Chanyuan Wang, Raghda A. M. Attia, Suleman H. Alfalqi, Jameel F. Alzaidi, Mostafa M. A. Khater
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By incorporating fractional derivatives, this model introduces non-locality and memory effects into the classical <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝕂</mi><mi>𝔻</mi><mi>𝕂</mi><mi>𝕂</mi></math></span><span></span> equations, commonly utilized in phenomena such as shallow water waves, nonlinear optics, and plasma physics. The fractional approach enhances mathematical representations, allowing for a more realistic depiction of the intricate behaviors observed in numerous modern physical systems. This study focuses on the development of accurate and efficient numerical techniques tailored for the computationally demanding <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝔾</mi><mi>𝔽</mi><mi>𝕂</mi><mi>𝔻</mi><mi>𝕂</mi><mi>𝕂</mi></math></span><span></span> model, leveraging the Khater II and generalized rational approximation methods. These methodologies facilitate stable time-integration, effectively addressing the model’s stiffness and multi-dimensional nature. Through numerical analysis, insights into the stability and convergence of the algorithms are gained. Simulations conducted validate the performance of these methods against established solutions while also uncovering novel capabilities for exploring complex wave dynamics in scenarios involving complete fractional formulations. 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引用次数: 0
摘要
(3+1)-dimensional generalized nonlinear fractional Konopelchenko-Dubrovsky-Kaup-Kupershmidt(𝔾𝔽𝔽𝕂𝔻𝕂𝕂)模型表示了非线性波在复杂多维物理介质中的传播和相互作用,这些介质具有反常色散和耗散现象。通过加入分数导数,该模型在经典的𝕂𝔻𝕂𝕂方程中引入了非定位和记忆效应,常用于浅水波、非线性光学和等离子物理学等现象。分数方法增强了数学表达,可以更真实地描述在众多现代物理系统中观察到的复杂行为。本研究的重点是利用 Khater II 和广义有理近似方法,为计算要求极高的𝔾𝔽𝕂𝔻𝕂𝕂模型开发精确高效的数值技术。这些方法促进了稳定的时间积分,有效地解决了模型的刚性和多维性问题。通过数值分析,可以深入了解算法的稳定性和收敛性。所进行的模拟验证了这些方法与既定解决方案的性能,同时也发现了在涉及完整分数公式的情况下探索复杂波浪动力学的新功能。研究结果强调了将分数微积分融入高维非线性偏微分方程的潜力,为推进当代物理学科复杂波现象的建模和计算分析提供了一条大有可为的途径。
Stability analysis and conserved quantities of analytic nonlinear wave solutions in multi-dimensional fractional systems
The (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt model represents the propagation and interaction of nonlinear waves in complex multi-dimensional physical media characterized by anomalous dispersion and dissipation phenomena. By incorporating fractional derivatives, this model introduces non-locality and memory effects into the classical equations, commonly utilized in phenomena such as shallow water waves, nonlinear optics, and plasma physics. The fractional approach enhances mathematical representations, allowing for a more realistic depiction of the intricate behaviors observed in numerous modern physical systems. This study focuses on the development of accurate and efficient numerical techniques tailored for the computationally demanding model, leveraging the Khater II and generalized rational approximation methods. These methodologies facilitate stable time-integration, effectively addressing the model’s stiffness and multi-dimensional nature. Through numerical analysis, insights into the stability and convergence of the algorithms are gained. Simulations conducted validate the performance of these methods against established solutions while also uncovering novel capabilities for exploring complex wave dynamics in scenarios involving complete fractional formulations. The findings underscore the potential of integrating fractional calculus into higher-dimensional nonlinear partial differential equations, offering a promising avenue for advancing the modeling and computational analysis of complex wave phenomena across a spectrum of contemporary physical disciplines.
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