分数阶椭圆方程反边值问题的分数Tikhonov正则化方法  

Fractional Tikhonov Regularization Method for an Inverse Boundary Value Problem of the Fractional Elliptic Equation

作　　者：张潇 张宏武 Zhang Xiao;Zhang Hongwu(School of Mathematics and Information Science,North Minzu University,Yinchuan 750021)

机构地区：[1]北方民族大学数学与信息科学学院,银川750021

出　　处：《数学物理学报（A辑）》2024年第4期978-993,共16页Acta Mathematica Scientia

基　　金：宁夏自然科学基金(2022AAC03234);国家自然科学基金(11761004);宁夏高等教育一流学科建设基金(NXYLXK2017B09)。

摘　　要：该文研究了Tricomi-Gellerstedt-Keldysh型分数阶椭圆方程的反边值问题.对于该不适定问题,建立了条件稳定性结果.基于问题的不适定性,构造了分数Tikhonov正则化方法,以恢复解对测量数据的连续依赖性.在正则化参数的先验和后验选取规则下,分别给出并证明了相应的Hölder型收敛性结果.最后,通过两个数值例子验证了分数Tikhonov正则化方法的模拟效果.数值结果表明,该方法能稳定有效地处理文中反问题.In this paper,we study an inverse boundary value problem for fractional elliptic equation of Tricomi-Gellerstedt-Keldysh-type.For this ill-posed problem,a conditional stability result is established.Based on the ill-posedness analysis,a fractional Tikhonov regularization method was constructed to recover the continuous dependence of the solution on the measurement data.Under the a-priori and a-posteriori selection rules for regularization parameter,the corresponding convergence results of Hölder type are derived and proved,respectively.Finally,the simulation effectiveness of the fractional Tikhonov method is verified by two numerical examples.The numerical results show that the method works stably and effectively in dealing with the inverse problem in the text.

关 键 词：反边值问题 分数阶椭圆方程 分数Tikhonov正则化 先验和后验收敛性估计 数值模拟 

分 类 号：O175[理学—数学]