计算数学
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计算数学  2019, Vol. 41 Issue (2): 170-190    DOI:
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求解Riesz空间分数阶扩散方程的一种新的数值方法
杨晋平1, 李志强1, 闫玉斌2
1. 吕梁学院 数学系, 吕梁 033001;
2. 切斯特大学 数学系, 英国 CH1 4BJ
A NEW NUMERICAL METHOD FOR SOLVING RIESZ SPACE-FRACTIONAL DIFFUSION EQUATION
Yang Jinping1, Li Zhiqiang1, Yan Yubin2
1. Department of Mathematics, Luliang University, Lvliang 033001, China;
2. Department of Mathematics, University of Chester, Chester CH1 4BJ, UK
 全文: PDF (418 KB)   HTML (1 KB)   输出: BibTeX | EndNote (RIS)      背景资料
摘要 本文利用Diethelm方法构造了一种逼近Riesz空间分数阶导数的O(△x3-α)格式,其中1 < α < 2,△x是空间步长.进一步对一阶时间导数采用Crank-Nicolson方法离散,得到了求解Riesz空间分数阶扩散方程的一种新的有限差分格式,并用矩阵方法证明了稳定性和收敛性,其误差估计为O(△t2+△x3-α),其中△t为时间步长.最后,数值算例验证了差分格式的正确性和有效性.
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关键词Riesz导数   Crank-Nicolson方法   稳定性   收敛性     
Abstract: By using Diethelm's method, we construct an approximate scheme to the Riesz space fractional derivative with order O(△x3-α), where 1 < α < 2 and △x denotes the space step size. Further we discretize the time derivative with the Crank-Nicolson method and obtain a new finite difference method for solving Riesz space fractional diffusion equation. The stability and convergence are proved by the matrix method and the error estimate in the maximum norm is O(△t2 + △x3-α), where △t denotes the time step size. Finally, some numerical examples are given to illustrate their correctness and efficiency.
Key wordsRiesz derivative   Crank-Nicolson method   Stability   Convergence   
收稿日期: 2017-07-28; 出版日期: 2019-05-18
基金资助:

山西省自然科学基金(201801D121010)和吕梁学院校内基金(ZRXN201511)资助项目.

引用本文:   
. 求解Riesz空间分数阶扩散方程的一种新的数值方法[J]. 计算数学, 2019, 41(2): 170-190.
. A NEW NUMERICAL METHOD FOR SOLVING RIESZ SPACE-FRACTIONAL DIFFUSION EQUATION[J]. Mathematica Numerica Sinica, 2019, 41(2): 170-190.
 
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