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

该项目建议开发一种新的微和纳米流体数学模型，它与工程，生物化学，纳米技术和生物技术交叉。微纳流控技术的研究具有极大的潜力，可以彻底改变生物和化学应用中的方法，这在设计系统中具有广泛的应用，在该系统中，处理小体积的流体以实现多路复用、自动化和高通量筛选。微流体和纳米流体广泛用于喷墨打印头、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，并直接为国家的知识基础设施做出贡献。非局部模型，如分数阶偏微分方程（FPDE），分数阶拉普拉斯算子和周波函数，正在成为模拟具有挑战性的现象的强大工具，包括生物学，物理学，化学，金融、工程和地下水中的溶质运移。这些模型比整数阶偏微分方程模型更能描述实际应用中的许多重要问题。非局部模型至今没有得到广泛应用的两个主要原因是：（1）它们产生的数值格式具有稠密矩阵和强局部行为的解，这比传统的整数阶PDE模型的数值求解成本要高得多。用中等数量的网格点对三维线性问题进行简单的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)非局部模型引入了数学上的困难，这在整数阶偏微分方程的上下文中没有遇到。现建议在这一关头有效地处理这两点。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。对一个简单的三维分数阶偏微分方程的初步数值实验表明，该方法将传统方法的CPU时间从2个月零25天减少到5.74秒，并显著减少了存储量。所提出的数学和数值分析将为非局部模型和相关的数值近似提供坚实的理论基础。快速准确的数值计算方法和严格的数学分析结果将应用于微纳米流体的新的数学模型的发展。由此产生的数学模型将用于微和纳米流体的研究。

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.