Pub Date : 2024-05-10DOI: 10.3390/fractalfract8050286
Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat, Ashraf Al-Quran
In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian’s and He’s polynomials, thereby offering a more efficient way of solving nonlinear problems.
{"title":"A Novel Technique for Solving the Nonlinear Fractional-Order Smoking Model","authors":"Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat, Ashraf Al-Quran","doi":"10.3390/fractalfract8050286","DOIUrl":"https://doi.org/10.3390/fractalfract8050286","url":null,"abstract":"In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian’s and He’s polynomials, thereby offering a more efficient way of solving nonlinear problems.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":null,"pages":null},"PeriodicalIF":5.4,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140990607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-10DOI: 10.3390/fractalfract8050285
Rehan Akram, J. Hong, Seung Gu Kim, Haseeb Sultan, Muhammad Usman, Hafiz Ali Hamza Gondal, Muhammad Hamza Tariq, Nadeem Ullah, Kang Ryoung Park
The segmentation of crops and weeds from camera-captured images is a demanding research area for advancing agricultural and smart farming systems. Previously, the segmentation of crops and weeds was conducted within a homogeneous data environment where training and testing data were from the same database. However, in the real-world application of advancing agricultural and smart farming systems, it is often the case of a heterogeneous data environment where a system trained with one database should be used for testing with a different database without additional training. This study pioneers the use of heterogeneous data for crop and weed segmentation, addressing the issue of degraded accuracy. Through adjusting the mean and standard deviation, we minimize the variability in pixel value and contrast, enhancing segmentation robustness. Unlike previous methods relying on extensive training data, our approach achieves real-world applicability with just one training sample for deep learning-based semantic segmentation. Moreover, we seamlessly integrated a method for estimating fractal dimensions into our system, incorporating it as an end-to-end task to provide important information on the distributional characteristics of crops and weeds. We evaluated our framework using the BoniRob dataset and the CWFID. When trained with the BoniRob dataset and tested with the CWFID, we obtained a mean intersection of union (mIoU) of 62% and an F1-score of 75.2%. Furthermore, when trained with the CWFID and tested with the BoniRob dataset, we obtained an mIoU of 63.7% and an F1-score of 74.3%. We confirmed that these values are higher than those obtained by state-of-the-art methods.
{"title":"Crop and Weed Segmentation and Fractal Dimension Estimation Using Small Training Data in Heterogeneous Data Environment","authors":"Rehan Akram, J. Hong, Seung Gu Kim, Haseeb Sultan, Muhammad Usman, Hafiz Ali Hamza Gondal, Muhammad Hamza Tariq, Nadeem Ullah, Kang Ryoung Park","doi":"10.3390/fractalfract8050285","DOIUrl":"https://doi.org/10.3390/fractalfract8050285","url":null,"abstract":"The segmentation of crops and weeds from camera-captured images is a demanding research area for advancing agricultural and smart farming systems. Previously, the segmentation of crops and weeds was conducted within a homogeneous data environment where training and testing data were from the same database. However, in the real-world application of advancing agricultural and smart farming systems, it is often the case of a heterogeneous data environment where a system trained with one database should be used for testing with a different database without additional training. This study pioneers the use of heterogeneous data for crop and weed segmentation, addressing the issue of degraded accuracy. Through adjusting the mean and standard deviation, we minimize the variability in pixel value and contrast, enhancing segmentation robustness. Unlike previous methods relying on extensive training data, our approach achieves real-world applicability with just one training sample for deep learning-based semantic segmentation. Moreover, we seamlessly integrated a method for estimating fractal dimensions into our system, incorporating it as an end-to-end task to provide important information on the distributional characteristics of crops and weeds. We evaluated our framework using the BoniRob dataset and the CWFID. When trained with the BoniRob dataset and tested with the CWFID, we obtained a mean intersection of union (mIoU) of 62% and an F1-score of 75.2%. Furthermore, when trained with the CWFID and tested with the BoniRob dataset, we obtained an mIoU of 63.7% and an F1-score of 74.3%. We confirmed that these values are higher than those obtained by state-of-the-art methods.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":null,"pages":null},"PeriodicalIF":5.4,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140991081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-09DOI: 10.3390/fractalfract8050284
Jinwoo Choi, Daecheon Lim, Sungjoon Lim
In stealth applications, there is a growing emphasis on the development of radar-absorbing structures that are efficient, flexible, and optically transparent. This study proposes a screen-printed metamaterial absorber (MMA) on polyethylene terephthalate (PET) substrates using indium tin oxide (ITO) as the grounding layer, which achieves both optical transparency and flexibility. These materials and methods enhance the overall flexibility and transparency of MMA. To address the limited transparency caused by the silver nanoparticle ink for the top pattern, a metal mesh was incorporated to reduce the area ratio of the printed patterns, thereby enhancing transparency. By incrementing the fractal order of the structure, we optimized the operating frequency to target the X-band, which is most commonly used in radar detection. The proposed MMA demonstrates remarkable performance, with a measured absorption of 91.99% at 8.85 GHz and an average optical transmittance of 46.70% across the visible light spectrum (450 to 700 nm), indicating its potential for applications in transparent windows or drone stealth.
{"title":"Screen-Printed Metamaterial Absorber Using Fractal Metal Mesh for Optical Transparency and Flexibility","authors":"Jinwoo Choi, Daecheon Lim, Sungjoon Lim","doi":"10.3390/fractalfract8050284","DOIUrl":"https://doi.org/10.3390/fractalfract8050284","url":null,"abstract":"In stealth applications, there is a growing emphasis on the development of radar-absorbing structures that are efficient, flexible, and optically transparent. This study proposes a screen-printed metamaterial absorber (MMA) on polyethylene terephthalate (PET) substrates using indium tin oxide (ITO) as the grounding layer, which achieves both optical transparency and flexibility. These materials and methods enhance the overall flexibility and transparency of MMA. To address the limited transparency caused by the silver nanoparticle ink for the top pattern, a metal mesh was incorporated to reduce the area ratio of the printed patterns, thereby enhancing transparency. By incrementing the fractal order of the structure, we optimized the operating frequency to target the X-band, which is most commonly used in radar detection. The proposed MMA demonstrates remarkable performance, with a measured absorption of 91.99% at 8.85 GHz and an average optical transmittance of 46.70% across the visible light spectrum (450 to 700 nm), indicating its potential for applications in transparent windows or drone stealth.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":null,"pages":null},"PeriodicalIF":5.4,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140996159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-08DOI: 10.3390/fractalfract8050282
F. Salama
In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation.
近年来，各种复杂系统和现实世界的现象已被证明包含随时间、空间或其他变量变化的记忆和遗传特性。因此，包含可变阶分数算子的分数偏微分方程被广泛用于精确模拟这类现象。在本文中，我们考虑了在时间方向上具有 Caputo 变阶分数导数的二维分数电缆方程，它更适合用于描述生物系统中的神经元动力学。我们提出了一个点向方案，即 Crank-Nicolson 有限差分法，以及一个称为显式解耦分组法的分组方案来解决所考虑的问题。对数值方案的稳定性和收敛性进行了详细分析。为了证明所提方法的有效性，还给出了数值模拟结果，并以表格和图形形式表示。基于 CPU 时序、迭代计数和最大绝对误差的定量分析表明，在求解变阶分式方程时，显式解耦分组法比 Crank-Nicolson 有限差分方案更有效。
{"title":"On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics","authors":"F. Salama","doi":"10.3390/fractalfract8050282","DOIUrl":"https://doi.org/10.3390/fractalfract8050282","url":null,"abstract":"In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":null,"pages":null},"PeriodicalIF":5.4,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140998795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-08DOI: 10.3390/fractalfract8050283
Sumei Zhang, Haiyang Xiao, Hongquan Yong
This paper aims to provide an effective method for pricing forward starting options under the double fractional stochastic volatilities mixed-exponential jump-diffusion model. The value of a forward starting option is expressed in terms of the expectation of the forward characteristic function of log return. To obtain the forward characteristic function, we approximate the pricing model with a semimartingale by introducing two small perturbed parameters. Then, we rewrite the forward characteristic function as a conditional expectation of the proportion characteristic function which is expressed in terms of the solution to a classic PDE. With the affine structure of the approximate model, we obtain the solution to the PDE. Based on the derived forward characteristic function and the Fourier transform technique, we develop a pricing algorithm for forward starting options. For comparison, we also develop a simulation scheme for evaluating forward starting options. The numerical results demonstrate that the proposed pricing algorithm is effective. Exhaustive comparative experiments on eight models show that the effects of fractional Brownian motion, mixed-exponential jump, and the second volatility component on forward starting option prices are significant, and especially, the second fractional volatility is necessary to price accurately forward starting options under the framework of fractional Brownian motion.
{"title":"Forward Starting Option Pricing under Double Fractional Stochastic Volatilities and Jumps","authors":"Sumei Zhang, Haiyang Xiao, Hongquan Yong","doi":"10.3390/fractalfract8050283","DOIUrl":"https://doi.org/10.3390/fractalfract8050283","url":null,"abstract":"This paper aims to provide an effective method for pricing forward starting options under the double fractional stochastic volatilities mixed-exponential jump-diffusion model. The value of a forward starting option is expressed in terms of the expectation of the forward characteristic function of log return. To obtain the forward characteristic function, we approximate the pricing model with a semimartingale by introducing two small perturbed parameters. Then, we rewrite the forward characteristic function as a conditional expectation of the proportion characteristic function which is expressed in terms of the solution to a classic PDE. With the affine structure of the approximate model, we obtain the solution to the PDE. Based on the derived forward characteristic function and the Fourier transform technique, we develop a pricing algorithm for forward starting options. For comparison, we also develop a simulation scheme for evaluating forward starting options. The numerical results demonstrate that the proposed pricing algorithm is effective. Exhaustive comparative experiments on eight models show that the effects of fractional Brownian motion, mixed-exponential jump, and the second volatility component on forward starting option prices are significant, and especially, the second fractional volatility is necessary to price accurately forward starting options under the framework of fractional Brownian motion.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":null,"pages":null},"PeriodicalIF":5.4,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140998486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers the possibility of using monofractal and multifractal analysis of acoustic signals to detect water leaks through gate valves. Detrended fluctuation analysis (DFA) and multifractal detrended fluctuation analysis (MF-DFA) were used. Experimental studies were conducted on a ½-inch nominal diameter wedge valve, which was fitted to a ¾-inch nominal diameter steel pipeline. The water leak was simulated by opening the valve. The resulting leakage rates for different valve opening conditions were 5.3, 10.5, 14, 16.8, and 20 L per minute (L/min). The Hurst exponent for acoustic signals in a hermetically sealed valve is at the same level as a deterministic signal, while the width of the multifractal spectrum closely matches that of a monofractal process. When a leak occurs, turbulent flow pulsations appear, and with small leak sizes, the acoustic signals become anticorrelated with a high degree of multifractality. As the leakage increases, the Hurst exponent also increases and the width of the multifractal spectrum decreases. The main contributor to the multifractal structure of leak signals is small, noise-like fluctuations. The analysis of acoustic signals using the DFA and MF-DFA methods enables determining the extent of water leakage through a non-sealed gate valve. The results of the experimental studies are in agreement with the numerical simulations. Using the Ansys Fluent software (v. 19.2), the frequencies of flow vortices at different positions of gate valve were calculated. The k-ω SST turbulence model was employed for calculations. The calculations were conducted in a transient formulation of the problem. It was found that as the leakage decreases, the areas with a higher turbulence eddy frequency increase. An increase in the frequency of turbulent fluctuations leads to enhanced energy dissipation. Some of the energy from ordered processes is converted into the energy of disordered processes.
{"title":"Detection of Gate Valve Leaks through the Analysis Fractal Characteristics of Acoustic Signal","authors":"Ayrat Zagretdinov, Shamil Ziganshin, Eugenia Izmailova, Yuri Vankov, Ilya Klyukin, Roman Alexandrov","doi":"10.3390/fractalfract8050280","DOIUrl":"https://doi.org/10.3390/fractalfract8050280","url":null,"abstract":"This paper considers the possibility of using monofractal and multifractal analysis of acoustic signals to detect water leaks through gate valves. Detrended fluctuation analysis (DFA) and multifractal detrended fluctuation analysis (MF-DFA) were used. Experimental studies were conducted on a ½-inch nominal diameter wedge valve, which was fitted to a ¾-inch nominal diameter steel pipeline. The water leak was simulated by opening the valve. The resulting leakage rates for different valve opening conditions were 5.3, 10.5, 14, 16.8, and 20 L per minute (L/min). The Hurst exponent for acoustic signals in a hermetically sealed valve is at the same level as a deterministic signal, while the width of the multifractal spectrum closely matches that of a monofractal process. When a leak occurs, turbulent flow pulsations appear, and with small leak sizes, the acoustic signals become anticorrelated with a high degree of multifractality. As the leakage increases, the Hurst exponent also increases and the width of the multifractal spectrum decreases. The main contributor to the multifractal structure of leak signals is small, noise-like fluctuations. The analysis of acoustic signals using the DFA and MF-DFA methods enables determining the extent of water leakage through a non-sealed gate valve. The results of the experimental studies are in agreement with the numerical simulations. Using the Ansys Fluent software (v. 19.2), the frequencies of flow vortices at different positions of gate valve were calculated. The k-ω SST turbulence model was employed for calculations. The calculations were conducted in a transient formulation of the problem. It was found that as the leakage decreases, the areas with a higher turbulence eddy frequency increase. An increase in the frequency of turbulent fluctuations leads to enhanced energy dissipation. Some of the energy from ordered processes is converted into the energy of disordered processes.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":null,"pages":null},"PeriodicalIF":5.4,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141000732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-08DOI: 10.3390/fractalfract8050281
S. M. M. Rasouli, Samira Cheraghchi, Paulo Moniz
Considering the Friedmann–Lemaître–Robertson–Walker (FLRW) metric and the Einstein scalar field system as an underlying gravitational model to construct fractional cosmological models has interesting implications in both classical and quantum regimes. Regarding the former, we just review the most fundamental approach to establishing an extended cosmological model. We demonstrate that employing new methodologies allows us to obtain exact solutions. Despite the corresponding standard models, we cannot use any arbitrary scalar potentials; instead, it is determined from solving three independent fractional field equations. This article concludes with an overview of a fractional quantum/semi-classical model that provides an inflationary scenario.
{"title":"Fractional Scalar Field Cosmology","authors":"S. M. M. Rasouli, Samira Cheraghchi, Paulo Moniz","doi":"10.3390/fractalfract8050281","DOIUrl":"https://doi.org/10.3390/fractalfract8050281","url":null,"abstract":"Considering the Friedmann–Lemaître–Robertson–Walker (FLRW) metric and the Einstein scalar field system as an underlying gravitational model to construct fractional cosmological models has interesting implications in both classical and quantum regimes. Regarding the former, we just review the most fundamental approach to establishing an extended cosmological model. We demonstrate that employing new methodologies allows us to obtain exact solutions. Despite the corresponding standard models, we cannot use any arbitrary scalar potentials; instead, it is determined from solving three independent fractional field equations. This article concludes with an overview of a fractional quantum/semi-classical model that provides an inflationary scenario.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":null,"pages":null},"PeriodicalIF":5.4,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140998311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-07DOI: 10.3390/fractalfract8050279
Dalei Jing, Peng Qi
Building upon the efficient transport capabilities observed in the fractal tree-like convergent structures found in nature, this paper numerically studies the transport process of the combined electroosmotic and pressure-driven flow within a fractal tree-like convergent microchannel (FTCMC) with uniform channel height. The present work finds that the flow rate of the combined flow first increases and then decreases with the increasing branch width convergence ratio under the fixed voltage difference and pressure gradient along the FTCMC, which means that there is an optimal branch width convergence ratio to maximize the transport efficiency of the combined flow within the FTCMC. The value of the optimal branch convergence ratio is highly dependent on the ratio of the voltage difference and pressure gradient to drive the combined flow. By adjusting the structural and dimensional parameters of the FTCMC, the dependencies of the optimal branch convergence ratio of the FTCMC on the branching level convergence ratio, the length ratio, the branching number, and the branching level are also investigated. The findings in the present work can be used for the optimization of FTCMC with high transport efficiency for combined electroosmotic and pressure-driven flow.
{"title":"The Optimal Branch Width Convergence Ratio to Maximize the Transport Efficiency of the Combined Electroosmotic and Pressure-Driven Flow within a Fractal Tree-like Convergent Microchannel","authors":"Dalei Jing, Peng Qi","doi":"10.3390/fractalfract8050279","DOIUrl":"https://doi.org/10.3390/fractalfract8050279","url":null,"abstract":"Building upon the efficient transport capabilities observed in the fractal tree-like convergent structures found in nature, this paper numerically studies the transport process of the combined electroosmotic and pressure-driven flow within a fractal tree-like convergent microchannel (FTCMC) with uniform channel height. The present work finds that the flow rate of the combined flow first increases and then decreases with the increasing branch width convergence ratio under the fixed voltage difference and pressure gradient along the FTCMC, which means that there is an optimal branch width convergence ratio to maximize the transport efficiency of the combined flow within the FTCMC. The value of the optimal branch convergence ratio is highly dependent on the ratio of the voltage difference and pressure gradient to drive the combined flow. By adjusting the structural and dimensional parameters of the FTCMC, the dependencies of the optimal branch convergence ratio of the FTCMC on the branching level convergence ratio, the length ratio, the branching number, and the branching level are also investigated. The findings in the present work can be used for the optimization of FTCMC with high transport efficiency for combined electroosmotic and pressure-driven flow.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":null,"pages":null},"PeriodicalIF":5.4,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141002792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-07DOI: 10.3390/fractalfract8050278
Yuhang Yao, Jiaxin Yuan, Tao Chen, Chen Zhang, Hui Yang
Based on an adaptive neural control scheme, this paper investigates the consensus problem of random Markov jump multi-agent systems with full state constraints. Each agent is described by the fractional-order random nonlinear uncertain system driven by random differential equations, where the random noise is the second-order stationary stochastic process. First, in order to deal with the unknown functions with Markov jump parameters, a radial basis function neural network (RBFNN) structure is introduced to achieve approximation. Second, for the purpose of keeping the agents’ states from violating the constraint boundary, the tan-type barrier Lyapunov function is employed. By using the stochastic stability theory and adopting the backstepping technique, a novel adaptive neural control design method is presented. Furthermore, to cope with the differential explosion problem in the design course, the extended state observer (ESO) is developed instead of neural network (NN) approximation or command filtering techniques. Finally, the exponentially noise-to-state stability in the mean square is analyzed rigorously by the Lyapunov method, which guarantees the consensus of the considered multi-agent systems and all the agents’ outputs are bounded in probability. Two simulation examples are provided to verify the effectiveness of the suggested control strategy.
本文基于自适应神经控制方案，研究了具有完全状态约束的随机马尔可夫跃迁多代理系统的共识问题。每个代理都由随机微分方程驱动的分数阶随机非线性不确定系统来描述，其中随机噪声为二阶静态随机过程。首先，为了处理具有马尔可夫跳跃参数的未知函数，引入了径向基函数神经网络（RBFNN）结构来实现逼近。其次，为了防止代理状态违反约束边界，采用了 tan 型屏障 Lyapunov 函数。利用随机稳定性理论和反步进技术，提出了一种新的自适应神经控制设计方法。此外，为了应对设计过程中的微分爆炸问题，还开发了扩展状态观测器（ESO），而不是神经网络（NN）逼近或指令滤波技术。最后，利用 Lyapunov 方法严格分析了均方差中指数噪声到状态的稳定性，从而保证了所考虑的多代理系统的一致性，并且所有代理的输出在概率上都是有界的。本文提供了两个仿真实例来验证所建议的控制策略的有效性。
{"title":"Adaptive Neural Control for a Class of Random Fractional-Order Multi-Agent Systems with Markov Jump Parameters and Full State Constraints","authors":"Yuhang Yao, Jiaxin Yuan, Tao Chen, Chen Zhang, Hui Yang","doi":"10.3390/fractalfract8050278","DOIUrl":"https://doi.org/10.3390/fractalfract8050278","url":null,"abstract":"Based on an adaptive neural control scheme, this paper investigates the consensus problem of random Markov jump multi-agent systems with full state constraints. Each agent is described by the fractional-order random nonlinear uncertain system driven by random differential equations, where the random noise is the second-order stationary stochastic process. First, in order to deal with the unknown functions with Markov jump parameters, a radial basis function neural network (RBFNN) structure is introduced to achieve approximation. Second, for the purpose of keeping the agents’ states from violating the constraint boundary, the tan-type barrier Lyapunov function is employed. By using the stochastic stability theory and adopting the backstepping technique, a novel adaptive neural control design method is presented. Furthermore, to cope with the differential explosion problem in the design course, the extended state observer (ESO) is developed instead of neural network (NN) approximation or command filtering techniques. Finally, the exponentially noise-to-state stability in the mean square is analyzed rigorously by the Lyapunov method, which guarantees the consensus of the considered multi-agent systems and all the agents’ outputs are bounded in probability. Two simulation examples are provided to verify the effectiveness of the suggested control strategy.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":null,"pages":null},"PeriodicalIF":5.4,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141004583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-07DOI: 10.3390/fractalfract8050277
Muhammad Nadeem, LOREDANA-FLORENTINA Iambor
This study provides an innovative and attractive analytical strategy to examine the numerical solution for the time-fractional Schrödinger equation (SE) in the sense of Caputo fractional operator. In this research, we present the Elzaki transform residual power series method (ET-RPSM), which combines the Elzaki transform (ET) with the residual power series method (RPSM). This strategy has the advantage of requiring only the premise of limiting at zero for determining the coefficients of the series, and it uses symbolic computation software to perform the least number of calculations. The results obtained through the considered method are in the form of a series solution and converge rapidly. These outcomes closely match the precise results and are discussed through graphical structures to express the physical representation of the considered equation. The results showed that the suggested strategy is a straightforward, suitable, and practical tool for solving and comprehending a wide range of nonlinear physical models.
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