• 论文 •

### 二维空间时间分数阶色散方程的差分方法

1. 北京航空航天大学 数学与系统科学学院, 北京 100191
• 收稿日期:2014-12-24 修回日期:2015-02-13 出版日期:2015-12-20 发布日期:2016-01-04
• 通讯作者: 杨小远(1964-),女,江苏淮安人,教授,xiaoyuanyang@vip.163.com,主要研究方向为分数阶随机偏微分方程. E-mail:xiaoyuanyang@vip.163.com
• 作者简介:张英晗(1986-),男,河北邢台人,博士研究生,zhangyinghan007@126.com
• 基金资助:
国家自然科学基金(61271010);北京市自然科学基金(4152029);北京航空航天大学博士研究生创新基金

### Difference methods for two-dimensional space-time fractional dispersion equation

ZHANG Yinghan, YANG Xiaoyuan

1. School of Mathematics and Systems Science, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
• Received:2014-12-24 Revised:2015-02-13 Online:2015-12-20 Published:2016-01-04

Abstract: The two-dimensional space-time fractional dispersion equation is obtained from the standard two-dimensional dispersion equation by replacing the first order time derivative by the Caputo fractional derivative, and the two second order space derivatives by the Riemann-Liouville fractional derivatives, respectively. Base on the shifted Grünwald finite difference approximation for the two space fractional derivatives, an implicit difference method and a practical alternate direction implicit difference method were proposed to approximate the fractional dispersion equation. The consistency, stability, and convergence of the two implicit difference methods were analyzed. By using mathematical induction method, it was proven that the two implicit difference methods were all unconditionally stable and convergent and the order of convergence were obtained. The convergence speed and computational complexity of the two implicit difference methods were compared. A numerical simulation for a space-time fractional dispersion equation with known exact solution was also presented, and correctness of the theoretical analysis was verified by the numerical results.