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

本计划拟发展一种微型及奈米流体学的数学模型，涵盖工程、生物化学、奈米科技及生物科技。微流体和纳米流体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，广泛应用于小体积流体处理系统的设计，以实现多路复用、自动化和高通量筛选。微纳米流体学广泛应用于喷墨打印头、DNA芯片、片上实验室技术、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，并直接促进国家的智力基础设施。非局部模型，如分数阶偏微分方程（FPDEs）、分数阶拉普拉斯方程（分数阶拉普拉斯方程）和周期动力学，正在成为模拟具有挑战性的现象的强大工具，这些现象包括异常输运、长时间记忆或空间相互作用，在生物、物理、化学、金融、工程和地下水溶质输运等广泛领域。这些模型比整阶PDE模型更适合描述应用程序中的许多重要问题。迄今为止，非局部模型没有得到广泛应用的主要原因有两个：(1)非局部模型生成的数值格式具有密集的矩阵和具有强局部行为的解，与传统的整数阶PDE模型相比，其数值求解成本要高得多。一个简单的三维线性问题的模拟，一个中等数量的网格点，可能需要最先进的超级计算机几百年才能完成，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没有遇到的数学困难。建议在这个关键时刻有效地解决这两个问题。本文提出的快速数值方法将为非局部模型提供显著的计算优势，并将促进其应用。简单三维分数PDE的初步数值实验表明，该方法将传统方法的2个月25天的CPU时间缩短到5.74秒，并显著降低了存储空间。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谨的数学分析结果将应用于微纳米流体的新型数学模型的发展。所建立的数学模型将用于微、纳米流体学的研究。

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.