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已广泛应用于科学、工程和工业领域。在这个项目中，我们考虑一类复杂现象的模型，这些模型表现出记忆效应和长程相互作用，在粘弹性材料的设计和制造、反常扩散传输、天然气和石油采收中的水力压裂、多孔材料的生物堵塞、以及某些材料的变形，例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。该项目包括研究生的培训机会。分数阶偏微分方程（FPDE）以幂律衰减尾部为特征，已被证明可以准确地模拟非局部性质的复杂现象。然而，与整数阶 PDE 相比，目前对变阶 FPDE 的严格数学和数值分析知之甚少。例如，众所周知，对具有平滑数据的平滑域施加的线性椭圆和抛物线 FPDE 表现出较弱的初始或边界奇异性，这与它们的整数阶类似物形成鲜明对比。这使得基于真实解的（通常不真实的）平滑度假设来对 FPDE 进行数值近似的误差估计是不现实的。在该项目中，研究人员开发了变阶 FPDE 的准确且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，而无需对其真实解进行任何人为规律性假设，而仅在其系数、变量阶数和其他相关数据的规律性假设下进行。该奖项反映了 NSF 的法定要求使命，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。

项目成果

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

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

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

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