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流体和纳米流体学数学建模，该模型与工程，生物化学，纳米技术和生物技术相交。对微荧光学的研究具有巨大的潜力，可以彻底改变生物和化学应用中的方法，这些方法在设计的系统设计中广泛应用，在这些系统的设计中，处理了低体积的流体以实现多重，自动化和高通量筛选。微流体和纳米流体学被广泛用于喷墨打印头，DNA芯片，芯片上实验室技术，微螺旋杆和微型技术技术的开发中。该项目还将为研究生和本科生提供高级跨学科培训。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。诸如部分偏微分方程（FPDES），诸如部分偏微分方程（FPDES），laplacian和Peridynelanic等单位模型成为了在包括型和长期范围的范围内部范围内的范围内部互动，以构建型号的范围，并成为有力的工具地下水的化学，金融，工程和溶质运输。与整数PDE模型相比，这些模型对应用程序中许多重要问题的描述更为适当。到目前为止，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和具有强烈局部行为的密集矩阵和解决方案的数值方案，与传统的整数PDE模型相比，在数字上求解的昂贵得多。对三维线性问题的幼稚模拟具有中等数量的网格点可能需要数百年的最高计算机才能完成，因此被认为是不现实的。 （2）非局部模型引入了数学困难，在整数阶PDE的背景下未遇到这些困难。建议在此关头有效解决这两个要点。本文提出的快速数值方法将为非本地模型提供显着的计算益处，并将促进其应用。简单的三维分数PDE的初步数值实验表明，所提出的方法将CPU时间从2个月和25天的传统方法缩短为5.74秒，并大大降低了存储。所提出的数学和数值分析将为非本地模型和相关的数值近似值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微型和纳米流体学的新型数学建模。所得的数学模型将用于微流体和纳米流体学研究。