时间分数阶偏微分方程高效数值方法及其解的性质研究

In recent years, fractional partial differential equations have been widely applied in many fields, such as signal processing, earthquake analysis, quantum economic, et al. The fractional derivative is defined through a singular integral, the analytic expression of the solution is often difficult to obtain, so it becomes very important to study efficient numerical algorithms and properties of the solution for fractional partial differential equations. The first goal of this project is to construct efficient finite difference algorithms for time fractional partial differential equations. The high order approximations are used to discretize the time fractional derivative, for spatial direction, the weighted average compact operator and the high order exponential scheme are adopted. Another aspect to be studied in this project is the quenching problem for fractional models. The Fourier transform method and the characteristic function theory will be used to estimate the quenching time for fractional models, this project will also design efficient numerical algorithms to show this phenomenon. Problems studied in this project always have nonsymmetric structure, it is the selling point of the project to use the energy method combined with the matrix decomposition technique to ensure the stability and convergence of the scheme.

近年来，分数阶偏微分方程已被广泛应用于信号处理、地震分析和量子经济等领域。由于分数阶导数是一个带有奇性核的积分，分数阶偏微分方程解的解析表达式往往难以求得，因此，探求分数阶偏微分方程的高效数值算法及其解的性质是当前十分重要的科学任务。本项目拟对时间分数阶偏微分方程构建高效有限差分算法，时间分数阶导数主要采用申请者及其他学者提出的高阶离散格式，空间方向主要利用加权平均紧格式、指数型格式来构造高阶格式；本项目关注的另一方面是分数阶模型的猝灭问题，拟借助Fourier变换、特征函数理论等技巧估计出分数阶模型猝灭发生的时刻，并设计高效的数值算法来展示这一现象。本项目所研究的问题大都具有非对称结构，采用能量法与矩阵分解技巧相结合来保证格式的稳定性和收敛性是本项目的特色。

近年来，分数阶微分方程的数值求解因其广泛的应用背景而备受关注。本项目主要研究时间分数阶偏微分方程的高效数值方法。通过项目组的努力，基本完成了预定的研究目标，发表了15篇学术论文，其中SCI收录12篇，含高被引论文1篇，培养了1名博士后，先后培养了8名硕士研究生。主要针对时间分数阶扩散方程、时间分数阶积分微分方程、时间分数阶电报方程、非线性时间分数阶波方程、时间分数阶KDV-Burgers方程、以及非线性分数阶方程的quenching现象等问题进行了研究，通过降阶、加权平均、非均匀网格离散、差分法、正交样条配置等方法或技术设计算法，通过能量法、Fourier方法证明算法的收敛性与稳定性，并通过数值实验来验证理论结果。