空间分数阶扩散方程的Multiquadric拟插值解法  

Multiquadric Quasi-interpolation for Space Fractional Diffusion Equations

作　　者：王自强[1] 曹俊英[1,2] 

机构地区：[1]贵州民族大学理学院,贵州贵阳550025 [2]厦门大学数学科学学院,福建厦门361005

出　　处：《厦门大学学报（自然科学版）》2015年第3期358-363,共6页Journal of Xiamen University：Natural Science

基　　金：国家重点基础研究发展计划(973计划)(2012(B025904));国家自然科学基金(11426074);贵州省科学技术基金([2014]2098;[2013]2144);贵州省教育厅项目([2013]405)

摘　　要：基于拟插值算子对空间分数阶扩散方程构造了一个新的数值格式.首先在散落点上用三次Multiquadric(MQ)函数的平移构造了一个拟插值算子,分析了此拟插值算子的再生性、保形性和对分数阶导数的收敛性,最后利用上述拟插值算子并结合时间差分格式构造了空间分数阶扩散方程的计算格式.收敛性分析显示:当时间方向用Crank-Nicolson格式时,精度为O(Δt2+h4-α),当时间方向用向后Euler格式时,精度为O(Δt+h4-α),其中Δt为时间步长,h为空间步长.数值结果表明MQ拟插值方法是构造数值格式的一个有效工具.In this article,we apply a new quasi-interpolation operator to solve the space fractional diffusion equation.We first construct a new univariate quasi-interpolation operator based on scattered points by cubic multiquadric functions.We discuss the polynomial reproduction,shape-preserving properties,and convergence for fractional derivative of this quasi-interpolation operator.Based on this quasi-interpolation,a spatial approximation is proposed to discretize partial differential equations.By combining the quasi-interpolation in space and finite difference schemes in time,we construct an efficient method to solve the space fractional diffusion equation.Numerical experiments show that the accuracy of our method is of order O（Δt2＋h4-α）if the Crank-Nicholson scheme is used,and order O（Δt＋h4-α）if Backward Euler is used,whereΔt is the time step size,and his the space mesh size.Numerical results show that MQ quasi-interpolation method is an effective tool for constructing numerical schemes.

关 键 词：Multiquadric拟插值 分数阶扩散方程 保形性 逼近性分析 

分 类 号：O241.82[理学—计算数学]