重构具有Wentzell边界条件的抛物型方程的热源  

Reconstruction of Heat Sources for Parabolic Equations With Wentzell Boundary Conditions

作　　者：伊海鸿 杨柳[1] 田瑜 YI Haihong;YANG Liu;TIAN Yu(School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,P.R.China)

机构地区：[1]兰州交通大学数理学院,兰州730070

出　　处：《应用数学和力学》2025年第4期505-518,共14页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(61663018,11961042);甘肃省自然科学基金(25JRRA163,25JRRA952)。

摘　　要：该文主要研究以Wentzell边界条件为背景,利用终端温度测量值在抛物热传导方程中重构与空间相关源项的反问题,这一研究在热传导工程问题中经常出现.该研究的难点是对Wentzell边界条件的处理,通过应用散度定理使得边界条件可以与抛物方程相结合,而在不同的边界条件下,极值原理的证明也有所区别.由于原问题的不适定性,基于最优控制理论的框架,对原问题进行优化,建立了正则化解的存在性和所满足的必要条件,并且在极值原理成立的情形下,证明了正则化解的唯一性和稳定性.The inverse problem of reconstructing spatially related source terms in parabolic heat conduction equations was studied under the Wentzell boundary conditions and with the terminal temperature measurements.This study has important applications in determining the source terms in heat conduction engineering problems,and the difficulty lies in the handling of the Wentzell boundary conditions.Based on the divergence theorem,the boundary conditions were combined with parabolic equations.The extremum principle was proved differently under various boundary conditions.Due to the ill-posedness of the original problem,based on the framework of the optimal control theory,the original problem was optimized,and the existence and necessary conditions for the regularization solution were established.Furthermore,under the validness of the extremum principle,the uniqueness and stability of the regularization solution were proved.

关 键 词：反源问题 Wentzell边界条件 最优控制 散度定理 

分 类 号：O175.26[理学—数学]