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

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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.