一个时间分数阶扩散方程的参数反演问题

Parameters inversion for a time fractional diffusion equation

研究了一维时间分数阶扩散方程中同时确定分数微分阶数与扩散系数的数值反演问题.基于对Caputo意义下时间导数的离散,提出了一个求解正问题的隐式差分格式.应用最佳摄动量正则化算法对所提参数反问题进行了数值模拟,讨论了正则参数、数值微分步长的选取对反演结果的影响.计算结果表明所提的参数反演问题具有数值唯一性.

An inverse problem of determining differential order and diffusion coefficient simultaneously in a one-dimensional time fractional order diffusion equation is investigated. An implicit difference scheme for solving the forward problem is presented based on discretization of Caputo derivative. Furthermore, simulations of numerical inversion for the parameters determination are carried out by applying an optimal perturbation regularization algorithm and impacts of regularization parameter and numerical differential step on the inversion results are discussed. Inversion re sults show that the inverse problem studied in this paper is of numerical uniqueness.