积分微分方程和反常扩散问题的高效谱方法

Integro-differential equations and fractional differential equations arise in many branches of science and nature covering physics, biology, chemistry, earth science, environmental science, and control theory etc. Theoretical and numerical investigations of these equations have been attracting the close attention of researchers. These equations share a common feature: they all contain non-local operators. On one side, this feature makes them more suitable for describing the behavior of materials with memory, which can be found in modeling many complex systems. On the other side, this feature makes the theoretical study and the numerical treatment of these equations more complicated than classical local problems. The main purpose of this project is to construct highly efficient numerical methods for a class of integro-differential equations and fractional differential equations. The goals of the project include: 1) investigate well-posedness of a family of Volterra integral equations, including existence, uniqueness, and regularity (singularity) of the solution etc.; 2) develop new spectral/spectral -element methods for solving a class of high-dimensional Volterra integral/ fractional PDEs; 3) investigate weak formulations and existence of weak solutions of a kind of fractional PDEs; 4) design and analyze high order numerical methods for time space abnormal diffusion equations, and apply them to some emerging applications in fluid dynamics and materials science, such as fractional Fokker-Planck equations, fractional cable equations, fractional Dias-Dutykh equations，and so on.

积分微分方程和分数阶微分方程在物理、化学、生物、工程、自然界等许多领域有应用背景。这类方程有一个共同点，即都具有非局部算子。一方面，这一特点使之更适合用以描述具有记忆性质的材料行为，因此可以在一些复杂系统的建模中找到应用。另一方面，非局部算子的存在使得这类方程的理论研究和数值求解更为困难。本项目旨在研究一类积分微分方程和分数阶微分方程的基本性质和高阶算法设计。主要研究内容包括：1）研究几类积分方程和分数阶微分方程解的性质，特别是研究解的正则性，刻划解的奇性特征，为设计高阶算法提供理论支持；2）设计和分析Volterra型积分方程的谱/谱元法；3）考察几个分数阶偏微分方程初边值问题的变分形式和适定性理论，获得一些解的存在唯一性结果，为基于变分的算法设计搭建基础框架；4) 设计和分析复杂区域上时间-空间反常扩散方程的基于区域分解的有限元/谱元法， 并将之应用于一些具有实际背景的分数阶模型问题。

本项目研究了几个主要问题：1）几类积分方程和分数阶微分方程解的性质，特别是研究了解的正则性，刻划了解的奇性特征。这些结果的取得为设计高阶算法提供理论支持；2）设计和分析了Volterra型积分方程的谱/谱元法；3）几个分数阶偏微分方程初边值问题的变分形式和适定性理论，获得了一些解的存在唯一性结果，为基于变分的算法设计搭建了基础框架；4) 设计和分析了复杂区域上时间-空间反常扩散方程的基于区域分解的有限元/谱元法， 并将之应用于一些具有实际背景的分数阶模型问题。