用Tikhonov正则化方法同时反演对流扩散方程的对流速度和源函数  

Simultaneously inverting the convection velocity and source function of convection-diffusion equations by using the Tikhonov regularization method

作　　者：周子融 杨柳[1] 王清艳 ZHOU Zi-Rong;YANG Liu;WANG Qing-Yan(College of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China)

机构地区：[1]兰州交通大学数理学院,兰州730070

出　　处：《四川大学学报（自然科学版）》2024年第1期15-24,共10页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(61663018,11961042);甘肃省自然科学基金(22JR5RA341)。

摘　　要：在给定两个附加观测数据的条件下,本文基于Tikhonov正则化方法研究了对流扩散方程的对流速度和源函数的同时反演问题.鉴于原问题是一个初始值非零的对流扩散方程,本文通过将初始值转化为源项得到了一个组合源项,首先将原问题转化为一个具有齐次条件的对流扩散问题.由于所得问题是不适定的,本文进而利用Tikhonov正则化方法构建了相应的极小化目标泛函,得到了问题最优解的存在性和应满足的必要条件.最后,对终端时刻较小的特殊情形,本文证明了最优解的唯一性和稳定性.In this paper,given additionally two observation data,the inverse problem of simultaneously inverting the convection velocity and source function of the convection diffusion equations is studied.The original problem belongs to a class of convection-diffusion equations with non-zero initial value.First,by transforming the information of the initial value into a source function and then combining it with the source function,we transform the original problem into a convection-diffusion problem with homogeneous conditions.Further,to handle the ill-posedness of the new problem,we construct the corresponding minimization objective functional by using the Tikhonov regularization method,and the existence and necessary conditions for the optimal solution are discussed.Finally,for the special case of small terminal time,the uniqueness and stability of the optimal solution are obtained.

关 键 词：对流扩散方程 反问题 源函数 TIKHONOV正则化方法 

分 类 号：O241.82[理学—计算数学]