Numerical and Analytical Investigations on Nonlocal Dispersive Wave Equations

非局部色散波动方程的数值与分析研究

• 批准号: 1620465
• 负责人: Yanzhi Zhang
• 金额: $ 18万
• 依托单位: Missouri University of Science and Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620465&HistoricalAwards=false
• 关键词: Numerical; Analytical; Investigations; Nonlocal; Dispersive

Nonlocal dispersive wave equations have been recently applied in many areas such as electromagnetism, acoustics, cosmology, elasticity, biology, hydrodynamics, viscoelasticity, seismics, water wave, plasma, quantum mechanics, brain and consciousness, and so on. However, their nonlocality introduces considerable challenges in both mathematical analysis and numerical simulations. This project seeks to address fundamental issues related to mathematical modeling and numerical simulations of nonlocal dispersive wave equations as well as their solution properties. The proposed project will bridge the gap between different areas, enhance interdisciplinary research, and advance the application of fractional differential equations in practice. Since nonlocal wave equations have broad applications in physics, chemistry, biology and engineering, the research in this project has great potentials to advance the research and technology in relevant areas.The main objectives of this research are to build mathematical and numerical treatments for the nonlocal Schroedinger wave equations, and to provide a deeper understanding of the modeling with long-range interactions, so as to advance their application to problems with nonlocality. In this project, both the discrete nonlinear Schroedinger (DNLS) equation with long-range interactions and the fractional nonlinear Schroedinger (fNLS) equation with the fractional Laplacian will be investigated. Integrating the discrete and continuous models offers a new opportunity for a deeper understanding of the nonlocality of the Schroedinger wave equations. On the one hand, accurate algorithms will be developed to improve the efficiency and reduce the computational costs in simulating the DNLS with large lattice sites, especially in two- or three-dimensional lattices. On the other hand, efficient and accurate numerical methods for discretizing the fractional Laplacian will be designed and applied to study the properties of the stationary states and dynamics of fNLS. The study on the DNLS and fNLS will provide a deeper understanding on modeling and properties of long-range interactions, as well as is benefitting the development of numerical algorithms for fractional differential equations.

最近在许多领域应用了非局部分散波方程，例如电磁，声学，宇宙学，弹性，弹性，生物学，水动力学，粘弹性，地震，水波，等离子体，等离子体，量子力学，大脑和意识等等。但是，他们的非局部性在数学分析和数值模拟中都引入了相当大的挑战。该项目旨在解决与非局部分散波方程的数学建模和数值模拟有关的基本问题及其解决方案属性。拟议的项目将弥合不同领域之间的差距，增强跨学科研究，并推动分数微分方程在实践中的应用。由于非局部波动方程在物理，化学，生物学和工程学中具有广泛的应用，因此该项目的研究具有促进相关领域的研究和技术的巨大潜力。这项研究的主要目标是为非局部schroedinger波动方程式建立数学和数值处理，并与远程互动相处，以便在不远程互动方面进行更深入的了解，以便在不远程互动方面进行不断的互动，以便在不远程互动方面进行问题。在这个项目中，将研究具有长距离相互作用的离散非线性Schroedinger（DNLS）方程，以及与分数Laplacian的分数非线性Schroedinger（FNLS）方程。整合离散模型为对Schroedinger波方程的非局部性的深入了解提供了一个新的机会。一方面，将开发准确的算法来提高效率并降低使用大晶格位点模拟DNL的计算成本，尤其是在二维或三维晶格中。另一方面，将设计和应用有效且准确的数值方法来分散分数laplacian，以研究固定状态和FNL的动力学的性质。关于DNL和FNLS的研究将对远程相互作用的建模和特性提供更​​深入的了解，并使分数微分方程的数值算法的发展受益。