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一維拋物型方程反問(wèn)題的變分迭代解法

2016-06-04 08:30:33黃得建李艷青

黃得建,李艷青

(瓊州學(xué)院 數(shù)學(xué)系,海南 三亞 572022)

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一維拋物型方程反問(wèn)題的變分迭代解法

黃得建,李艷青*

(瓊州學(xué)院 數(shù)學(xué)系,海南 三亞 572022)

摘要:應(yīng)用變分迭代法研究了第一邊值條件下拋物型偏微分方程反問(wèn)題的數(shù)值解法. 在第一邊值條件的基礎(chǔ)上,利用附加條件確定拋物型偏微分方程中的一個(gè)未知參數(shù)和方程的精確解. 兩個(gè)例子說(shuō)明了這種方法的有效性.

關(guān)鍵詞:變分迭代法;反問(wèn)題;拋物型方程;拉格朗日乘子;未知參數(shù)

一維反拋物型方程在數(shù)學(xué)、物理和工程技術(shù)中有很廣泛的應(yīng)用[1-3],文獻(xiàn)[4-6]給出這類問(wèn)題解的存在性、唯一性以及這種模型在許多問(wèn)題中的應(yīng)用,正是因?yàn)檫@種模型的重要性,近年來(lái),許多數(shù)學(xué)工作者對(duì)拋物型反問(wèn)題的數(shù)值解法做了許多研究[7-10].

本文考慮下面的一維拋物型方程反問(wèn)題[11]

(1)

其中φ(x),μ1(t)和μ2(t)為初始和邊界條件,f(x,t)為已知函數(shù),p(t)為未知系數(shù).如何根據(jù)適當(dāng)?shù)母郊訔l件來(lái)確定未知函數(shù)p(t)在理論研究和實(shí)際應(yīng)用方面都有很重要的意義. 如果u(x,t)表示溫度分布函數(shù),那么問(wèn)題(1)可視為確定未知系數(shù)p(t)的控制問(wèn)題.

為了確定未知系數(shù)p(t),需要附加條件,選取

(2)

(3)

其中s(t),E(t)且|E(t)|>0為已知函數(shù),x*是區(qū)間(0,1)內(nèi)固定的一點(diǎn). 文獻(xiàn)[1,5,11]給出問(wèn)題(1)~(2)或(1)~(3)的解的存在唯一性.

1變分迭代法

變分迭代法是何吉?dú)g提出的[12-13],這種方法被成功的應(yīng)用到Burger’s方程[14]、雙曲型偏微分方程[15]、強(qiáng)非線性方程[16]、分?jǐn)?shù)階非線性方程[17]和廣義KdV方程[18].本文將應(yīng)用變分迭代法求出問(wèn)題(1)~(2)或(1)~(3)的精確解和未知參數(shù)p(t).

應(yīng)用以下兩個(gè)變換[19],設(shè)

(4)

(5)

則問(wèn)題(1)~(2)或(1)~(3)能化為如下形式:

(6)

以及

(7)

(8)

由式(4),(5)可知

(9)

(10)

顯然,原問(wèn)題(1)~(2)或(1)~(3)同輔助問(wèn)題(6)~(7)或(6)~(8)是等價(jià)的. 文獻(xiàn)[4-6]已給出問(wèn)題(6)~(7)或(6)~(8)解的存在性和唯一性.

根據(jù)變分迭代法,對(duì)式(6)中的第一個(gè)方程構(gòu)造t方向上的校正泛函如下形式

(11)

(12)

對(duì)式(11)或(12)兩邊變分,整理可得

利用分步積分公式,可得

所以有

將λ(t)≡-1代入(11)或(12),可得到如下解的迭代公式

(13)

(14)

由式(13)或(14)可得到 (6) 的解w(x,t) ,再由式(9)和(10)可得原問(wèn)題的解及未知參數(shù)p(t).

2實(shí)例

例1考慮問(wèn)題(1)~(2),其條件如下:

附加條件為:

設(shè)

由迭代公式(13)可得:

……

所以,有

由式(9)和(10)可得原問(wèn)題的解和參數(shù)p(t):

例2考慮問(wèn)題(1)~(3),其條件如下:

附加條件為:

設(shè)

由迭代公式(14)可得

……

所以,有

由式(9)和(10)可得原問(wèn)題的解和參數(shù)p(t):

3結(jié)論

本文成功地將變分迭代法運(yùn)用于一維拋物型方程反問(wèn)題的求解,運(yùn)用這種方法不需要分離變量,方便使用數(shù)學(xué)軟件進(jìn)行編程,不占用太多存儲(chǔ)空間,收斂速度也比較快,是一種非常方便實(shí)用的方法.

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The variational iteration method for solving an inverse problem of one-dimensional parabolic equation

HUANG Dejian1, LI Yanqing*

(Department of Mathematics, Qiongzhou University, Sanya 572022, China)

Abstract:In this paper ,the variational iteration method is used to study the exact solution of an inverse parabolic problem. Using the additional given data , this method gets the exact solution and unknown parameter of parabolic partial difference with equation with the first boundary conditions. Two examples show the efficiency of the variational iteration method.

Key words:variational iteration method; inverse problem; parabolic equation; Lagrange multipliers; unknown parameter

中圖分類號(hào):O241

文獻(xiàn)標(biāo)志碼:A

文章編號(hào):1671-9476(2016)02-0034-05

DOI:10.13450/j.cnki.jzknu.2016.02.007

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