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被广泛用于科学，工程和行业。在这个项目中，我们考虑了一类复杂现象模型，这些模型表现出记忆效果和远距离相互作用，并在粘弹性材料的设计和制造中应用，异常的扩散运输，气体和油回收率中的水电，多孔材料的生物封存以及某些材料的生物销售以及某些材料的变形，例如在整形植入型植入型植入物和形状的存储器中。重点放在分数计算上，特别是变性阶分数部分微分方程，其中分数顺序可能是空间，时间甚至未知溶液的函数。研究活动将有助于分数演算的分析，模拟，建模和应用，并为学生提供高级的跨学科培训。该项目包括研究生的培训机会。以幂律衰减尾巴为特征的分数部分微分方程（FPDE）已证明可以准确地对非本地性质的复杂现象进行建模。但是，可变级FPDE的严格数学和数值分析目前不比整数阶PDE所知。例如，众所周知，在光滑的域上施加的线性椭圆形和抛物线FPDE具有光滑数据表现出弱的初始或边界奇异性，这与它们的整数类似物形成了鲜明的对比。这使得基于其真实解决方案的（通常是不真实的）平滑度假设对FPDE进行数值近似值的误差估计是不现实的。在这个项目中，研究人员对可变阶FPDES及其快速解决方案算法开发了准确稳定的数值近似值，并证明了它们的适当性和平滑性能。研究者还将证明对可变级FPDE的数值近似值的最佳级误差估计值，而无需任何人为的定期假设，但仅在其系数，可变订单和其他相关数据的规律性假设下，该奖项就会反映出NSF的法定任务，并通过评估范围来进行评估，并反映了概念的支持。

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