一类时间分布阶和空间Riesz分数阶扩散方程的差分方法

发布时间：2018-05-31 04:02

本文选题：时间分布阶和空间Riesz分数阶扩散方程 + L2-1_σ插值格式　；参考：《湘潭大学》2017年硕士论文

【摘要】：本文考虑一类时间分布阶和空间Riesz分数阶扩散方程初边值问题的数值求解。首先,对分布阶中的积分项用复化中矩形公式进行离散,这样分布阶分数阶微分方程转变成了多项的时间Caputo分数阶扩散方程。然后,将方程中的多项时间Caputo导数用L2- 1σ插值格式进行离散,对空间Riesz分数阶导数采用线性插值和分数阶中心差商逼近,从而构造了数值方法,并证明了方法的稳定性和收敛性。该数值方法的收敛阶为OO(τ2 +h2+ (△α)2),其中τ、h、△α分别表示时间步长、空间步长、分布阶步长。数值实验表明数值方法的理论分析和结果是一致的,也表明数值方法的有效性。
[Abstract]:In this paper, we consider the numerical solution of the initial boundary value problem for a class of diffusion equations of time distribution order and space Riesz fractional order. First, the integral term in the distribution order is discretized by the rectangular formula in the complex form, so that the fractional order differential equation of the distributed order is transformed into a multiterm Caputo fractional diffusion equation. Then, the multi-time Caputo derivatives in the equation are discretized by L _ 2-1 蟽 interpolation scheme, and the linear interpolation and fractional center difference quotient are used to approximate the fractional derivative of space Riesz, and the numerical method is constructed. The stability and convergence of the method are proved. The convergence order of the numerical method is O _ O (蟿 _ 2h _ 2 (伪 ~ + _ 2), where 蟿 _ h, 伪 denotes the time step, the space step and the distribution order step, respectively. Numerical experiments show that the theoretical analysis of the numerical method is consistent with the results, and the effectiveness of the numerical method is also demonstrated.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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