两类时间分数阶偏微分方程的谱配置法及收敛性分析

发布时间：2018-06-14 15:33

  本文选题：时间分数阶波动方程 + 时间分数阶Fokker-Planck方程　； 参考：《湘潭大学》2017年硕士论文

【摘要】：分数阶积分微分方程是指含有分数阶积分或分数阶导数的方程,是传统的微积分方程的推广.主要包括分数阶波动方程,分数阶Fokker-Planck方程等.由于解析地求解此类分数阶偏微分方程很困难甚至不可能,所以研究这类方程的数值求解方法是重要的,有价值的.本文主要是利用Jacobi谱配置法求解时间分数阶波动方程和时间分数阶Fokker-Planck方程.先利用Caputo, Riemann-Liouville分数阶导数的定义及其相关性质将原问题转化为求解带弱奇异核的第二类Volterra型积分方程,然后利用适当的线性变换将方程转化为新的Volterra型积分方程,使得方程具有更好的正则性,再分别从时间,空间上采用Jacobi谱配置法,即以Jacobi-Gauss点为配置点,用高斯积分公式逼近积分项,得到全离散格式进行求解.最后从理论上严格证明了,在L∞和L2ω范数意义下,原方程的真解与数值解之间的误差均具有指数收敛性.同时,我们也给出了具体的数值算例,数值结果证实了谱配置法求解这两类方程的有效性以及理论结果的正确性.
[Abstract]:Fractional integrodifferential equation refers to the equation with fractional integral or fractional derivative. It is a generalization of traditional calculus equation. It mainly includes fractional wave equation, fractional Fokker-Planck equation and so on. It is very difficult or even impossible to solve this kind of fractional partial differential equation analytically, so it is important and valuable to study the numerical solution of this kind of equation. In this paper, Jacobi spectrum collocation method is used to solve time fractional wave equation and time fractional Fokker-Planck equation. Firstly, by using the definition of Caputo, Riemann-Liouville fractional derivative and its related properties, the original problem is transformed into the Volterra type integral equation of the second kind with weak singular kernels, and then the equation is transformed into a new Volterra integral equation by proper linear transformation. The method of Jacobi spectrum collocation is used in time and space, that is, the Jacobi-Gauss point is used as collocation point, the integral term is approximated by Gao Si integral formula, and the full discrete scheme is obtained. Finally, it is strictly proved theoretically that the errors between the true solution and the numerical solution of the original equation are exponentially convergent in the sense of L 鈭，

本文编号：2017943