拉普拉斯方程矩形域上Robin系数的反演
发布时间：2018-08-27 16:07
【摘要】：在很多自然科学和工程应用领域里,人们经常会遇到微分方程边界上Robin系数的反演问题,这是一类经典的具有广泛应用背景的反问题,在现实工程领域(例如金属腐蚀探测)中经常用到.在进行金属表面的腐蚀性探测时,由于客观条件的限制,某些部分的边界数据是很难直接测量得到的,这时我们需要利用可测量部分的边界数据来检测未知边界的腐蚀性情况.以此应用问题为背景,本文我们主要讨论矩形区域上的Robin系数反演问题,具体是Laplace方程边值问题对应的反问题.我们假定在矩形的部分边界{(x,0)|x∈(0,7π)}上的Cauchy数据是可以测量得到,即u(x,0)和uy(x,0)已知,但是另一部分边界{(x,1)|x∈(0,π)}上的数据是无法直接测量的,而边界{(x,1)|x∈(0,7r)}上的腐蚀性程度由定义于其上的Robin系数γ(x)来刻画,我们的任务就是通过部分边界{(x,0)|x∈(0,π)}上的可测量的数据来反演定义于另一部分边界上的γ(x).本文的结构如下.第二章介绍了在求解反问题时需要的一些预备知识,包括Fourier级数展开、Tikhonov正则化等内容.第三章研究了反问题在一定条件下的唯一性和条件收敛性,并提出了正则化的求解方案以及正则化参数的选取策略.最后两章给出了求解正问题和反问题的数值方案和算例,说明了正则化方案的有效性.
[Abstract]:In many fields of natural science and engineering applications, people often encounter the inverse problem of Robin coefficients on the boundary of differential equations, which is a classical inverse problem with extensive application background. It is often used in practical engineering fields such as metal corrosion detection. The boundary data of some parts are difficult to be measured directly because of the limitation of objective conditions. In this case, we need to use the boundary data of measurable parts to detect the corrosion of unknown boundaries. In this paper, we mainly discuss the inverse problem of Robin coefficient in rectangular region, which is the inverse problem of the boundary value problem of Laplace equation. We assume that the Cauchy data on the partial boundary {(x0) x 鈭，
本文编号：2207770
[Abstract]:In many fields of natural science and engineering applications, people often encounter the inverse problem of Robin coefficients on the boundary of differential equations, which is a classical inverse problem with extensive application background. It is often used in practical engineering fields such as metal corrosion detection. The boundary data of some parts are difficult to be measured directly because of the limitation of objective conditions. In this case, we need to use the boundary data of measurable parts to detect the corrosion of unknown boundaries. In this paper, we mainly discuss the inverse problem of Robin coefficient in rectangular region, which is the inverse problem of the boundary value problem of Laplace equation. We assume that the Cauchy data on the partial boundary {(x0) x 鈭，
本文编号：2207770
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