【摘要】：在很多自然科學(xué)和工程應(yīng)用領(lǐng)域里,人們經(jīng)常會遇到微分方程邊界上Robin系數(shù)的反演問題,這是一類經(jīng)典的具有廣泛應(yīng)用背景的反問題,在現(xiàn)實工程領(lǐng)域(例如金屬腐蝕探測)中經(jīng)常用到.在進(jìn)行金屬表面的腐蝕性探測時,由于客觀條件的限制,某些部分的邊界數(shù)據(jù)是很難直接測量得到的,這時我們需要利用可測量部分的邊界數(shù)據(jù)來檢測未知邊界的腐蝕性情況.以此應(yīng)用問題為背景,本文我們主要討論矩形區(qū)域上的Robin系數(shù)反演問題,具體是Laplace方程邊值問題對應(yīng)的反問題.我們假定在矩形的部分邊界{(x,0)|x∈(0,7π)}上的Cauchy數(shù)據(jù)是可以測量得到,即u(x,0)和uy(x,0)已知,但是另一部分邊界{(x,1)|x∈(0,π)}上的數(shù)據(jù)是無法直接測量的,而邊界{(x,1)|x∈(0,7r)}上的腐蝕性程度由定義于其上的Robin系數(shù)γ(x)來刻畫,我們的任務(wù)就是通過部分邊界{(x,0)|x∈(0,π)}上的可測量的數(shù)據(jù)來反演定義于另一部分邊界上的γ(x).本文的結(jié)構(gòu)如下.第二章介紹了在求解反問題時需要的一些預(yù)備知識,包括Fourier級數(shù)展開、Tikhonov正則化等內(nèi)容.第三章研究了反問題在一定條件下的唯一性和條件收斂性,并提出了正則化的求解方案以及正則化參數(shù)的選取策略.最后兩章給出了求解正問題和反問題的數(shù)值方案和算例,說明了正則化方案的有效性.
[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，

本文編號：2207770