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

变阶分数阶偏微分方程：计算、分析与应用

• 批准号: 2012291
• 负责人: Hong Wang
• 金额: $ 20万
• 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: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012291&HistoricalAwards=false
• 关键词: Variable; Order; Fractional; Partial; Differential

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.

数学建模和仿真技术在科学、工程和工业中有着广泛的应用。在这个项目中，我们考虑了一类具有记忆效应和远程相互作用的复杂现象的模型，应用于粘弹性材料的设计和制造，异常扩散输送，天然气和石油开采中的水力压裂，多孔材料的生物堵塞，以及一些材料的变形，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。这些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训练。该项目包括为研究生提供培训机会。分数阶偏微分方程（FPDEs）具有幂律衰减尾的特征，可以准确地模拟非局部性质的复杂现象。然而，对于变阶FPDEs的严格的数学和数值分析目前还不如对整阶FPDEs的了解。例如，众所周知，具有光滑数据的光滑域上的线性椭圆型和抛物型FPDEs表现出弱的初始奇异性或边界奇异性，这与它们的整数阶类似物形成鲜明对比。这使得基于（通常不真实的）真实解的平滑假设对FPDEs进行数值近似的误差估计是不现实的。在本项目中，研究人员开发了变阶FPDEs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性。研究人员还将证明变阶FPDEs数值近似的最优阶误差估计，而不需要对其真解进行任何人为的正则性假设，但仅在其系数、变阶和其他相关数据的正则性假设下。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

A time-fractional diffusion equation with space-time dependent hidden-memory variable order: analysis and approximation

• DOI: 10.1007/s10543-021-00861-4
• 发表时间: 2021-04
• 期刊: BIT Numerical Mathematics
• 影响因子: 1.5
• 作者: Xiangcheng Zheng;Hong Wang

Analysis of a time-fractional substantial diffusion equation of variable order

变阶时间分数实质扩散方程的分析

• DOI: 10.3390/fractalfract60201
• 发表时间: 2022
• 期刊: Fractal and fractional
• 影响因子: 5.4
• 作者: Zheng, Xiangcheng;Wang, Hong;Guo, Xu
• 通讯作者: Guo, Xu

An Error Estimate of a Modified Method of Characteristics Modeling Advective-Diffusive Transport in Randomly Heterogeneous Porous Media

随机异质多孔介质中平流扩散传输特征模型修正方法的误差估计

• DOI: 10.4208/csiam-am.2020-0216
• 发表时间: 2021
• 期刊: CSIAM Transactions on Applied Mathematics
• 作者: null, Xiangcheng Zheng;Wang, Hong
• 通讯作者: Wang, Hong

Analysis and numerical approximation to time-fractional diffusion equation with a general time-dependent variable order

• DOI: 10.1007/s11071-021-06353-y
• 发表时间: 2021-05
• 期刊: Nonlinear Dynamics
• 影响因子: 5.6
• 作者: Xiangcheng Zheng;Hong Wang

Well-posedness and numerical approximation of a fractional diffusion equation with a nonlinear variable order

• DOI: 10.1051/m2an/2020072
• 发表时间: 2020-10
• 期刊: ESAIM: Mathematical Modelling and Numerical Analysis
• 作者: Buyang Li;Hong Wang;Jilu Wang