含三个时间分数阶导数的反常扩散方程求解与微分阶数反演  

The solution to three time-fractional anomalous diffusion equation and numerical inversion of the fractional orders

作　　者：孙春龙[1] 李功胜[1] 贾现正[1] 杜殿虎[1] 

机构地区：[1]山东理工大学理学院,山东淄博255049

出　　处：《山东理工大学学报（自然科学版）》2015年第3期1-7,共7页Journal of Shandong University of Technology:Natural Science Edition

基　　金：国家自然科学基金资助项目(11071148;11371231);山东省自然科学基金资助项目(ZR2011AQ014)

摘　　要：对于一类带有三个时间分数阶的一维反常扩散问题,基于Caputo意义下时间分数阶导数的离散,给出了一个有限差分求解格式,并利用分离变量法及Laplace变换得到该问题的解析解.进一步应用同伦正则化算法,根据内点处的浓度观测数据对确定微分阶数的反问题进行数值反演,并讨论时间-空间步长及数据扰动等因素对反演算法的影响.A finite difference scheme is introduced to solve the 1-D three-term time-fractional a- nomalous diffusion equation based on Caputo＇s discretization to the time fractional derivatives. U- sing the method of separation of variables and Laplace transform, the analytical solution of the forward problem is obtained. Furthermore, the homotopy regularization algorithm is applied to determine the three time-fractional orders by the additional measurements at an interior point in the domain. Numerical inversions are performed to demonstrate effectiveness of the proposed al- gorithm, and influences of the time-space mesh grids and the data noises on the inversion algorithm are discussed.

关 键 词：时间分数阶导数 含三个时间分数阶的扩散 反问题 同伦正则化算法 数值反演 

分 类 号：O175[理学—数学]