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基于分数阶扩散方程的离散线性代数方程组迭代方法研究

邵新慧, 亢重博   

  1. 东北大学理学院, 沈阳 110819
  • 收稿日期:2020-08-07 出版日期:2022-02-14 发布日期:2022-02-14
  • 基金资助:
    中央高校基本业务费(N2005013)资助

邵新慧, 亢重博. 基于分数阶扩散方程的离散线性代数方程组迭代方法研究[J]. 计算数学, 2022, 44(1): 107-118.

Shao Xinhui, Kang Chongbo. RESEARCH ON ITERATIVE METHOD FOR DISCRETE LINEAR ALGEBRAIC EQUATIONS FROM FRACTIONAL DIFFUSION EQUATIONS[J]. Mathematica Numerica Sinica, 2022, 44(1): 107-118.

RESEARCH ON ITERATIVE METHOD FOR DISCRETE LINEAR ALGEBRAIC EQUATIONS FROM FRACTIONAL DIFFUSION EQUATIONS

Shao Xinhui, Kang Chongbo   

  1. College of Science, Northeastern University, Shenyang 110819, China
  • Received:2020-08-07 Online:2022-02-14 Published:2022-02-14
本文构建一类双参数拟Toeplitz分裂(TQTS)迭代方法求解变系数非定常空间分数阶扩散方程.TQTS迭代法是基于QTS迭代法引入双参技术建立而成,通过选取适当的参数使迭代矩阵谱半径变得更小,从而有效提升收敛的速度.然后对TQTS迭代法进行收敛性分析,获得相应的收敛区域,并对迭代法中涉及的参数进行讨论,获得使迭代矩阵谱半径上界达到最小的最优参数的表达式.最后通过数值仿真实验验证TQTS迭代法的有效性,实验结果表明TQTS迭代法改进效果十分突出,在迭代时间和步数上均有明显的减小.
This paper constructs a class of two-parameter quasi-Toeplitz splitting (TQTS) iteration methods to solve the unsteady space-fractional diffusion equations with variable coefficients. The TQTS iterative method is established based on the QTS iterative method by introducing double-parameter technology, and by selecting appropriate parameters to make the spectral radius of the iterative matrix smaller, thereby effectively increasing the speed of convergence. Then, the convergence of the TQTS iterative method is analyzed to obtain the corresponding convergence area, and the parameters involved in the iterative method are discussed to obtain the expression of the optimal parameters that minimizes the upper bound of the spectral radius of the iterative matrix. Finally, the effectiveness of the TQTS iterative method is verified by numerical simulation experiments. The experimental results show that the improvement effect of the TQTS iterative method is very prominent, and the iteration time and the number of steps are significantly reduced.

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