Fractional Partial Differential Equations and Related Nonlocal Models: Fast Numerical Methods, Analysis, and Application

分数阶偏微分方程及相关非局部模型：快速数值方法、分析和应用

• 批准号: 1620194
• 负责人: Hong Wang
• 金额: $ 25万
• 依托单位: University of South Carolina at Columbia
• 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: 2016-10-01 至 2020-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620194&HistoricalAwards=false
• 关键词: Fractional; Partial; Differential; Equations; Related

The project proposes to develop a novel mathematical modeling of micro- and nano-fluidics, which intersects engineering, biochemistry, nanotechnology, and biotechnology. The study of micro-and nano-fluidics has great potential to revolutionize the methods in biological and chemical applications, which has wide applications to the design of systems in which low volumes of fluids are processed to achieve multiplexing, automation, and high-throughput screening. Micro- and nano-fluidics is used widely in the development of inkjet printheads, DNA chips, lab-on-a-chip technology, micro-propulsion, and micro-thermal technologies. The project will also provide advanced interdisciplinary training to graduate and undergraduate students. All of these activities will have broad and long-lasting impacts and contribute directly to the intellectual infrastructure of the nation.Nonlocal models such as fractional partial differential equations (FPDEs), fractional Laplacian, and peridynamics are emerging as powerful tools for modeling challenging phenomena including anomalous transport and long-range time memory or spatial interactions in a wide range of fields such as biology, physics, chemistry, finance, engineering, and solute transport in groundwater. These models provide more appropriate description of many important problems in applications than integer-order PDE models do. Two of the main reasons that nonlocal models have not been used extensively so far are as follows: (1) They generate numerical schemes with dense matrices and solutions with strongly local behavior, which are significantly more expensive to solve numerically than traditional integer-order PDE models. A naive simulation of a three-dimensional linear problem with a moderate number of grid points may take state of the art supercomputers hundreds of years to finish and so deemed unrealistic. (2) Nonlocal models introduce mathematical difficulties, which were not encountered in the context of integer-order PDEs. It is proposed to effectively address both points at this juncture. The fast numerical methods proposed herein will provide significant computational benefits for nonlocal models, and will facilitate their applications. Preliminary numerical experiments of a simple three-dimensional fractional PDE showed that the proposed method reduced the CPU time from 2 months and 25 days by a traditional method to 5.74 seconds and reduced storage significantly. The proposed mathematical and numerical analysis will provide a solid theoretical foundation for nonlocal models and related numerical approximations. The fast and accurate numerical methods and rigorous mathematical analysis results will be applied in the development of a novel mathematical modeling of micro- and nano-fluidics. The resulting mathematical model will be utilized in the study of micro- and nano-fluidics.

该项目提议开发一种新型的微纳米流体数学模型，它涉及工程学、生物化学、纳米技术和生物技术。微流体和纳米流体的研究具有彻底改变生物和化学应用方法的巨大潜力，在处理少量流体以实现多重、自动化和高通量筛选的系统设计中具有广泛的应用。微纳米流体广泛应用于喷墨打印头、DNA 芯片、芯片实验室技术、微推进和微热技术的开发。该项目还将为研究生和本科生提供先进的跨学科培训。所有这些活动都将产生广泛而持久的影响，并直接为国家的智力基础设施做出贡献。分数偏微分方程（FPDE）、分数拉普拉斯和近场动力学等非局部模型正在成为建模具有挑战性的现象的强大工具，包括生物学、物理、化学、金融、工程和地下水中溶质输运等广泛领域的异常输运和长程时间记忆或空间相互作用。与整数阶偏微分方程模型相比，这些模型可以更恰当地描述应用中的许多重要问题。迄今为止，非局部模型尚未广泛使用的两个主要原因如下：（1）它们生成具有密集矩阵的数值格式和具有强局部行为的解，这些数值求解的成本比传统的整数阶偏微分方程要昂贵得多模型。对具有适量网格点的三维线性问题的简单模拟可能需要最先进的超级计算机数百年才能完成，因此被认为是不现实的。 (2) 非局部模型引入了整数阶偏微分方程中不会遇到的数学困难。建议此时有效解决这两点。本文提出的快速数值方法将为非局部模型提供显着的计算优势，并将促进其应用。简单三维分数阶偏微分方程的初步数值实验表明，该方法将CPU时间从传统方法的2个月零25天减少到5.74秒，并显着减少了存储空间。所提出的数学和数值分析将为非局部模型和相关数值近似提供坚实的理论基础。快速准确的数值方法和严格的数学分析结果将应用于开发微纳米流体的新型数学模型。由此产生的数学模型将用于微流体和纳米流体的研究。