含两个时间分数阶导数的二维反常扩散方程求解与微分阶数反演  

The solution to the 2D two-term time-fractional diffusion equation and numerical inversion for the fractional orders

作　　者：王凤丹[1] 孙春龙[1] 李功胜[1] 贾现正[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）》2017年第2期1-7,共7页Journal of Shandong University of Technology:Natural Science Edition

基　　金：国家自然科学基金项目(11371231;11071148)

摘　　要：对于一类含两个时间分数阶导数的二维反常扩散方程,基于对时间分数阶导数在Caputo意义下的离散,得到一个有限差分格式;利用分离变量法与Laplace变换得到该问题的解析解,并将两种方法得到的解进行数值比较.进一步,给定终值时刻数据,应用同伦正则化算法对扩散方程中的两个时间微分阶数进行数值反演,并给出反演算例.数值结果表明随着数据扰动水平的降低,解误差逐步变小,所用的反演算法对微分阶数反问题是有效的.A finite difference scheme is introduced to solve the 2D two-term time-fractional diffusion equation based on Caputors discretization to the time fractional derivatives. Using the method of separation of variables and Laplace transform, analytical solution to the forward problem is obtained, and numerical test is presented to compare the finite difference solution with the analytical solution. Furthermore, the homotopy regularization algorithm is applied to solve the inverse problem of determining the fractional orders given additional data at the final time. Numerical inversions with noisy data are performed, and the inversion solutions error becomes small as the noise level goes to small demonstrating the effectiveness of the proposed algorithm.

关 键 词：时间分数阶导数 二维分数阶扩散 解析解 微分阶数反问题 数值反演 

分 类 号：O175[理学—数学]