反向熱傳導(dǎo)問題的擬可逆正則化方法

劉利平

(甘肅政法學(xué)院 信息工程學(xué)院，甘肅 蘭州　730070)

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反向熱傳導(dǎo)問題的擬可逆正則化方法

劉利平

(甘肅政法學(xué)院 信息工程學(xué)院，甘肅 蘭州730070)

摘要：反向熱傳導(dǎo)問題是一類嚴(yán)重的不適定問題，它給數(shù)值處理帶來了很大的不便。通過擬可逆正則化方法，恢復(fù)了解對數(shù)據(jù)的連續(xù)依賴性，并給出誤差估計(jì)。通過提高先驗(yàn)光滑性假設(shè)得到了t=0時(shí)的收斂性。

關(guān)鍵詞：反向熱傳導(dǎo)問題；不適定問題；擬可逆方法；誤差估計(jì)

0引言

由熱導(dǎo)體在某一時(shí)刻t=T>0時(shí)的溫度測量值來確定物體在0≤t

考慮如下反向熱傳導(dǎo)問題：

(1)

以下用‖·‖和(·)分別表示L2(R)空間中的范數(shù)和內(nèi)積，本文將通過擬可逆正則化方法，恢復(fù)解對數(shù)據(jù)的連續(xù)依賴性。

對方程(1)利用分離變量法得到該問題的精確解為：

(2)

‖w(·,0)‖≤E

(3)

1擬可逆正則化及其誤差估計(jì)

考慮對應(yīng)于問題(1)，正如文獻(xiàn)[9,10]提出的問題：

(4)

同樣利用分離變量法可以求得問題(4)解的為：

(5)

(6)

證明由(5)式，有

(7)

(8)

由‖φ1-φ2‖≤δ，即

(9)

(10)

于是可以得到

‖v1(·,t)-v2(·,t)‖2

(11)

(12)

證明由(3) (5)式，有

(13)

和

‖w(·,t)-v(·,t)‖=

(14)

(15)

注意到(n2-θ2)(T-t)≥0,n=0,1,2,…，于是

利用不等式1-e-r≤r(r≥0)可得

A(n)≤e-n2t(n2-θ2)(T-t)

(16)

(17)

于是A(n)≤e-n2tn4α(T-t)，對于函數(shù)h(n)=e-n2tn4，令h′(n)=0求得其最大值為：

(18)

從而得到這一節(jié)的主要結(jié)果。

(19)

證明設(shè)v(·,t)是由(5)式給出的問題(4)對應(yīng)于精確數(shù)據(jù)φT(x)的解，則

‖w(·,t)-vδ(·,t)‖≤‖w(·,t)-

v(·,t)‖+‖v(·,t)-vδ(·,t)‖。

(20)

結(jié)合定理1，定理2以及式子 (13)，(20)可以得到估計(jì)(19)。

2零點(diǎn)處理

然而注意到當(dāng)t=0時(shí)上面的收斂性將不再成立。因而為了得到t=0時(shí)正則解的收斂性，我們務(wù)必需要更強(qiáng)的先驗(yàn)假設(shè)條件：

(21)

其中p>0為整數(shù)。

(22)

(23)

設(shè)v(·,0)是由式(5)給出的問題(4)對應(yīng)于精確數(shù)據(jù)φT(x)的解。由(2)和(5)式有：

‖w(·,0)-vδ(·,0)‖≤‖w(·,0)-v(·,0)‖+‖v(·,0)-vδ(·,0)‖

(24)

(25)

(26)

當(dāng)n≥1時(shí)，分情況考慮如下2種情況：

(27)

(28)

(29)

(b)當(dāng)p≥2，注意到n2≥1有B(n)≤αT。

(30)

綜合(27)、(28)、(29)以及(30)式，容易得到

(31)

結(jié)合(24)、(25)和(31)式，就得到了(22)。

(32)

3小結(jié)

反向熱傳導(dǎo)問題是一類嚴(yán)重的不適定性問題，已經(jīng)出現(xiàn)了大量的方法用來解決此問題。研究考慮了利用擬可逆正則化方法去解決這個(gè)問題，恢復(fù)了解對初始數(shù)據(jù)的連續(xù)依賴性。

參考文獻(xiàn)：

[1] 劉繼軍.不適定問題的正則化方法及應(yīng)用[M].北京:科學(xué)出版社,2005:108-109.

[2] PAYNE L E.Improperly posed problems in partial differential equations[C/OL]. Philadelphia: SIAM,1975,187-189.[2015-08-25].https://books.glgoo.com/books?hl=zh-CN&lr=&id=JgmRU0DO2soC&oi=fnd&pg=PP2&dq=Improperly+posed+problems+in+partial+differential+equations&ots=Xgdq9TItIM&sig=b7uZKf7WKvoe2HjE_67h6G9udZc&redir_esc=y#v=onepage&q=Improperly%20posed%20problems%20in%20partial%20differential%20equations&f=false.

[3] HADAMARD J.Lectures on Cauchy’s problem in linear partial differential equations[M/OL].Courier Corporation,New Heavens:Yale University Press,2014,96-97.[2015-08-22].https://books.glgoo.com/books?hl=zh-CN&lr=&id=9RNeBAAAQBAJ&oi=fnd&pg=PA3&dq=Lectures+on+Cauchy+problems+in+linear+partial+differential+equations&ots=EpbmEem6iM&sig=5MKE0tKkJi8_dixvp_2cdSZS0PQ&redir_esc=y#v=onepage&q=Lectures%20on%20Cauchy%20problems%20in%20linear%20partial%20differential%20equations&f=false.

[4] MERA N S,ELLIOTT L,INGHAM D B,et al.An iterative boundary element method for solving the one dimensional backward heat conduction problem[J].International Journal of Heat and Mass Transfer,2001,44(10):1973-1946.

[5] MIRANKER W L.A well posed problem for the backward heat equation[J].Proceedings of the American Mathematical Society,1961,12(2):243-247.

[6] LESNIC D,ELLIOTT J,INGHAM D B.An iterative boundary element method for solving the backward heat conduction problem using an elliptic approximation[J]. Inverse Problems in Engineering,1998,6(4):255-279.

[7] LIU C S.Group preserving scheme for backward heat conduction problems[J]. International Journal of Heat and Mass Transfer,2004,47(12):2567-2576.

[8] JOURHMANE M,MERA N S.An iterative algorithm for the backward heat conduction problem based on variable relaxation factors[J].Inverse Problems in Engineering，2002,10(4):293-308.

[9] QIAN Z,FU C L,XIONG X T.Fourth-order modified method for the Cauchy problem for the Laplace equation[J].Journal of Computational and Applied Mathematics,2006,192(2):205-218.

[10]KIRSCH A.An introduction to the mathematical theory of inverse problem[M/OL].Springer Science & Business Media,2011,110-112. [2015-09-03].https://books.glgoo.com/books?hl=zh-CN&lr=&id=RTo9ZFaSSugC&oi=fnd&pg=PR3&dq=An+Introduction+to+the+Mathematical+Theory+of+Inverse+Problem&ots=hOWY_HHIn2&sig=aNlhPmZu0jlCl2nNcy-cfAGUl1k&redir_esc=y#v=onepage&q=An%20Introduction%20to%20the%20Mathematical%20Theory%20of%20Inverse%20Problem&f=false.

文章編號(hào)：1004—5570(2016)01-0073-04

收稿日期：2015-10-06

作者簡介：劉利平(1984-)，女，講師，理學(xué)碩士，研究方向：數(shù)學(xué)物理反問題理論及計(jì)算研究，E-mail: 13494766@qq.com.

中圖分類號(hào)：O551.3；O175

文獻(xiàn)標(biāo)識(shí)碼：A

The perturbation method for a backward heat equation

LIU Liping

(School of Information Engineering, Gansu Institute of Political Science and Law University,Lanzhou, Gansu 730070, China)

Abstract:The backward heat conduction problem (BHCP) is a severely ill-posed problem and it seems impossible to compute numerically on the data does not depend the solution when it continuously. In this paper, the stability of this problem is recovered by a perturbation method and the error estimates is given. The other improved by prior smoothness assumptions.

Key words:backward heat conduction problem; ill-posed problem; the perturbation method; error estimate