变系数双侧空间回火分数阶对流-扩散方程的隐式中点法

doi:10.12286/jssx.j2021-0876

计算数学 ›› 2023, Vol. 45 ›› Issue (2): 160-176.DOI: 10.12286/jssx.j2021-0876

变系数双侧空间回火分数阶对流-扩散方程的隐式中点法

殷学芬, 曹学年

湘潭大学数学与计算科学学院, 湘潭 411105

收稿日期:2021-10-13 出版日期:2023-05-14 发布日期:2023-05-13
基金资助:
国家自然科学基金(12071403)资助.

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殷学芬, 曹学年. 变系数双侧空间回火分数阶对流-扩散方程的隐式中点法[J]. 计算数学, 2023, 45(2): 160-176.

Yin Xuefen, Cao Xuenian. THE IMPLICIT MIDPOINT METHOD FOR VARIABLE-COEFFICIENT TWO-SIDED SPACE TEMPERED FRACTIONAL DIFFUSION EQUATION[J]. Mathematica Numerica Sinica, 2023, 45(2): 160-176.

THE IMPLICIT MIDPOINT METHOD FOR VARIABLE-COEFFICIENT TWO-SIDED SPACE TEMPERED FRACTIONAL DIFFUSION EQUATION

Yin Xuefen, Cao Xuenian

School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China

Received:2021-10-13 Online:2023-05-14 Published:2023-05-13

针对带非线性源项的变系数双侧空间回火分数阶对流-扩散方程, 采用隐式中点法离散一阶时间偏导数, 中心差商公式离散对流项, 用二阶回火加权移位差分算子逼近左、右 Riemann-Liouville 空间回火分数阶偏导数, 构造了一类新的数值格式. 证明了数值方法的稳定性和收敛性, 且方法在时间和空间均为二阶收敛. 数值试验验证了数值方法的理论分析结果.

关键词： 回火分数阶对流-扩散方程, 变系数, 隐式中点法, 稳定性, 收敛性

In this paper, an implicit midpoint method is applied to discretize the first order time partial derivative, the convection term discretized by central difference formula and the second order tempered and weighted and shifted Grünwald difference operator is used to approximate the left and right Riemann-Liouville space tempered fractional partial derivative, the numerical scheme is constructed for solving variable-coefficient two-sided space tempered fractional diffusion equation with a nonlinear source term. The energy method is used to prove the stability and convergence of the numerical scheme, and the time and space convergence order of the numerical scheme is second order. Numerical experiments demonstrate the effetiveness of the numerical schemes.

Key words: Tempered fractional convection-diffision equations, Variable-coefficient, The implicit midpoint method, Stability, Convergence

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摘要

变系数双侧空间回火分数阶对流-扩散方程的隐式中点法

THE IMPLICIT MIDPOINT METHOD FOR VARIABLE-COEFFICIENT TWO-SIDED SPACE TEMPERED FRACTIONAL DIFFUSION EQUATION

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