非光滑解分数阶(偏)微分方程的数值方法

Most of numerical algorithms for fractional (partial) differential equations(FDE/FPDE) assume that true solution of equations are sufficiently smooth. The numerical scheme for FDE/FPDE with non-smooth solutions is relatively scare and needs investigation and enrichment.. In this project, we will first probe highly accurate algorithms for boundary value problem of FDEs. Our endeavor will be devoted to investigate the behavior of true solution of the problem near singular points under general assumptions. Thereby,the true solution will be decomposed into the product of a singular factor and a relatively smooth factor. Based upon the decomposition, the original equation can be converted into a new one through a nonlinear transformation. Since the smoothness of solution of the new equation is abundantly improved, spectral method for the new equation yields a highly accurate algorithm for the problem. Secondly, we will investigate a fully discrete scheme for time fractional diffusion equation with either non-smooth initial value or non-smooth force term. Finite difference schemes on nonuniform mesh in time and finite element method in space will be applied. Convergence analysis and numerical stability of all algorithms regarding to different problems will be carried out.

在现有分数阶(偏)微分方程的数值算法中，大部分算法都假设方程的解充分光滑,而对于具有非光滑解的方程，其高精度数值算法的设计和理论都亟待完善和丰富。. 本项目首先将研究分数阶常微分方程边值问题的高精度算法。在一般的假设条件下，该类型问题的解在奇点附近的性态将建立起来，从而原问题的解将分解为奇异因子与相对光滑因子的乘积。基于此分解，通过非线性变换，将原方程转化为新的方程。而新方程解的光滑性有大幅提升。用谱方法逼近新方程的解，从而获得原问题的高精度算法。继而，本项目将研究具有非光滑初值或右端的分数阶扩散方程的时间非均匀网格差分、空间有限元法的全离散格式。探讨算法在对应问题上的稳定性和收敛性等。

该项目主要研究了具有非光滑解的分数阶常微分方程的谱配置法和谱元法以及具有非光滑解的分数阶偏微分方程的差分\有限元法。. 通过将分数阶常微分方程转化为积分方程的形式, 我们得到了源项光滑的条件下,方程的解在奇点附近的具体表达形式，从而设计出合适的谱逼近格式，并证明了最优收敛阶。特别地，我们分别设计了具有良条件数的多项式和非多项式谱配置格式，大大降低了由格式离散得到的代数系统的条件数。. 同样地，对于具有一般源项的分数阶Fokker-Planck方程，我们首先探索了其真解在奇点附近的正则性，并以此对时间差分-空间有限元的全离散格式证明了最优收敛阶。而且,当该方程的分数阶指标趋于整数使得方程趋于整数阶Fokker-Planck方程时，我们的收敛阶与整数阶方程收敛阶相一致。最后，我们考虑了Cole-Cole色散模型的数值格式。通过引入新变量消除分数阶项和极化过程，我们将原模型降维为带弱奇异核的积分偏微分方程。探索方程解的奇异性后，我们为之设计恰当的谱元\有限元格式，证明了格式的最优收敛阶。