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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

邱泽山, 曹学年   

  1. 湘潭大学数学与计算科学学院, 湘潭 411105
  • 收稿日期:2019-07-25 出版日期:2021-04-15 发布日期:2021-05-13
  • 通讯作者: 曹学年,cxn@xtu.edu.cn.
  • 基金资助:
    国家自然科学基金(12071403)资助.

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.

Qiu Zeshan, Cao Xuenian. CRANK-NICOLSON QUASI-COMPACT SCHEMES FOR ONE-SIDED NORMALIZED TEMPERED FRACTIONAL DIFFUSION EQUATIONS WITH DRIFT[J]. Mathematica Numerica Sinica, 2021, 43(2): 210-226.

CRANK-NICOLSON QUASI-COMPACT SCHEMES FOR ONE-SIDED NORMALIZED TEMPERED FRACTIONAL DIFFUSION EQUATIONS WITH DRIFT

Qiu Zeshan, Cao Xuenian   

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
  • Received:2019-07-25 Online:2021-04-15 Published:2021-05-13
基于已有的针对单侧正规化回火分数阶扩散方程的三阶拟紧算法,将该算法的思想应用于带漂移的单侧正规化回火分数阶扩散方程的数值模拟,并结合Crank-Nicolson方法导出数值格式.证明了数值格式的稳定性与收敛性,且数值格式的时间收敛阶和空间收敛阶分别是二阶和三阶.通过数值试验验证了数值格式的有效性和理论结果.
Based on the existed third-order quasi-compact algorithm for one-sided normalized tempered fractional diffusion equations, the idea of the algorithm is applied to the numerical simulation of the one-sided normalized tempered fractional diffusion equations with drift, and combined the Crank-Nicolson method to derive the numerical schemes. The stability and convergence of the numerical schemes are proved, and the time and space convergence orders of the numerical schemes are second and third order, respectively. The effectiveness of the numerical schemes and the theoretical results are verified by numerical experiments.

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[1] Cartea Á, del-Castillo-Negrete D. Fluid limit of the continuous-time random walk with general Lévy jump distribution functions[J]. Physical Review E, 2007, 76(4):041105.
[2] Meerschaert M M, Zhang Y, Baeumer B. Tempered anomalous diffusion in heterogeneous systems[J]. Geophysical Research Letters, 2008, 35(17).
[3] Baeumer B, Meerschaert M M. Tempered stable Lévy motion and transient super-diffusion[J]. Journal of Computational and Applied Mathematics, 2010, 233(10):2438-2448.
[4] Li C, Deng W H. High order schemes for the tempered fractional diffusion equations[J]. Advances in Computational Mathematics, 2016, 42:543-572.
[5] Qu W, Liang Y. Stability and convergence of the Crank-Nicolson scheme for a class of variablecoefficient tempered fractional diffusion equations[J]. Advances in Difference Equations, 2017, 108:1-11.
[6] Çelik C, Duman M. Finite element method for a symmetric tempered fractional diffusion equation[J]. Applied Numerical Mathematics, 2017, 120:270-286.
[7] Yu Y Y, Deng W H, Wu Y J. High-order quasi-compact difference schemes for fractional diffusion equations[J]. Communications in Mathematical Sciences, 2017, 15:1183-1209.
[8] Yu Y Y, Deng W H, Wu Y J, Wu J. Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations[J]. Applied Numerical Mathematics, 2017, 112:126-145.
[9] Zhang Y X, Li Q, Ding H F. High-order numerical approximation formulas for Riemann-Liouville (Riesz) tempered fractional derivatives:construction and application (I)[J]. Applied Mathematics and Computation, 2018, 329:432-443.
[10] Hu D D, Cao X N. The implicit midpoint method for Riesz tempered fractional diffusion equation with a nonlinear source term[J]. Advances in Difference Equations, 2019, 1:66.
[11] Hu D D, Cao X N. A fourth-order compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equation[J]. International Journal of Computer Mathematics, 2019:1-21.
[12] Qiu Z S, Cao X N. Second-order numerical methods for the tempered fractional diffusion equations[J]. Advances in Difference Equations, 2019, 2019(1):485.
[13] Zhang H, Liu F, Turner I, Chen S. The numerical simulation of the tempered fractional BlackScholes equation for European double barrier option[J]. Applied Mathematics Modelling, 2016, 40:5819-5834.
[14] Sabzikar F, Meerschaert M M, Chen J H. Tempered fractional calculus[J]. Journal of Computational Physics, 2015, 293:14-28.
[15] Ding H F, Li C P. A High-Order Algorithm for Time-Caputo-Tempered Partial Differential Equation with Riesz Derivatives in Two Spatial Dimensions[J]. Journal of Scientific Computing, 2019:1-29.
[16] Deng W H, Zhang Z J. Numerical schemes of the time tempered fractional Feynman-Kac equation[J]. Computers and Mathematics with Applications, 2017, 73:1063-1076.
[17] Dehghan M, Abbaszadeh M. A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation[J]. Computers and Mathematics with Applications, 2018, 75:2903-2914.
[18] Moghaddam B P, Machado J A T, Babaei A. A computationally efficient method for tempered fractional differential equations with application[J]. Computational and Applied Mathematics, 2018, 37:3657-3671.
[19] Hanert E, Piret C. A Chebyshev pseudospectral method to solve the space-time tempered fractional diffusion equation[J]. SIAM Journal on Scientific Computing, 2014, 36:A1797-A1812.
[20] Sun X R, Zhao F Q, Chen S P. Numerical algorithms for the time-space tempered fractional Fokker-Planck equation[J]. Advances in Differential Equations, 2017, 2017:259.
[21] Chan R H F, Jin X Q. An introduction to iterative Toeplitz solvers[M]. SIAM, 2007.
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