(Miersemann)Partial Differential Equations

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1、PartialDifferentialEquationsLectureNotesErichMiersemannDepartmentofMathematicsLeipzigUniversityVersionOctober,20122Contents1Introduction91.1Examples.............................111.2Equationsfromvariationalproblems..............151.2.1Ordinarydifferentialequations.............151.2

2、.2Partialdifferentialequations..............161.3Exercises.............................222Equationsoffirstorder252.1Linearequations.........................252.2Quasilinearequations......................312.2.1Alinearizationmethod.................322.2.2InitialvalueproblemofCauchy.

3、............332.3Nonlinearequationsintwovariables..............402.3.1InitialvalueproblemofCauchy.............482.4NonlinearequationsinRn....................512.5Hamilton-Jacobitheory.....................532.6Exercises.............................593Classification633.1Linearequati

4、onsofsecondorder................633.1.1Normalformintwovariables..............693.2Quasilinearequationsofsecondorder..............733.2.1Quasilinearellipticequations..............733.3Systemsoffirstorder.......................743.3.1Examples.........................763.4Systemsofse

5、condorder.....................823.4.1Examples.........................833.5TheoremofCauchy-Kovalevskaya................843.5.1Appendix:Realanalyticfunctions...........9034CONTENTS3.6Exercises.............................1014Hyperbolicequations1074.1One-dimensionalwaveequation....

6、.............1074.2Higherdimensions........................1094.2.1Casen=3.........................1124.2.2Casen=2........................1154.3Inhomogeneousequation.....................1174.4AmethodofRiemann......................1204.5Initial-boundaryvalueproblems...............

7、..1254.5.1Oscillationofastring..................1254.5.2Oscillationofamembrane................1284.5.3Inhomogeneouswaveequations.............1314.6Exercises.............................1365Fouriertransform1415.1Definition,properties.......................1415.1.1Pseudodifferentialo

8、perators...............1465.2Exercises..