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      有關常系數(shù)非齊次三階偏微分方程在工程中的解法及推廣

      2014-12-22 14:43:41韓志偉
      科技與創(chuàng)新 2014年23期

      韓志偉

      摘? 要:在工程應用中,有關偏微分方程解的具體形式往往能夠使復雜問題簡單化。在許多工程的實際問題中,雖然不同偏微分方程代表的實際意義各不相同,但卻具有完全相同形式的數(shù)學規(guī)律,因此,研究一般意義上的方程有助于解決實際應用問題。主要研究了一般形式的常系數(shù)非齊次三階偏微分方程的解,并探討了常系數(shù)非齊次N階偏微分方程的特殊情況,得到了不同條件下解的形式。

      關鍵詞:三階偏微分方程;常系數(shù);非齊次;余函數(shù)

      中圖分類號:O175.14?????????? 文獻標識碼:A?? ????????????文章編號:2095-6835(2014)23-0104-02

      近年來,有關偏微分方程解一直是熱點研究問題。在實際工程應用中,對一般意義上的偏微分方程解的研究可以讓原本復雜的工程計算變得簡單。因此,本文討論了常系數(shù)非齊次三階偏微分方程的一般解,進而研究了N階常系數(shù)非齊次偏微分方程,從而得到了具體解的形式。

      1? 常系數(shù)非齊次三階偏微分方程的一般解

      三階偏微分方程的一般式為:

      ???????????????????????? (1)

      方程式(1)可簡記為:

      fD,D′)z=φx,t).??????????????????????? (2)

      方程(1)的解由兩部分構成,通解zn和余函數(shù)zp,可記為式(3):

      z=zn+zp.?????????????????????????????????????? (3)

      當方程(2)的右端φxt)=0時,通過解其對應的齊次方程可得到通解:

      fDD′)z=0.?????????????????????????????????? (4)

      不妨假定齊次方程的通解形式為zn=cehx+k,其中,ch,k為待定常數(shù),代入方程(4)中可得:

      cfh,kehx+kt=0.???????????????????????????? (5)

      其中,對應的特征方程為:

      fh,k)=0.?????????????????????????????????? (6)

      因此,齊次方程的通解zn必定具備該形式cehx+k.

      在特征方程(6)中,如果能解出常數(shù)k的值,那么,(4)式中的D′必為r階的(r≥2),通解的表達式如下所示:

      .??????? (7)

      同理,在方程(6)中,如果能解出常數(shù)h的值,那么(4)式中的D′必為r階的(r≥2),通解的表達式如下所示:

      .?????? (8)

      根據方程(2),考慮其余函數(shù)的形式為:

      .?????????????????????????? (9)

      或者

      .??????????? (10)

      根據式(6)可得:

      a1h3+a2h2k+a3hk2+a4k3+b1h2+b2hk+b3k2+c1h+c2k=0.??? (11)

      其變形式為:

      a1h3+(a2k+b1)h2+(a3k2+b2k+c1)h+(a4k3+b3k2+c2k)=0.????????????????? ????????????????????? (12)

      令:A=(a2k+b1)-3a1(a3k3+b2k2+c1k);

      B=(a2k+b1)(a3k2+b2k+c1)-9a1(a4k3+b3k2+c2k);

      C=(a3k2+b2k+c1)2-3(a2k+b1)(a4k3+b3k2+c2k).

      根據盛金定理可知,記Δ=B2-4AC. 由此可以得到以下結論。

      情形1:當式(12)中A=B=0時,可解得:

      .????????????????????????? (13)

      情形2:當Δ>0時,則有:

      ,

      .????????????????????????????? (14)

      其中,,i2=-1.

      情形3:當Δ=0時,此時的形式較為簡單:

      .????????? ??(15)

      其中,A≠0).

      情形4:當Δ<0時,解得:

      ,

      .????? (16)

      其中,θ=arccosT,A>0,-1<T<1.

      將上述結果代入式(7)和(10)中,可以得到一般三階方程解的一般形式。同理可得:

      a4k3+(a3h+b3)h2+(a2h2+b2h+c2)k+(a1h3+b1h2+c1h)=0.

      (17)

      令:A1=(a3h+b3)-3a4(a1h3+b1h2+c1h);

      B1=(a3h+b3)(a2h2+b2h+c2)-9a4(a1h3+b1h2+c1h);

      C1=(a2h2+b2h+c2)2-3(a3h+b3)(a1h3+b1h2+c1h).

      根據盛金定理可知,記Δ1=B12-4A1C1. 由此可以得到以下結論。

      情形1′:在式(17)中,當A1=B1=0時,解得:

      .??????????????????????? (18)

      情形2′:當Δ1>0時,解得k的值為:

      ,

      .???????? (19)

      其中,i2=-1.

      情形3′:當Δ1=0時,此時可以解出:

      .???????? (20)

      其中,A1≠0).

      情形4′:當Δ1<0時,此時可以得到:

      ,

      .?????????? (21)

      其中,θ1=arccosT1,A1>0,

      -1<T1<1.

      根據不同的情形,將結果代入式(8)和(10)中,可以得到一般方程解的形式。

      2? 常系數(shù)非齊次N階偏微分方程的一般解

      對常系數(shù)非齊次N階偏微分方程而言,要得出其一般意義上的解并不容易。但是,當φx,t)取一些特殊函數(shù)時,可以得到其解的具體形式。常系數(shù)非齊次N階偏微分方程的一般形式如下:

      .

      (22)

      根據φx,t)的不同取值,討論以下5種特殊情形。

      情形1′′:當φx,t)=cn1時,如果z具有zn1=An1xn形式的解,代入式(22)中求解。通過na0An1=cn1,可得解得系數(shù)

      .

      情形2′′:當φxt)=cn2x時,如果z具有zn2=An2xn+1形

      式的解,代入式(22)中,由(n+1)!a0An2x=cn2x.

      情形3′′:當φxt)=cn3t時,如果z具有zn3=An3tn+1形式的

      解,代入式(22)中,由(n+1)!anAn3t=cn3t可得.

      情形4′′:當φx,t)=cn4x+cn5t時,如果z具有zn4= An4xnt+An5xn+1形式的解,代入式(22)中,則可以通過

      得到.

      情形5′′:當φxt)=cn6xt時,如果z具有zn5= An6xn+1t+An7xn+2

      形式的解,代入式(22)中可以由

      解得.

      由于常系數(shù)非齊次N階偏微分方程解具有復雜性,所以,本文僅討論了5種解得的具體形式。當遇到具體工程問題時,可根據具體情形求解。

      3? 結束語

      本文僅研究了一般形式的常系數(shù)非齊次三階偏微分方程的解的一般式,在相應的常系數(shù)非齊N階偏微分方程中,得到了部分函數(shù)對應的特殊解的情況。在具體的工程應用中,對更多不同情況的求解一定可以得到更多對現(xiàn)實問題有幫助的結果。

      參考文獻

      [1]Devi J Vasundhara.Generalized monotone method for periodic boundary value problems of Caputo fractional differential equations[J].Commun.Appl.Anal,2008,12(4):399-406.

      [2]Yusufjon P.Apakov,Stasys Rutkauskas.On a boundary value problem to third order PDE with multiple characteristics[J].Nonlinear Analysis:Modelling and Control,2011,16(3):255-269.

      〔編輯:白潔〕

      The Methods and Generalizations of the Nonhomogeneous Three Order artial Differential

      P Equation with Constant Coefficients in Engineering

      Han Zhiwei

      AbstractIn engineering application, the specific form of partial differential equations is often able to simplify the complex problem. Although the actual significance questions represented the different partial differential equations are not identical, the laws of mathematics has exactly the same form of many practical problems in engineering. The research on the general sense of the equation may contribute to the solution of practical problems. This paper mainly studies the general form of the constant coefficient non-homogeneous three order partial differential equation, and discusses the special condition of non-homogeneous Nth order partial differential equation with constant coefficients, and has obtained the solution under the different conditions.

      Key words: third order partial differential equation; constant coefficient; non-homogeneous; complementary function

      [2]Yusufjon P.Apakov,Stasys Rutkauskas.On a boundary value problem to third order PDE with multiple characteristics[J].Nonlinear Analysis:Modelling and Control,2011,16(3):255-269.

      〔編輯:白潔〕

      The Methods and Generalizations of the Nonhomogeneous Three Order artial Differential

      P Equation with Constant Coefficients in Engineering

      Han Zhiwei

      AbstractIn engineering application, the specific form of partial differential equations is often able to simplify the complex problem. Although the actual significance questions represented the different partial differential equations are not identical, the laws of mathematics has exactly the same form of many practical problems in engineering. The research on the general sense of the equation may contribute to the solution of practical problems. This paper mainly studies the general form of the constant coefficient non-homogeneous three order partial differential equation, and discusses the special condition of non-homogeneous Nth order partial differential equation with constant coefficients, and has obtained the solution under the different conditions.

      Key words: third order partial differential equation; constant coefficient; non-homogeneous; complementary function

      [2]Yusufjon P.Apakov,Stasys Rutkauskas.On a boundary value problem to third order PDE with multiple characteristics[J].Nonlinear Analysis:Modelling and Control,2011,16(3):255-269.

      〔編輯:白潔〕

      The Methods and Generalizations of the Nonhomogeneous Three Order artial Differential

      P Equation with Constant Coefficients in Engineering

      Han Zhiwei

      AbstractIn engineering application, the specific form of partial differential equations is often able to simplify the complex problem. Although the actual significance questions represented the different partial differential equations are not identical, the laws of mathematics has exactly the same form of many practical problems in engineering. The research on the general sense of the equation may contribute to the solution of practical problems. This paper mainly studies the general form of the constant coefficient non-homogeneous three order partial differential equation, and discusses the special condition of non-homogeneous Nth order partial differential equation with constant coefficients, and has obtained the solution under the different conditions.

      Key words: third order partial differential equation; constant coefficient; non-homogeneous; complementary function

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