Variable-Order Fractional Partial Differential Equations: Computation, Analysis, and Application

数学建模和仿真技术在科学、工程和工业中得到了广泛的应用。在这个项目中，我们考虑了一类表现出记忆效应和长程相互作用的复杂现象模型，其应用于粘弹性材料的设计和制造，异常扩散输运，油气开采中的水力压裂，多孔材料的生物堵塞，以及某些材料的变形，如骨科植入物和形状记忆聚合物。重点是分数阶微积分，特别是变阶分数阶偏微分方程，其中分数阶可能是空间，时间甚至未知解的函数。研究活动将有助于分数阶微积分的分析，模拟，建模和应用，并为学生提供先进的跨学科培训。该项目包括为研究生提供培训机会。分数阶偏微分方程（FPDE），其特点是幂律衰减的尾巴，已被证明可以准确地模拟复杂的非局部现象。然而，严格的数学和数值分析的可变阶的有限差分方程是目前已知的比整数阶偏微分方程。例如，众所周知，线性椭圆和抛物型有限差分方程施加在光滑区域与光滑数据表现出弱的初始或边界奇异性，这是在其整数阶类似物形成鲜明对比。这使得它不切实际的进行误差估计的数值逼近的FPDE的基础上（往往是不真实的）光滑性假设的真实解决方案。在这个项目中，研究人员开发了变阶FPDE的精确和稳定的数值近似及其快速求解算法，并证明了它们的适定性和平滑特性。研究人员还将证明变阶FPDE数值近似的最优阶误差估计，而不需要对其真实解进行任何人为的正则性假设，而只需要对其系数、变阶和其他相关数据进行正则性假设。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Mathematical modeling and simulation techniques have been widely used in science, engineering, and industry. In this project, we consider a class of models of complex phenomena which exhibit memory effects and long range interactions, with applications in design and manufacturing of visco-elastic materials, anomalous diffusive transport, hydrofracking in gas and oil recovery, bioclogging of porous materials, and the deformation of some materials such as in orthopedic implants and shape memory polymers. The focus is on fractional calculus and specifically on variable order fractional partial differential equations, in which the fractional order may be a function of space, time and even unknown solutions. The research activities will contribute to the analysis, simulation, modeling and application of fractional calculus, and provide advanced interdisciplinary training to students. The project includes training opportunities for graduate students. Fractional partial differential equations (FPDEs), which are characterized by power-law decaying tails, have shown to accurately model complex phenomena of nonlocal nature. However, rigorous mathematical and numerical analysis of variable-order FPDEs is currently less known than that for integer-order PDEs. For instance, it is well known that linear elliptic and parabolic FPDEs imposed on smooth domains with smooth data exhibit weak initial or boundary singularity, which is in sharp contrast to their integer-order analogues. This makes it unrealistic to carry out error estimates of numerical approximations to FPDEs based on the (often untrue) smoothness assumptions of their true solutions. In this project the investigators develop accurate and stable numerical approximations to variable-order FPDEs and their fast solution algorithms, as well as prove their well-posedness and smoothing properties. The investigators will also prove optimal-order error estimates of numerical approximations to variable-order FPDEs without any artificial regularity assumption of their true solutions, but only under the regularity assumptions of their coefficients, variable orders and other related data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.