mixed time-varying delays (MTDs), including both discrete and distributed delays. Firstly, we establish an appropriate transformation variable to reformulate FCIFNNs with MTD into a differential first-order system. Then, utilizing the finite-time stability (FETS) theory and Lyapunov functionals (LFs), we establish some new effective criteria for achieving FDTS of the response system (RS) and drive system (DS). Eventually, we offer two numerical examples to display the effectiveness of our proposed synchronization strategies. Moreover, we also demonstrate the benefits of our approach through an application in image encryption. Full article
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23 pages, 2057 KiB  
Article
Navigating Choppy Waters: Interplay between Financial Stress and Commodity Market Indices
by Haji Ahmed, Faheem Aslam and Paulo Ferreira
Fractal Fract. 2024, 8(2), 96; https://doi.org/10.3390/fractalfract8020096 - 04 Feb 2024
Viewed by 1372
Abstract
Financial stress can have significant implications for individuals, businesses, asset prices and the economy as a whole. This study examines the nonlinear structure and dynamic changes in the multifractal behavior of cross-correlation between the financial stress index (FSI) and four well-known commodity indices, [...] Read more.
Financial stress can have significant implications for individuals, businesses, asset prices and the economy as a whole. This study examines the nonlinear structure and dynamic changes in the multifractal behavior of cross-correlation between the financial stress index (FSI) and four well-known commodity indices, namely Commodity Research Bureau Index (CRBI), Baltic Dry Index (BDI), London Metal Index (LME) and Brent Oil prices (BROIL), using multifractal detrended cross correlation analysis (MFDCCA). For analysis, we utilized daily values of FSI and commodity index prices from 16 June 2016 to 9 July 2023. The following are the most important empirical findings: (I) All of the chosen commodity market indices show cross correlations with the FSI and have notable multifractal characteristics. (II) The presence of power law cross-correlation implies that a noteworthy shift in FSI is likely to coincide with a considerable shift in the commodity indices. (III) The multifractal cross-correlation is highest between FSI and Brent Oil (BROIL) and lowest with LME. (IV) The rolling windows analysis reveals a varying degree of persistency between FSI and commodity markets. The findings of this study have a number of important implications for commodity market investors and policymakers. Full article
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29 pages, 3707 KiB  
Article
Investigating a Fractal–Fractional Mathematical Model of the Third Wave of COVID-19 with Vaccination in Saudi Arabia
by Fawaz K. Alalhareth, Mohammed H. Alharbi, Noura Laksaci, Ahmed Boudaoui and Meroua Medjoudja
Fractal Fract. 2024, 8(2), 95; https://doi.org/10.3390/fractalfract8020095 - 02 Feb 2024
Viewed by 925
Abstract
The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is responsible for coronavirus disease-19 (COVID-19). This virus has caused a global pandemic, marked by several mutations leading to multiple waves of infection. This paper proposes a comprehensive and integrative mathematical approach to the third [...] Read more.
The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is responsible for coronavirus disease-19 (COVID-19). This virus has caused a global pandemic, marked by several mutations leading to multiple waves of infection. This paper proposes a comprehensive and integrative mathematical approach to the third wave of COVID-19 (Omicron) in the Kingdom of Saudi Arabia (KSA) for the period between 16 December 2022 and 8 February 2023. It may help to implement a better response in the next waves. For this purpose, in this article, we generate a new mathematical transmission model for coronavirus, particularly during the third wave in the KSA caused by the Omicron variant, factoring in the impact of vaccination. We developed this model using a fractal-fractional derivative approach. It categorizes the total population into six segments: susceptible, vaccinated, exposed, asymptomatic infected, symptomatic infected, and recovered individuals. The conventional least-squares method is used for estimating the model parameters. The Perov fixed point theorem is utilized to demonstrate the solution’s uniqueness and existence. Moreover, we investigate the Ulam–Hyers stability of this fractal–fractional model. Our numerical approach involves a two-step Newton polynomial approximation. We present simulation results that vary according to the fractional orders (γ) and fractal dimensions (θ), providing detailed analysis and discussion. Our graphical analysis shows that the fractal-fractional derivative model offers more biologically realistic results than traditional integer-order and other fractional models. Full article
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18 pages, 337 KiB  
Article
Decay Properties for Transmission System with Infinite Memory and Distributed Delay
by Hicham Saber, Abdelkader Braik, Noureddine Bahri, Abderrahmane Beniani, Tariq Alraqad, Yousef Jawarneh and Khaled Zennir
Fractal Fract. 2024, 8(2), 94; https://doi.org/10.3390/fractalfract8020094 - 31 Jan 2024
Viewed by 863
Abstract
We consider a damped transmission problem in a bounded domain where the damping is effective in a neighborhood of a suitable subset of the boundary. Using the semigroup approach together with Hille–Yosida theorem, we prove the existence and uniqueness of global solution. Under [...] Read more.
We consider a damped transmission problem in a bounded domain where the damping is effective in a neighborhood of a suitable subset of the boundary. Using the semigroup approach together with Hille–Yosida theorem, we prove the existence and uniqueness of global solution. Under suitable assumption on the geometrical conditions on the damping, we establish the exponential stability of the solution by introducing a suitable Lyapunov functional. Full article
(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)
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