二维热传导方程源项反问题的一类正则化方法  

A Kind of Regularization Method for an Inverse Source Problem of Two-Dimensional Heat Equation

作　　者：彭建梅[1] 胡彬[1] 王泽文[1] 

机构地区：[1]东华理工大学理学院,南昌330013

出　　处：《江西科学》2017年第1期57-63,98,共8页Jiangxi Science

基　　金：国家自然科学基金(11561003;11661004);江西省高校科技落地计划(No.KJLD14051);江西省自然科学基金(20151BAB201018)

摘　　要：研究了一类二维热传导方程源项反问题,它是一个典型的不适定问题。通过将方程的终值时刻的温度场作Fourier展开,构造出源项反问题的正则化近似问题,从而获得源项的正则化解,并给出了正则化解的稳定性和收敛性结论。随后,给出了先验选取正则化参数时正则化解的收敛率。与之前的正则化方法相比,收敛率有所提高。最后,分别利用先验与后验选取正则化参数进行数值模拟,模拟结果表明本文提出的正则化方法是可行的。This paper considers an inverse source problem of two-dimensional heat equation, which is a typical ill-posed problem. By the Fourier expansion of the final temperature, a kind of regularized approximate problem is constructed for obtaining a regularization solution of the source. Then the sta- bility and the convergence for the regularization solution are presented. Furthermore, the converge rate of the regularized solution is analyzed in the prior selection of regularization parameter. Com- pared with the previous regularization methods, the convergence rate is improved. Finally, numerical simulations are done by the choice of regularization parameters with the prior and posterior strategies respectively, and numerical results show that the proposed regularization method is very stable.

关 键 词：热传导方程 源项反问题 正则化方法 不适定问题 正则化参数 

分 类 号：O175.26[理学—数学]