空间-时间分数阶变系数对流扩散方程微分阶数的数值反演  

NUMERICAL INVERSION OF THE FRACTIONAL ORDERS IN THE SPACE-TIME FRACTIONAL ADVECTION-DIFFUSION EQUATION WITH VARIABLE COEFFICIENTS

作　　者：贾现正[1] 张大利[1] 李功胜[1] 池光胜[2] 李慧玲[1] 

机构地区：[1]山东理工大学理学院应用数学研究所,山东淄博255049 [2]山东凯文科技职业学院本科教育学院,济南250200

出　　处：《计算数学》2014年第2期113-132,共20页Mathematica Numerica Sinica

基　　金：国家自然科学基金资助项目(11071148);山东省自然科学基金资助项目(ZR2011AQ014)

摘　　要：考虑终值数据条件下一维空间-时间分数阶变系数对流扩散方程中同时确定空间微分阶数与时间微分阶数的反问题.基于对空间-时间分数阶导数的离散,给出求解正问题的一个隐式差分格式,通过对系数矩阵谱半径的估计,证明差分格式的无条件稳定性和收敛性.联合最佳摄动量算法和同伦方法引入同伦正则化算法,应用一种单调下降的Sigmoid型传输函数作为同伦参数,对所提微分阶数反问题进行精确数据与扰动数据情形下的数值反演.结果表明同伦正则化算法对于空间-时问分数阶反常扩散的参数反演问题是有效的.This article deals with an inverse problem of determining the space and time fractional orders in the 1D space-time fractional advection-diffusion equation with variable coefficients by using final observations. An implicit finite difference scheme for solving the forward prob- lem is given, and the unconditional stability and convergence are proved with the help of estimation to the spectrum radius of the coefficient matrix. Furthermore, the homotopy regularization algorithm with a Sigmoid-type function as the homotopy parameter is intro- duced by combining the optimal perturbation algorithm with the homotopy method, and numerical inversions are performed not only with accurate data but also with noisy data. The inversion solutions give good approximations to the exact solutions demonstrate that the proposed algorithm is efficient for the parameters inversion arising from the anomalous diffusion.

关 键 词：空间-时间分数阶扩散 差分格式 稳定性与收敛性 微分阶数反问题 同伦正则化算法 数值反演 

分 类 号：O241.82[理学—计算数学]