阮周生，張文，王澤文

(1.東華理工大學(xué) 放射性地質(zhì)與勘探技術(shù)國(guó)防重點(diǎn)學(xué)科實(shí)驗(yàn)室，江西 南昌　330013；

2.東華理工大學(xué) 理學(xué)院，江西 南昌　330013)

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帶周期邊界條件時(shí)間分?jǐn)?shù)階擴(kuò)散方程逆時(shí)反問(wèn)題的條件穩(wěn)定性

阮周生1，2，張文1，2，王澤文2

(1.東華理工大學(xué) 放射性地質(zhì)與勘探技術(shù)國(guó)防重點(diǎn)學(xué)科實(shí)驗(yàn)室，江西 南昌330013；

2.東華理工大學(xué) 理學(xué)院，江西 南昌330013)

摘要：基于伴隨思想，利用分離變量方法研究了一類帶周期邊界條件時(shí)間分?jǐn)?shù)階擴(kuò)散方程，首先在弱解意義下推得了正問(wèn)題解的正則性，然后基于對(duì)初值的光滑性假設(shè)推得了逆時(shí)反問(wèn)題條件穩(wěn)定性結(jié)論.

關(guān)鍵詞：時(shí)間分?jǐn)?shù)階擴(kuò)散方程; 逆時(shí)反問(wèn)題；條件穩(wěn)定性

MSC 2010：35K05

第一作者：阮周生(1980)，江西吉安人，東華理工大學(xué)講師，博士，主要從事偏微分方程正反問(wèn)題理論與數(shù)值方法研究.

E-mail:zhshruan@126.com

本文考慮初邊界問(wèn)題

(1)

(2)

Γ(1-α)為伽馬函數(shù).

上面定解問(wèn)題中邊界條件含有周期邊界條件，是非局部的.對(duì)帶有非局部邊界條件定解問(wèn)題的數(shù)值研究始于文獻(xiàn)[1]. 帶非局部邊界的定解問(wèn)題具有特殊的性質(zhì)，因?yàn)閷?duì)應(yīng)的空間微分算子不是自伴算子，故對(duì)應(yīng)的特征函數(shù)系統(tǒng)不是完備的系統(tǒng)，需要通過(guò)伴隨函數(shù)來(lái)對(duì)特征函數(shù)系統(tǒng)完備化.

Ionkin等[2]利用分離變量法證明了二維帶非局部邊界條件熱傳導(dǎo)方程定解問(wèn)題解的存在性、唯一性及解對(duì)初值的穩(wěn)定性.Sergei[3]考慮了一類帶積分非局部邊界條件的初邊值波動(dòng)方程問(wèn)題，通過(guò)對(duì)定解條件的合理假設(shè)，利用Fourier分析方法證明了該問(wèn)題經(jīng)典解的存在性與唯一性. 整數(shù)階帶非局部邊界條件定解問(wèn)題數(shù)值方法研究可以參考文獻(xiàn)[4-7].Benchohra等[8]研究了Caputo分?jǐn)?shù)階微分方程帶非局部邊界條件定解問(wèn)題解的存在性條件.文獻(xiàn)[9-11]分別利用擬逆法、優(yōu)化方法和數(shù)據(jù)正則化方法研究帶Dirichlet邊界條件的時(shí)間分?jǐn)?shù)階擴(kuò)散方程逆時(shí)反問(wèn)題，帶非局部邊界條件分?jǐn)?shù)階微分方程反問(wèn)題的研究可參考文獻(xiàn)[12-13].

上面所述帶非局部邊界條件分?jǐn)?shù)階微分方程正問(wèn)題的研究是基于經(jīng)典解意義下考慮的，帶非局部邊界條件逆時(shí)問(wèn)題的研究?jī)H僅局限于反問(wèn)題解的存在性、唯一性研究，針對(duì)逆時(shí)反問(wèn)題條件穩(wěn)定性問(wèn)題不曾考慮. 本文對(duì)以上的研究結(jié)果做進(jìn)一步研究，首先在弱解的意義下研究解的正則性，而后推導(dǎo)出逆時(shí)反問(wèn)題的條件穩(wěn)定結(jié)果.

1弱解及基本引理

引理1[12-13]Mittag-Leffler函數(shù)滿足

給出弱解定義.

已知特征系統(tǒng)

X″(x)+λX=0,x∈(0,1),X(0)=0,X′(0)=X′(1).

(3)

對(duì)應(yīng)的特征值和特征函數(shù)系為

(4)

(5)

可推得伴隨問(wèn)題對(duì)應(yīng)的特征函數(shù)系統(tǒng)為

(6)

(7)

(8)

(9)

同理，對(duì)伴隨特征系統(tǒng)重排可得

(10)

f0=(f,Y0),f1,k=(f,Y2k-1),f2,k=(f,Y2k),k=1,2,….

(11)

為了證明方便， 給出輔助函數(shù)Hk(t)定義并證明其相關(guān)性質(zhì).

定義3

(12)

引理3

2正問(wèn)題解的正則性及逆時(shí)反問(wèn)題條件穩(wěn)定性

利用分離變量法求問(wèn)題(1)的級(jí)數(shù)形式解，設(shè)問(wèn)題(1)的形式解為

(13)

將式(13)代入問(wèn)題(1)得

(14)

(15)

為了后續(xù)表示方便，形式上記解算子K:φ→u(.,T)， 故有

(16)

給出問(wèn)題(1)解的正則性結(jié)論.

定理1當(dāng)φ(x)∈L2[0,1],可推得問(wèn)題(1)存在唯一的弱解u(x,t)，使得u∈C([0,T];L2[0,1])∩C((0,T];H2[0,1]),且存在常數(shù)C,滿足

形式推導(dǎo)有

其中

經(jīng)計(jì)算得

由Lebesgue控制收斂定理，

接下來(lái)證明弱解的唯一性，由弱解定義，只需要證明當(dāng)φ(x)=0時(shí)，問(wèn)題只有平凡解.

由分?jǐn)?shù)階常微分方程解的存在唯一性結(jié)論[15]得Ti,k(t)=0,i=1,2,k=1,2,…,故u(x,t)≡0,(x,t)∈[0,1]×[0,T]，綜合可知，定理得證.

(17)

當(dāng)φ(x)∈H2[0,1]，可以推得

(18)

定理2(條件穩(wěn)定性)若φ(x)∈H2[0,1],且‖φ(x)‖H2[0,1]≤U0，則有下面條件穩(wěn)定性結(jié)果

(19)

證明：

3小結(jié)

本文基于分離變量法考慮了一類帶周期邊界條件的時(shí)間分?jǐn)?shù)階擴(kuò)散方程正反問(wèn)題，通過(guò)對(duì)初始條件的先驗(yàn)假設(shè)，推得了初值反問(wèn)題的條件穩(wěn)定性結(jié)論，從結(jié)論中看出帶周期邊界條件的時(shí)間分?jǐn)?shù)階逆時(shí)反問(wèn)題是中度不適定的，比整數(shù)階逆時(shí)反問(wèn)題不適定性要稍弱.

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(責(zé)任編輯：王蘭英)

Conditional stability of backward problem for a time fractional

diffusion equation with periodic boundary condition

RUAN Zhousheng1，2，ZHANG Wen1，2，WANG Zewen2

(1. Key Laboratory for Radioactive Geology and Exploration Technology, Fundamental Science for

National Defense，East China University of Technology，Nanchang 330013，China;

2.College of Science， East China University of Technology， Nanchang 330013，China)

Abstract:Based on the adjoint idea， a kind of time fractional diffusion equation with periodic boundary condition by method of separation of variables was considered. Firstly， the regularization result of the solution to the direct problem in the sense of weak solution was derived. Then based on the smoothing assumption for the initial data， the conditional stability for the backward problem was gived.

Key words:time fractional diffusion equation; backward problem; conditional stability

基金項(xiàng)目：國(guó)家高新技術(shù)研究發(fā)展計(jì)劃(2012AA061504)；國(guó)家自然科學(xué)基金資助項(xiàng)目(11561003)；放射性地質(zhì)與勘探技術(shù)國(guó)防重點(diǎn)學(xué)科實(shí)驗(yàn)室資助項(xiàng)目(RGET1513)；江西省高?？萍悸涞赜?jì)劃資助項(xiàng)目(KJLD14051)

收稿日期：2015-01-10

中圖分類號(hào)：O175

文獻(xiàn)標(biāo)志碼:A

文章編號(hào)：1000-1565(2015)06-0561-05

DOI：10.3969/j.issn.1000-1565.2015.06.001