时空分数阶扩散偏微分方程的谱方法  

Spectral method for time-space fractional partial differential equation

作　　者：党明杰 蒋利华[1] DANG Mingjie;JIANG Lihua(School of Mathematics and Computing Science,Guilin University of Electronic Technology,Guilin 541004,China;Nanning Research Institute,Guilin University of Electronic Technology,Nanning 530003,China)

机构地区：[1]桂林电子科技大学数学与计算科学学院,广西桂林541004 [2]桂林电子科技大学南宁研究院,南宁530003

出　　处：《桂林电子科技大学学报》2024年第1期98-104,共7页Journal of Guilin University of Electronic Technology

基　　金：广西自然科学基金(2021GXNSFAA220109)。

摘　　要：扩散方程是物理学建模最基本的方程之一。研究时空分数阶扩散偏微分方程的谱方法数值求解,时间方向采用Caputo分数阶导数的L1插值逼近格式,构造了原方程在时间方向上的半离散格式,证明了半离散格式解的存在唯一性和稳定性,并给出了误差分析方面结论的相关证明。在半离散格式的基础上,空间方向采用Legendre谱方法离散得到原方程的全离散格式,进一步证明了此全离散格式的解存在且唯一,而是无条件稳定的,并严格证明了数值解与精确解之间的误差方面的结论。The diffusion equation is one of the fundamental equations in physics.This paper investigates the numerical solution of spectral method for a time-space fractional diffusion equation.In the article,the temporal semi-discrete scheme is constructed using the L1 interpolation approximation scheme of the Caputo fractional order.The existence of uniqueness and stability of the solution in this semi-discrete scheme is demonstrated,and the error analysis of the semi-discrete scheme is rigorously discussed.On the basis of this semi-discrete scheme,the fully discrete scheme is obtained by discretizing it in the spatial direction using the Legendre spectral method.It is further shown that the solution of this fully discrete scheme exists,is unique,and unconditionally stable.Conclusions on the error between the numerical and exact solutions is given and rigorously discussed in the article.

关 键 词：时空分数阶扩散偏微分方程 谱方法 解的存在唯一性 稳定性 误差分析 

分 类 号：O241.82[理学—计算数学]