1、1-1 Partial Differential Equations (PDE) 1-1-1 Partial Differential Equations and Their Orders Introduction Ordinary differential equation (the ODE for short) is a differential equation that contains one or more derivatives of the dependent variable (in addition to the dependent variable and the ind

2、ependent variable). 含自變量，未知函數(shù)及其導(dǎo)數(shù) 應(yīng)變量 dependent variable 自變量 independent variable The order (階) of an ordinary differential equation is the order of the highest-ordered derivative appearing in the equation. A differential equation that contains, in addition to the dependent variable and the independ

3、ent variables, one or more partial derivatives of the dependent variable is called a partial differential equation, the PDE for short （偏微分方程） The order of the highest-ordered partial derivative appearing in the equation is called the order of the PDE Independent and dependent variables may not appea

4、r in a PDE . A PDE must contain, however, at least one partial derivative of the dependent variable. For an unknown function of two independent variables,1,0,20 xxyxxyxuuuuare the first-order, the second-order andthe third-order PDE, respectively.The k-th order PDE of an unknown function of independ

5、ent variables can be written in a general form 22212221112( ,)0knknnnuuuuuuF u x xxxxxx xxx 2222222( , , , ),ttuauf x y z txyz Three-dimensional wave equation :two-dimensional heat-conduction equation: two-dimensional and the three-dimensional Laplace equations: 22222( , , ),tuauf x y txy ( , )0,( ,

6、 , )0u x yu x y ztwo-dimensional and the three-dimensional Poisson equationsthe dual-phase-lagging heat-conduction equation:雙相滯熱傳導(dǎo)方程( , ),( , , )u x yfu x y zf220(, )tttuuaubuf M tt 1-1-2 Linear, Nonlinear and Quasi-Linear (擬線性) Equations An ODE can be linear, nonlinear A PDE can be linear, nonlinea

7、r , quasi-linear. A PDE is said to be linear if it is linear in the unknown function and all its derivatives; nonlinear equation ; A nonlinear equation is said to be quasi-linear if it is linear in all highest-ordered derivatives of the unknown function. Linear PDE:( , )( , )uua x yf x yxx22uyxx y 2

8、( , )uuu x yxx y Nonlinear PDE:22()()0uuxy2( , )uua x yuxyQuasi-linear PDE:222220uuuuuxxyy 22220uuuuxxy Quasi-linear PDE is nonlinear equation. PDE with constant coefficients PDE with variable coefficients ,11ijinnijx xixi jia ubucufThe general second-order linear PDE in n independent variables has

9、the form: 0,f Homogeneous （齊次的）（齊次的）PDE Else , Nonhomogeneous （非齊次的）（非齊次的）PDE Note that the definition of homogeneity is only for the linear ODE, PDE1-1-3 Solutions of Partial Differential Equations A function is called a classical solution of the PDE, a solution for short, if it has continuous part

10、ial derivatives of all orders appearing in a PDE and satisfies the equation. 此處要注意常微分方程和偏微分方程的解中常數(shù)的含義。 ODE: 常數(shù)是數(shù)， PDE: 常數(shù)是任意函數(shù)。2xyux y( , ),uu x y已知：求解：32121:( )( )6Solutionux yxy 此處注意復(fù)合函數(shù)求導(dǎo)； 該方程有無數(shù)個(gè)解，求解PDE的特解，不像ODE那樣先求通解，再求特解。而是直接從方程和物理?xiàng)l件出發(fā)求特解。( , ),uu x y已知：求解：0 xyuuxyxy變換：:( , )()Solutionu x yf x

11、y1-1-4 Classification of Linear Second-Order Equation2220AxBxyCyDxEyF2BAC elliptic, parabolic, or hyperbolic 0,0,0 1112221220 xxxyyyxya ua ua ubub ucuf2120011002200(,)(,)(,)axyaxy axy elliptic, parabolic, or hyperbolic 0,0,0 0 xxyyyuuelliptic, parabolic, or hyperbolic y Tricomi equation:從另一角度（二次型）看方

12、程的分類：221111212222( )2Aaaa 111221220aaaa11122122aaaa1221aacharacteristic equation： 211221122122121122()0()0aaa aa aaa elliptic, parabolic, or hyperbolic 0,0,0 兩個(gè)特征根同號；有0特征根；異號12112212,aa 3,1( )ijiji jAa0 xxyyzzuuu1231Example 1:2()txxyyua uu21230,a1000100001Example 2:Solution:2( , , , )ttuauf x y z t

13、212341,a 2200000000aaExample 3:Solution:22210000000000000aaa Solution: the second-order linear partial differential equation in independent variables is elliptic, parabolic, or hyperbolic at a point accordingly as 1. all n characteristic roots are with the same sign, 2. there is a vanished character

14、istic root, 3. or n characteristic roots have different signs but (n-1) characteristic roots are with the same sign 0,1( )()nijiji jAap1112221220 xxxyyyxya ua ua ubub ucuf( , )x y( , )x y1-1-5 Canonical Forms(見另一課件)1112221220A uA uA uBuB uCuF222121122121122AA AJaa awith as the coefficients of second

15、-order derivatives 1運(yùn)用復(fù)合函數(shù)求導(dǎo)算一下221111122220 xxyyAaaa 222211122220 xxyyAaaa 2211122220 xxyya za z za z上面兩式有相同的形式：constantz ddd0 xyzzxzy2111222dydy()20dxdxaaadydxxyzz 22111222(dy)2d d(dx)0aax ya1112221220 xxxyyyxya ua ua ubub ucufcharacteristic equation ：21212112211ddaaa ayxa21212112211ddaaa ayxachara

16、cteristic roots ：21212112211ddaaa ayxa21212112211ddaaa ayxa( , ),( , )x ycx ycd222dd222d2 ,2yyxcyxcxyyxcyxcxyxyx 例如：Hyperbolic type: 21211220aa a1( , , ,)0uu u u st1112221220 xxxyyyxya ua ua ubub ucuf( , )x y( , )x y2( , , ,)0ssttstuus t u u u1112221220A uA uA uBuB uCuF1122(0)AAParabolic type: 21211220aa a1112221220 xxxyyyxya ua ua ubub ucuf( , )x y( , )x y3( , , ,)0uu u u 4( , , ,)0uu u u 222121122121122120AA AJaa aA110A Or:220A只