• 论文 • 上一篇    下一篇

带非线性源项的双侧空间分数阶扩散方程的隐式中点方法

胡冬冬, 曹学年, 蒋慧灵   

  1. 湘潭大学数学与计算科学学院, 湘潭 411105
  • 收稿日期:2017-12-14 出版日期:2019-09-15 发布日期:2019-08-21

胡冬冬, 曹学年, 蒋慧灵. 带非线性源项的双侧空间分数阶扩散方程的隐式中点方法[J]. 计算数学, 2019, 41(3): 295-307.

Hu Dongdong, Cao Xuenian, Jiang Huiling. THE IMPLICIT MIDPOINT METHOD FOR TWO-SIDE SPACE FRACTIONAL DIFFUSION EQUATION WITH A NONLINEAR SOURCE TERM[J]. Mathematica Numerica Sinica, 2019, 41(3): 295-307.

THE IMPLICIT MIDPOINT METHOD FOR TWO-SIDE SPACE FRACTIONAL DIFFUSION EQUATION WITH A NONLINEAR SOURCE TERM

Hu Dongdong, Cao Xuenian, Jiang Huiling   

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
  • Received:2017-12-14 Online:2019-09-15 Published:2019-08-21
本文用隐式中点方法离散一阶时间偏导数,并用拟紧差分算子逼近Riemann-Liouville空间分数阶偏导数,构造了求解带非线性源项的空间分数阶扩散方程的数值格式.给出了数值方法的稳定性和收敛性分析.数值试验表明数值方法是有效的.
In this paper, the numerical scheme was constructed for solving the space fractional diffusion equation with a nonlinear source term where the implicit midpoint method was applied to discretize the first order time partial derivative, and the quasi-compact difference operator was utilized to approximate Riemann-Liouville space fractional partial derivative. Stability and convergence analysis of this numerical method were given. Numerical experiments show that the numerical method is effective.

MR(2010)主题分类: 

()
[1] Hao Z, Sun Z, Cao W. A fourth-order approximation of fractional derivatives with its applications[J]. Journal of Computational Physics, 2015, 281:787-805.

[2] Zhou H, Tian W, Deng W. Quasi-compact finite difference schemes for space fractional diffusion equations[J]. Journal of Scientific Computing, 2013, 56(1):45-66.

[3] Yu Y, Deng W, Wu Y. High-order quasi-compact difference schemes for fractional diffusion equations[J]. Communications in Mathematical Sciences, 2017, 15(5):1183-1209.

[4] Cao X, Cao X, Wen L. The implicit midpoint method for the modified anomalous sub-diffusion equation with a nonlinear source term[J]. Journal of Computational and Applied Mathematics, 2017, 318:199-210.

[5] Choi H, Chung S, Lee Y. Numerical solutions for space-fractional dispersion equations with nonlinear source terms[J]. Bulletin of the Korean Mathematical Society, 2010, 47(6):1225-1234.

[6] Moroney T, Yang Q. Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast poisson preconditioners[J]. Journal of Computational Physics, 2013, 246(246):304-317.

[7] Liu F, Chen S, Turner I, Burrage K, Anh V. Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term[J]. Central European Journal of Physics, 2013, 11(10):1221-1232.

[8] Chen S, Liu F, Jiang. X, Turner I, Anh V. A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients[J]. Applied Mathematics and Computation, 2015, 257:591-601.

[9] Bu W, Tang Y, Wu Y, Yang J. Crank-Nicolson ADI Galerkin finite element method for twodimensional fractional FitzHugh-Nagumo monodomain model[J]. Applied Mathematics and Computation, 2015, 257:355-364.

[10] Choi Y, Chung S. Finite element solutions for the space-fractional diffusion equation with a nonlinear source term[J]. Abstract and Applied Analysis, 2012, 2012:183-201.

[11] Li Y, Wang D. Improved efficient difference method for the modified anomalous sub-diffusion equation with a nonlinear source term[J]. International Journal of Computer Mathematics, 2017, 94(4):821-840.

[12] Chan R, Jin X. An Introduction to Iterative Toeplitz Solvers[M]. Philadelphia:SIAM, 2007.

[13] Kilbas A, Srivastava H, Trujillo J. Theory and Applications of Fractional Differential Equations[M]. Elsevier Science Limited, 2006.

[14] Meerschaert M, Tadjeran C. Finite difference approximations for fractional advection dispersion flow equations[J]. Journal of Computational and Applied Mathematics, 2004, 172(1):65-77.

[15] Liu F, Zhang H. Numerical simulation of the Riesz fractional diffusion equation with a nonlinear source term[J], Journal of Applied Mathematics and Informatics, 2008, 26(1-2):1-14.

[16] Pang H, Sun H. Fourth order finite difference schemes for time space fractional sub-diffusion equations[J]. Computers & Mathematics with Applications, 2016, 71(6):1287-1302.

[17] Liu Y, Du Y, Li H, He S, Gao W. Finite difference/finite element method for a nonlinear timefractional fourth-order reaction-diffusion problem[J]. Computers & Mathematics with Applications, 2015, 70(4):573-591.

[18] Liu Y, Du Y, Li H, Wang J. A two-grid finite element approximation for a nonlinear time-fractional Cable equation[J]. Nonlinear Dynamics, 2016, 85(4):2535-2548.

[19] Wang D, Xiao A, Yang W. Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative[J]. Journal of Computational Physics, 2013, 242(242):670-681.
[1] 余妍妍, 代新杰, 肖爱国. 非自治刚性随机微分方程正则EM分裂方法的收敛性和稳定性[J]. 计算数学, 2022, 44(1): 19-33.
[2] 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.
[3] 古振东. 非线性弱奇性Volterra积分方程的谱配置法[J]. 计算数学, 2021, 43(4): 426-443.
[4] 高兴华, 李宏, 刘洋. 分布阶扩散—波动方程的有限元解的误差估计[J]. 计算数学, 2021, 43(4): 493-505.
[5] 包学忠, 胡琳. 随机变延迟微分方程平衡方法的均方收敛性与稳定性[J]. 计算数学, 2021, 43(3): 301-321.
[6] 李旭, 李明翔. 连续Sylvester方程的广义正定和反Hermitian分裂迭代法及其超松弛加速[J]. 计算数学, 2021, 43(3): 354-366.
[7] 张丽丽, 任志茹. 改进的分块模方法求解对角占优线性互补问题[J]. 计算数学, 2021, 43(3): 401-412.
[8] 邱泽山, 曹学年. 带漂移的单侧正规化回火分数阶扩散方程的Crank-Nicolson拟紧格式[J]. 计算数学, 2021, 43(2): 210-226.
[9] 袁光伟. 非正交网格上满足极值原理的扩散格式[J]. 计算数学, 2021, 43(1): 1-16.
[10] 朱梦姣, 王文强. 非线性随机分数阶微分方程Euler方法的弱收敛性[J]. 计算数学, 2021, 43(1): 87-109.
[11] 李天怡, 陈芳. 求解一类分块二阶线性方程组的QHSS迭代方法[J]. 计算数学, 2021, 43(1): 110-117.
[12] 尹江华, 简金宝, 江羡珍. 凸约束非光滑方程组一个新的谱梯度投影算法[J]. 计算数学, 2020, 42(4): 457-471.
[13] 古振东, 孙丽英. 非线性第二类Volterra积分方程的Chebyshev谱配置法[J]. 计算数学, 2020, 42(4): 445-456.
[14] 尚在久, 宋丽娜. 关于辛算法稳定性的若干注记[J]. 计算数学, 2020, 42(4): 405-418.
[15] 洪庆国, 刘春梅, 许进超. 一种抽象的稳定化方法及在非线性不可压缩弹性问题上的应用[J]. 计算数学, 2020, 42(3): 298-309.
阅读次数
全文


摘要