山东大学工科研究生数学物理方法class数理方程的分类

1-1PartialDifferentialEquations(PDE)

1-1-1PartialDifferentialEquationsandTheirOrders

Introduction

Ordinarydifferentialequation(theODEforshort)isadifferentialequationthatcontainsoneormorederivativesofthedependentvariable(inadditiontothedependentvariableandtheindependentvariable).含自变量，未知函数及其导数应变量dependentvariable

自变量independentvariableTheorder(阶)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:双相滞热传导方程

1-1-2Linear,NonlinearandQuasi-Linear

(拟线性)Equations

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

nonlinearequation;Anonlinearequationissaidtobequasi-linearifitislinearinallhighest-orderedderivativesoftheunknownfunction.

LinearPDE:

NonlinearPDE:Quasi-linearPDE:Quasi-linearPDEisnonlinearequation.PDEwithconstantcoefficients

PDEwithvariablecoefficients

Thegeneralsecond-orderlinearPDEinnindependentvariableshastheform:

Homogeneous（齐次的）PDE

Else,Nonhomogeneous（非齐次的）PDE

NotethatthedefinitionofhomogeneityisonlyforthelinearODE,PDE1-1-3SolutionsofPartialDifferentialEquations

AfunctioniscalledaclassicalsolutionofthePDE,asolutionforshort,ifithascontinuouspartialderivativesofallordersappearinginaPDEandsatisfiestheequation.此处要注意常微分方程和偏微分方程的解中常数的含义。ODE:常数是数，

PDE:常数是任意函数。已知：求解：此处注意复合函数求导；该方程有无数个解，求解PDE的特解，不像ODE那样先求通解，再求特解。而是直接从方程和物理条件出发求特解。已知：求解：变换：

1-1-4ClassificationofLinearSecond-Order

Equation

elliptic,parabolic,orhyperbolic

elliptic,parabolic,orhyperbolic

elliptic,parabolic,orhyperbolic

Tricomiequation:从另一角度（二次型）看方程的分类：characteristicequation：

elliptic,parabolic,orhyperbolic

两个特征根同号；有0特征根；异号Example1:Example2:Solution:Example3:Solution:Solution:thesecond-orderlinearpartialdifferentialequationinindependentvariablesiselliptic,parabolic,orhyperbolicatapointaccordinglyas1.allncharacteristicrootsarewiththesamesign,2.thereisavanishedcharacteristicroot,3.orncharacteristicrootshavedifferentsignsbut(n-1)characteristicrootsarewiththesamesign

1-1-5CanonicalForms(见另一课件)withasthecoefficientsofsecond-orderderivatives

运用复合函数求导算一下上面两式有相同的形式：c