偏微分方程(二) 全16講 台灣國立交通大學 DVD 只於電腦播放

本課程是由交通大學應用數學系提供。

ThisisthecontinuouscourseofPartialDifferentialEquations(I).ThisistheGRADUATElevelpartialdifferentialequation.WewillfocusontherelationbetweenmathematicsandphysicsandshowthestudentshowtounderstandPDEsintuitively.
授課教師應用數學系林琦焜老師

授課時數每週3小時

授課學分3學分

授課學期97學年度

授課對象研究所學生

預備知識Calculus,AdvancedCalculus,LinearAlgebra,Ordinarydifferentialequation,ComplexAnalysisandRealanalysis
課程綱要
課程目標/概述

本課程屬研究所程度的微分方程課程，授課偏重於數學與物理間的連結，並且讓學生藉由此課程了解直觀地PDE概念。

課程章節
章節

主題
第五章HyperbolicEquationsinHigherDimensions
第六章Higher-OrderEllipticEquationswithConstantCoefficients
第七章ParabolicEquations
第八章H.LewysExampleofaLinearEquationwithoutSolutions

課程書目

*FritzJohn,PartialDifferentialEquations(4thEdition),AppliedMathematicalSciencesVol.1Springer-Verlag1982.
*A.Pazy,SemigroupsofLinearOperatorsandApplicationstoPartialDifferentialEquations,Springer-Verlag,1983
單元主題

內容綱要
第五章HyperbolicEquationsinHigherDimensions
5-1TheWaveEquationinn-DimensionalSpace
(1)Themethodofsphereicalmeans
(2)Hadmardsmethodofdescent
(3)DuhamelsprincipleandthegeneralCauchyproblem
(4)mixedproblem
5-2Higher-OrderHyperbolicEquationswithConstantCoefficients
(1)Standardformoftheinitial-valueproblem
(2)solutionbyFouriertransform,
(3)solutionofamixedproblembyFouriertransform
5-3SymmetricHyperbolicSystem
(1)Thebasicenergyinequality
(2)Finitedifferencemethod
(3)Schaudermethod
第六章Higher-OrderEllipticEquationswithConstantCoefficients
6-1TheFundamentalSolutionforOddnTravellingwave
6-2TheDirichletProblemLax-Milgramtheorem,Gardinginequality
6-3SobolevSpaceWeaksolutionandHibertspace
第七章ParabolicEquations
7-1TheHeatEquationSelf-Similarity,Heatkernel,maximumprinciple
7-2TheInitial-ValueProblemforGeneralSecond-OrderParabolicEquations
(1)Finitedifferenceandmaximumprinciple
(2)ExistenceofInitialValueProblem
第八章H.LewysExampleofaLinearEquationwithoutSolutions
8-1BriefintroductionofFunctionalAnalysisHilbertandBanachspaces,projectiontheorem,Leray-Schaudertheorem
8-2SemigroupsoflinearoperatorGeneration,representationandspectralproperties
8-3PerturbationsandApproximationsTheTrottertheorem
8-4TheabstractCauchyProblemBasictheory
8-5ApplicationtolinearpartialdifferentialequationsParabolicequation,WaveequationandSchrodingerequation
8-6ApplicationstononlinearpartialdifferentialequationsKdVequation,nonlinearheatequation,nonmlinearSchrodingerequation