Volterra方程与高维分数阶扩散方程的理论与数值研究

In this project, our work is focused on the theoretical investigation and numerical computation of the Volterra integral equations (VIEs) and multi-dimensional fractional diffusion equations (FDEs), which are of interest not only in their own right, but also in that they constitute the principal parts in many other fractional equations. The main contribution of this project is twofold:.First, we propose a parallel in time (called also time parareal) method.to solve Volterra integral equations of the second kind. Our main contributions in this work are two folds: (i) The time parareal method is combined with the spectral method for each sub-problem, leading to an algorithm of very high accuracy; (ii) a rigorous convergence analysis of the overall method will be provided. The overall computational cost is considerably reduced while the desired spectral rate of convergence can be obtained.. Second, we investigate initial boundary value problems of the multi-dimensional space-time fractional diffusion equation and its numerical resolutions. The main contribution of this work are twofold: (i) we establish the well-posedness of the weak solution. The variational formulation of the initial boundary value problems of FDEs will be developed, and the existence and uniqueness of the weak solution will be established by using classical theory for elliptic problems; (ii) based on the weak formulation, we construct a space-time parallel combined with spectral method for efficiently solving the space-time multi-dimensional fractional diffusion problem. Moreover, a convergence analysis of the overall method will be provided. Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the "global time dependence" can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.This will greatly promote the application of fractional partial differential equation in the fields of physics and biology.

本项目拟从理论和数值计算两方面对 Volterra 积分方程以及高维分数阶扩散方程进行深入研究，主要内容包括以下两个方面：.提出采用时间并行算法对 Volterra 积分方程进行数值求解。该工作主要包括两方面：第一，提出用时间并行算法结合谱方法对积分方程进行求解的高阶数值方法；第二, 对该方法的收敛性展开严格的理论分析。该方法将有效地解决在用低阶数值方法求解积分方程时遇到的巨大存储和计算时间问题。. 研究高维时间- - 空间分数阶扩散方程初边值问题的适定性及其数值解。该工作首先导出高维分数阶扩散方程初边值问题的弱形式及弱解的存在唯一性。其次，基于弱解理论,提出用并行算法结合时间- - 空间谱方法数值求解并开展收敛性分析。 适定性理论的建立，高效数值方法的提出，使得数值求解长时间高维分数阶偏微分方程初边值问题成为可能。这将有助于促进高维分数阶方程模型在物理、生物等科学领域中的应用。

本项目从理论和数值计算两方面对Volterra 积分方程, 时间空间变分数阶扩散方程，三角谱方法的LBB条件数，三角元上的二阶微分算子的特征值，分数阶算子的特征值进行深入研究，主要内容如下：.（1）提出采用时间并行算法对Volterra 积分方程进行数值求解。该工作主要包括两方面：第一，提出用时间并行算法结合谱方法对积分方程进行求解的高阶数值方法；第二, 对该方法的收敛性展开严格的理论分析。该方法有效地解决在用低阶数值方法求解积分方程时遇到的巨大存储和计算时间问题。. （2）提出采用Galerkin 方法求解具有弱奇异核Volterra 积分方程，并给出严格的收敛性证明。误差分析结果表明，当方程右端项具有一定正则度时，数值解与精确解之间的L^2-以及L^\infty-误差随多项式阶数的增加呈指数衰减。 . （3）对变分数阶的扩散-对流方程构造了具有高精度的、稳定的、相容的隐式数值离散格式，并给出相应的稳定性和收敛性证明和算例验证。. （4）从理论上严格证明了三角形上谱方法的最优Inf-Sup常数为N^{-1/2}, 同时导出压力误差估计。同时证明了三角元上的二阶算子的特征值为N^8。 . （5）研究分数阶扩散算子，分数阶对流扩散算子以及分数阶积分算子的离散特征值与离散阶数N之间的关系。