廖秋明,李丹霞
(肇慶學院 數(shù)學與統(tǒng)計學院,廣東 肇慶 526061)
具時間依賴邊界條件的熱傳導方程的近似解法研究
廖秋明,李丹霞
(肇慶學院 數(shù)學與統(tǒng)計學院,廣東 肇慶 526061)
研究了求解具時間依賴邊界條件的熱傳導方程的近似解。首先,對溫度邊界條件為時間的冪函數(shù)的情況,采用標準的多項式溫度近似函數(shù),結(jié)合熱平衡積分法及改進的熱平衡積分法,求得時間的次數(shù)和溫度近似函數(shù)的指數(shù)之間的關(guān)系,從而確定溫度近似解函數(shù);然后,對復雜的時間依賴的邊界條件,應用線性微分方程疊加原理,構(gòu)建近似解表達式。實驗結(jié)果表明,這種方法既簡便又具有良好的計算精度,能較好地模擬傳熱過程。
時間依賴邊界條件;熱傳導方程;熱平衡積分解法;疊加原理;近似解
熱平衡積分法(heat balance integral method,HBIM),是1958年由T.R.Goodman[1]提出的求解導熱問題的有效方法。由于其計算簡便、速度快,且計算精度較高,從而得到了廣泛應用。改進的熱平衡積分法(refined integral method,RIM)是N.Sadoun等[2]提出的一種改進方法。即通過對導熱微分方程關(guān)于空間變量積分2次,并結(jié)合傳統(tǒng)的熱平衡積分方程,得到一個新的積分方程,所求得的近似解具有更高的精確性。但是,用熱平衡積分法求解具時間依賴邊界條件的熱傳導問題時,往往難以得到計算精度良好的近似解。為此,人們提出了各種改進方法及應用[3-9]。如D.Langford[3]采用最小誤差測量函數(shù)來確定溫度近似函數(shù),顯示出更高的精確性。T.G.Myers[4-5]利用D.Langford 提出的函數(shù),求得了非整數(shù)形式的HBIM 和RIM 最優(yōu)多項式次數(shù),并將之應用到標準的導熱問題及相變導熱問題中。但標準的多項式溫度近似解,只能解決溫度邊界條件隨著時間遞增的情況。S.L.Mitchell等[6]提出了用對數(shù)表達式表示近似解以及時間依賴的指數(shù),以解決邊界溫度隨著時間遞減情況下的導熱近似。Ling Feng等[7]提出了一種優(yōu)化多項式溫度近似解指數(shù)的積分方法。J.Hristov[8-9]改進熱平衡積分法,并應用于非線性傳熱問題。
本文采用標準的多項式溫度近似函數(shù),當溫度邊界條件為時間的冪函數(shù)時,結(jié)合HBIM 和RIM獲得時間和溫度近似函數(shù)的指數(shù)之間的關(guān)系,從而確定溫度近似解表達式,并應用線性微分方程疊加原理,構(gòu)建近似解,以解決復雜的時間依賴邊界條件,以得到簡便又計算精度良好的近似解。
一個半無限大的平板,當時間t>0時,在邊界(x=0)處的溫度設為(t) ;在有限的時間內(nèi),邊界的熱擾動將滲透到平板有限厚度范圍內(nèi),即滲透深度記作(t)。半無限大的平板的熱傳導問題無量綱形式的數(shù)學模型為
式中:T表示溫度;
x表示空間坐標;
利用余誤差函數(shù),可得到精確解:
下面結(jié)合熱平衡積分法及改進的熱平衡積分法,求時間t的次數(shù)k和n之間的函數(shù)關(guān)系。
圖1 當(t)=1-t時,不同t值下的精確解曲線Fig.1 Exact solution curves under the condition of(t)=1-t with t the variable
對問題(1)關(guān)于空間變量x進行積分,得到熱平衡積分
下面以本文方法得到的近似解(12)和精確解以及HBIM,RIM的近似解進行實驗對比,結(jié)果如圖2和圖3所示。其中HBIM,RIM的近似解同樣采用標準的多項式,其指數(shù)n依賴時間變化[6],這里不失一般性取t=0.8。
圖2 當(t)=1-t , t = 0.8時,本文方法,HBIM,RIM近似解與精確解曲線的比較Fig.2 Curves for the approximate solution of HBIM, RIM and the presented approach as contrast to the exact solution curve when(t)=1-t with t = 0.8
圖3 當(t)=1-t,t=0.8時,本文方法,HBIM,RIM近似解與精確解之間的絕對誤差比較Fig.3 Curves for the approximate solution of HBIM,RIM and the presented approach as contrast to the absolute error curve when(t)=1-t with t = 0.8
從圖2和圖3可以看出,采用本文方法得到的近似解和精確解很接近,絕對誤差小于0.03,計算精度比HBIM和RIM的近似解更高。
在溫度邊界隨時間變化的條件下,采用標準的多項式溫度近似函數(shù),首先解決當溫度邊界條件為時間的基本冪函數(shù)時的近似解,結(jié)合HBIM和RIM,求得時間的次數(shù)和近似解函數(shù)的指數(shù)之間的關(guān)系,從而確定熱平衡積分溫度近似函數(shù);然后應用線性微分方程疊加原理,構(gòu)建近似解。該方法簡便,得到的近似解精度高。用積分方法求出來的溫度近似解不是唯一的,只有計算精度良好的近似解才具實用價值。本文的近似解法能較好地模擬當邊界溫度隨著時間遞減情況下傳熱過程中出現(xiàn)的拐點現(xiàn)象。在本文研究的基礎(chǔ)上,可以解決更復雜的傳熱問題,如相變傳熱問題、高維傳熱問題等。
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(責任編輯 :鄧光輝)
On an Approximate Solution of the Heat Equation Under Time-Dependent Boundary Conditions
LIAO Qiuming, LI Danxia
(School of Mathematics and Statistics,Zhaoqing University,Zhaoqing Guangdong 526061,China)
In order to find an approximate solution of the heat equation under time-dependent boundary conditions,a research has been conducted as shown in the following steps.Firstly, with the power function of time being the temperature boundary condition, combined with the heat balance and refined integral methods, the standard polynomial approximation solution to the temperature equation has been adopted so as to obtain the relationship between the power of the approximate solution and the power of the temperature boundary function, thus working out the approximate solution.Secondly, an approximate solution expression will be constructed based on the application of the principle of superposition of linear differential equation, with the complex time dependent boundary condition taken into consideration.The experimental results show that this simple method, characterized with a better calculation accuracy, proves to be very effective in simulating the heat transfer process.
time-dependent boundary condition ;equation of heat conduction ;heat balance integral method ;principle of superposition ;approximate solution
O175.29
A
1673-9833(2016)04-0078-04
10.3969/j.issn.1673-9833.2016.04.015
2016-06-08
廣東省自然科學基金資助項目(2015A030313704)
廖秋明(1972-),男,廣東肇慶人,肇慶學院副教授,主要研究方向為偏微分方程理論及應用,E-mail:lqmzqu@163.com