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[美] 赫斯赫 (Hsieh P.F.) 著 / 高等教育出版社 / 2007-07 / 平装
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常微分方程基础理论(影印版)
本书为开展常微分方程研究工作的读者提供必要的准备知识,可作为本科高年级和研究生常微分方程课程教材。
本书内容分为四部分:第一部分(第一、二、三章)的内容包括解的存在性、唯一性、对数据的光滑依赖性,以及解的非唯一性;第二部分(第四、六、七章)讨论线性常微分方程,书中用矩阵的S-N分解代替Jordan分解,前者的计算较后者更容易;第三部分(第八、九、十章)讨论非线性常微分方程的稳定性、渐近稳定性等几何理论;第四部分(第五、十一,十二、十三章)讨论常微分方程的幂级数解,包括线性常微分方程的奇点分类及非线性常微分方程当参数或自变量趋向某奇点时的渐近解等。
PrefaceChapterⅠ.FundamentalTheoremsofOrdinaryDifferentialEquationsⅠ-1.ExistenceanduniquenesswiththeLipschitzconditionⅠ-2.ExistencewithouttheLipschitzconditionⅠ-3.SomeglobalpropertiesofsolutionsⅠ-4.AnalyticdifferentialequationsExercisesⅠChapterⅡ.DependenceonDataⅡ-1.ContinuitywithrespecttoinitialdataandparametersⅡ-2.DifferentiabilityExercisesⅡChapterⅢ.NonuniquenessⅢ-1.ExamplesⅢ-2.TheKnesertheoremⅢ-3.SolutioncurvesontheboundaryofR(A)Ⅲ-4.MaximalandminimalsolutionsⅢ-5.AcomparisontheoremⅢ-6.SufficientconditionsforuniquenessExercisesⅢChapterⅣ.GeneralTheoryofLinearSystemsⅣ-1.SomebasicresultsconcerningmatricesⅣ-2.HomogeneoussystemsoflineardifferentialequationsⅣ-3.HomogeneoussystemswithconstantcoefficientsⅣ-4.SystemswithperiodiccoefficientsⅣ-5.LinearHamiltoniansystemswithperiodiccoefficientsⅣ-6.NonhomogeneousequationsⅣ-7.Higher-orderscalarequationsExercisesⅣChapterⅤ.SingularitiesoftheFirstKindⅤ-1.FormalsolutionsofanalgebraicdifferentialequationⅤ-2.ConvergenceofformalsolutionsofasystemofthefirstkindⅤ-3.TheS-NdecompositionofamatrixofinfiniteorderⅤ-4.TheS-NdecompositionofadifferentialoperatorⅤ-5.AnormalformofadifferentialoperatorⅤ-6.CalculationofthenormalformofadifferentialoperatorⅤ-7.ClassificationofsingularitiesofhomogeneouslinearsystemsExercisesⅤChapterⅥ.Boundary-ValueProblemsofLinearDifferentialEquationsoftheSecond-OrderⅥ-1.ZerosofsolutionsⅥ-2.Sturm-LiouvilleproblemsⅥ-3.EigenvalueproblemsⅥ-4.EigenfunctionexpansionsⅥ-5.JostsolutionsⅥ-6.ScatteringdataⅥ-7.RefiectionlesspotentialsⅥ-8.ConstructionofapotentialforgivendataⅥ-9.DifferentialequationssatisfiedbyreflectionlesspotentialsⅥ-10.PeriodicpotentialsExercisesⅥChapterⅦ.AsymptoticBehaviorofSolutionsofLinearSystemsⅦ-1.LiapounoffstypenumbersⅦ-2.LiapounoffstypenumbersofahomogeneouslinearsystemⅦ-3.CalculationofLiapounoffstypenumbersofsolutionsⅦ-4.AdiagonalizationtheoremⅦ-5.SystemswithasymptoticallyconstantcoefficientsⅦ-6.AnapplicationoftheFloquettheoremExercisesⅦChapterⅧ.StabilityⅧ-1.BasicdefinitionsⅧ-2.AsufficientconditionforasymptoticstabilityⅧ-3.StablemanifoldsⅧ-4.AnalyticstructureofstablemanifoldsⅧ-5.Two-dimensionallinearsystemswithconstantcoefficientsⅧ-6.AnalyticsystemsinR2Ⅷ-7.PerturbationsofanimpropernodeandasaddlepointⅧ-8.PerturbationsofapropernodeⅧ-9.PerturbationofaspiralpointⅧ-10.PerturbationofacenterExercisesⅧChapterⅨ.AutonomousSystemsⅨ-1.Limit-invariantsetsⅨ-2.LiapounoffsdirectmethodⅨ-3.OrbitalstabilityⅨ-4.ThePoincare-BendixsontheoremⅨ-5.IndicesofJordancurvesExercisesⅨChapterⅩ.TheSecond-OrderDifferentialEquation(d2x)/(dt2)+h(x)*(dx)/(dt)+g(x)=0Ⅹ-1.Two-pointboundary-valueproblemsⅩ-2.ApplicationsoftheLiapounofffunctionsⅩ-3.ExistenceanduniquenessofperiodicorbitsⅩ-4.MultipliersoftheperiodicorbitofthevanderPolequationⅩ-5.ThevanderPolequationforasmallε>0Ⅹ-6.ThevanderPolequationforalargeparameterⅩ-7.AtheoremduetoM.NagumoⅩ-8.AsingularperturbationproblemExercisesⅩChapterⅪ.AsymptoticExpansionsⅪ-1.AsymptoticexpansionsinthesenseofPoincareⅪ-2.GevreyasymptoticsⅪ-3.FlatfunctionsintheGevreyasymptoticsⅪ-4.BasicpropertiesofGevreyasymptoticexpansionsⅪ-5.ProofofLemmaⅪ-2-6ExercisesⅪChapterⅫ.AsymptoticExpansionsinaParameterⅫ-1.AnexistencetheoremⅫ-2.BasicestimatesⅫ-3.ProofofTheoremⅫ-1-2Ⅻ-4.Ablock-diagonalizationtheoremⅫ-5.GevreyasymptoticsolutionsinaparameterⅫ-6.AnalyticsimplificationinaparameterExercisesⅫChapterⅩⅢ.SingularitiesoftheSecondKindⅩⅢ-1.AnexistencetheoremⅩⅢ-2.BasicestimatesⅩⅢ-3.ProofofTheoremⅩⅢ-1-2ⅩⅢ-4.Ablock-diagonalizationtheoremⅩⅢ-5.Cyclicvectors(AlemmaofP.Deligne)ⅩⅢ-6.TheHukuhara-TurrittintheoremⅩⅢ-7.Ann-th-orderlineardifferentialequationatasingularpointofthesecondkindⅩⅢ-8.GevreypropertyofasymptoticsolutionsatanirregularsingularpointExercisesⅩⅢReferencesIndex
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