对称方法在偏微分方程中的应用

图书标准信息

• 作者 [加]布鲁曼（Bluman G.W.） 著
• 出版社 世界图书出版公司
• 出版时间 2015-01
• 版次 1
• ISBN 9787510086267
• 定价 79.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 389页
• 正文语种 英语

【内容简介】

　　ThisbookisasequeltoSymmetriesandIntegrationMethods(2002),byGeorgeW.BlumanandStephenC.Anco.ItincludesasignificantupdateofthematerialinthelastthreechaptersofSymmet'riesan,dDzjjerentialEqua-tions(1989;reprintedwithcorrections,1996),byGeorgeW.BlumanandSukeyukiKumei.Theemphasisinthepresentbookisonhowtofindsys-tematicallysymmetries(localandnonlocal)andconservationlaws(localandnonlocal)ofagivenPDEsystemandhowtousesystematicallysymmetriesandconservationlawsforrelatedapplications.Inparticular,foragivenPDEsystem,itisshownhowsystematically(1)tofindhigher-orderandnonlocalsymmetriesofthesystem;(2)toconstructbydirectmethodsitsconserva-tionlawsthroughfindingsetsofconservationlawmultipliersandformulastoobtainthefluxesofaconservationlawfromaknownsetofmultipliers;(3)todeterminewhetherithasalinearizationbyaninvertiblemappingandcon-structsuchalinearizationwhenoneexistsfromknowledgeofitssymmetriesandlorconservationlawmultipliers,inthecasewheiithegivenPDEsystemisnonlinear;(4)touseconservationlawstoconstructequivalentnonlocallyrelatedsystems;(5)tousesuchnonlocallyrelatedsystemstoobtainnonlo-calsymmetries,nonlocalconservationlawsandnon-invertiblemappingstolinearsystems;and(6)toconstructspecificsolutionsfromreductionsarisingfromitssymmetriesaswellasfromextensionsofsymmetrymethodstofindsuchreductions.
　　Thisbookisaimedatappliedmathematicians;scientistsandengineersinterestedinfindingsolutionsofpartialdifferentialequationsandiswritteninthestyleoftheabove-mentioned1989bookbyBlumanandKumei.Therearenumerousexamplesinvolvingvariouswell-knownphysicalandengineeringPDEsystems.

【目录】

Preface
Introduction
1LocalTransformationsandConservationLaws
1.1Introduction
1.2LocalTransformations
1.2.1Pointtransformations
1.2.2Contacttransformations
1.2.3Higher-ordertransformations
1.2.4One-parameterhigher-ordertransformations
1.2.5Pointsymmetries
1.2.6Contactandhigher-ordersymmetries
1.2.7Equivalencetransformationsandsymmetryclassification
1.2.8Recursionoperatorsforlocalsymmetries
1.3ConservationLaws
1.3.1Localconservationlaws
1.3.2Equivalentconservationlaws
1.3.3Multipliersforconservationlaws.Euleroperators
1.3.4Thedirectmethodforconstructionofconservationlaws.Cauchy-Kovalevskayaform
1.3.5Examples
1.3.6Linearizingoperatorsandadjointequations
1.3.7Determinationoffluxesofconservationlawsfrommultipliers
1.3.8Self-adjointPDEsystems
1.4Noether'sTheorem
1.4.1Euler-Lagrangeequations
1.4.2Noether'sformulationofNoether'stheorem
1.4.3Boyer'sformulationofNoether'stheorem
1.4.4LimitationsofNoether'stheorem
1.4.5Examples
1.5SomeConnectionsBetweenSymmetriesandConservationLaws
1.5.1Useofsymmetriestofindnewconservationlawsfromknownconservationlaws
1.5.2Relationshipsamongsymmetries，solutionsofadjointequations，andconservationlaws
1.6Discussion

2ConstructionofMappingsRelatingDifferentialEquations
2.1Introduction
2.2Notations；MappingsofInfinitesimalGenerators
2.2.1Theoremsoninvertiblemappings
2.3MappingofaGivenPDEtoaSpecificTargetPDE
2.3.1Constructionofnon-invertiblemappings
2.3.2Constructionofaninvertiblemappingbyapointtransformation
2.4InvertibleMappingsofNonlinearPDEstoLinearPDEsThroughSymmetries
2.4.1InvertiblemappingsofnonlinearPDEsystems（withatleasttwodependentvariables）tolinearPDEsystems
2.4.2InvertiblemappingsofnonlinearPDEsystems（withonedependentvariable）tolinearPDEsystems
2.5InvertibleMappingsofLinearPDEstoLinearPDEswithConstantCoefficients
2.5.1ExamplesofmappingvariablecoefficientlinearPDEstoconstantcoefficientlinearPDEsthroughinvertiblepointtransformations
2.5.2ExampleoffindingthemostgeneralmappingofagivenconstantcoefficientlinearPDEtosomeconstantcoefficientlinearPDE
2.6InvertibleMappingsofNonlinearPDEstoLinearPDEsThroughConservationLawMultipliers
2.6.1Computationalsteps
2.6.2ExamplesoflinearizationsofnonlinearPDEsthroughconservationlawmultipliers
2.7Discussion

3NonlocallyRelatedPDESystems
3.1Introduction
3.2NonlocallyRelatedPotentialSystemsandSubsystemsinTwoDimensions
3.2.1Potentialsystems
3.2.2Nonlocallyrelatedsubsystems
3.3TreesofNonlocallyRelatedPDESystems
3.3.1Basicprocedureoftreeconstruction
3.3.2Atreeforanonlineardiffusionequation
3.3.3Atreeforplanargasdynamics（PGD）equations
3.4NonlocalConservationLaws
3.4.1Conservationlawsarisingfromnonlocallyrelatedsystems
3.4.2Nonlocalconservationlawsfordiffusion-convectionequations
3.4.3Additionalconservationlawsofnonlineartelegraphequations
3.5ExtendedTreeConstructionProcedure
3.5.1Anextendedtreeconstructionprocedure
3.5.2Anextendedtreeforanonlineardiffusionequation
3.5.3Anextendedtreeforanonlinearwaveequation
3.5.4Anextendedtreefortheplanargasdynamicsequations
3.6Discussion

4ApplicationsofNonlocallyRelatedPDESystems
4.1Introduction
4.2NonlocalSymmetries
4.2.1Nonlocalsymmetriesofanonlineardiffusionequation
4.2.2NonlocALsymmetriesofanonlinearwaveequation
4.2.3Classificationofnonlocalsymmetriesofnonlineartelegraphequationsarisingfrompointsymmetriesofpotentialsystems
4.2.4Nonlocalsymmetriesofnonlineartelegraphequationswithpowerlawnonlinearities
4.2.5Nonlocalsymmetriesoftheplanargasdynamicsequations
4.3ConstructionofNon-invertibleMappingsRelatingPDEs
4.3.1Non-invertiblemappingsofnonlinearPDEsystemstolinearPDEsystems
4.3.2Non-invertiblemappingsoflinearPDEswithvariablecoefficientstolinearPDEswithconstantcoefficients.
4.4Discussion

5FurtherApplicationsofSymmetryMethods：MiscellaneousExtensions
5.1Introduction
5.2ApplicationsofSymmetryMethodstotheConstructionofSolutionsofPDEs
5.2.1Theclassicalmethod
5.2.2Thenonclassicalmethod
5.2.3Invariantsolutionsarisingfromnonlocalsymmetriesthatarelocalsymmetriesofnonlocallyrelatedsystems
5.2.4FutrtherextensionsofsymmetrymethodsforconstructionofsolutionsofPDEsconnectedwithnonlocaUyrelatedsystems
5.3NonlocallyRelatedPDESystemsinThreeorMoreDimensions
5.3.1Divergence-typeconservationlawsandresultingpotentialsystems
5.3.2Nonlocallyrelatedsubsystems
5.3.3Treeconstruction，nonlocalconservationlaws，andnonlocalsymmetries
5.3.4Lower-degreeconservationlawsandrelatedpotentialsystems
5.3.5Examplesofapplicationsofnonlocallyrelatedsystemsinhigherdimensions
5.3.6Symmetriesandexactsolutionsofthethree-dimensionalMHDequilibriumequations
5.4SymbolicSoftware
5.4.1Anexampleofsymboliccomputationofpointsymmetries
5.4.2Anexampleofpointsymmetryclassification
5.4.3Anexampleofsymboliccomputationofconservationlaws
5.5Discussion
References
Theorem，CorollaryandLemmaIndex
AuthorIndex
SubjectIndex