1-1PartialDifferentialEquations(PDE)

1-1-1PartialDifferentialEquationsandTheirOrders

Introduction

Ordinarydifferentialequation(theODEforshort)isadifferentialequationthatcontainsoneormorederivativesofthedependentvariable(inadditiontothedependentvariableandtheindependentvariable).含自變量，未知函數(shù)及其導(dǎo)數(shù)應(yīng)變量dependentvariable

自變量independentvariableTheorder(階)ofanordinarydifferentialequationistheorderofthehighest-orderedderivativeappearingintheequation.Adifferentialequationthatcontains,inadditiontothedependentvariableandtheindependentvariables,oneormorepartialderivativesofthedependentvariableiscalledapartialdifferentialequation,thePDEforshort（偏微分方程）Theorderofthehighest-orderedpartialderivativeappearingintheequationiscalledtheorderofthePDEIndependentanddependentvariablesmaynotappearinaPDE.APDEmustcontain,however,atleastonepartialderivativeofthedependentvariable.Foranunknownfunctionoftwoindependentvariables,arethefirst-order,thesecond-orderandthethird-orderPDE,respectively.

Thek-thorderPDEofanunknownfunctionofindependentvariablescanbewritteninageneralform

Three-dimensionalwaveequation:two-dimensionalheat-conductionequation:

two-dimensionalandthethree-dimensionalLaplaceequations:two-dimensionalandthethree-dimensionalPoissonequationsthedual-phase-laggingheat-conductionequation:雙相滯熱傳導(dǎo)方程

1-1-2Linear,NonlinearandQuasi-Linear

(擬線性)Equations

AnODEcanbelinear,nonlinearAPDEcanbelinear,nonlinear,quasi-linear.APDEissaidtobelinearifitislinearintheunknownfunctionandallitsderivatives;

nonlinearequation;Anonlinearequationissaidtobequasi-linearifitislinearinallhighest-orderedderivativesoftheunknownfunction.

LinearPDE:

NonlinearPDE:Quasi-linearPDE:Quasi-linearPDEisnonlinearequation.PDEwithconstantcoefficients

PDEwithvariablecoefficients

Thegeneralsecond-orderlinearPDEinnindependentvariableshastheform:

Homogeneous（齊次的）PDE

Else,Nonhomogeneous（非齊次的）PDE

NotethatthedefinitionofhomogeneityisonlyforthelinearODE,PDE1-1-3SolutionsofPartialDifferentialEquations

AfunctioniscalledaclassicalsolutionofthePDE,asolutionforshort,ifithascontinuouspartialderivativesofallordersappearinginaPDEandsatisfiestheequation.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)。已知：求解：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。已知：求解：變換：

1-1-4ClassificationofLinearSecond-Order

Equation

elliptic,parabolic,orhyperbolic

elliptic,parabolic,orhyperbolic

elliptic,parabolic,orhyperbolic

Tricomiequation:從另一角度（二次型）看方程的分類：characteristicequation：

elliptic,parabolic,orhyperbolic

兩個(gè)特征根同號；有0特征根；異號Example1:Example2:Solution:Example3:Solution:Solution:thesecond-orderlinearpartialdifferentialequationinindependentvariablesiselliptic,parabolic,orhyperbolicatapointaccordinglyas1.allncharacteristicrootsarewiththesamesign,2.thereisavanishedcharacteristicroot,3.orncharacteristicrootshavedifferentsignsbut(n-1)characteristicrootsarewiththesamesign

1-1-5CanonicalForms(見另一課件)withasthecoefficientsofsecond-orderderivatives

運(yùn)用復(fù)合函數(shù)求導(dǎo)算一下上面兩式有相同的形式：c