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微分方程与数学物理问题

微分方程与数学物理问题

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作者: [瑞典] 伊布拉基莫夫著

出版社: 高等教育出版社

出版时间: 2009-08

版次: 1

ISBN: 9787040276039

定价: 68.00

装帧: 精装

开本: 16开

纸张: 胶版纸

页数: 348页

字数: 255千字

正文语种: 英语

分类: 自然科学

1 张插图图片

25人买过

　　APracticalCourseinDifferentialEquationsandMathematicalModellingisauniqueblendofthetraditionalmethodsofordinaryandpartialdifferentialequationswithLiegroupanalysisenrichedbytheauthorsowntheoreticaldevelopments.Thebook——whichaimstopresentnewmathematicalcurriculabasedonsymmetryandinvarianceprinciples——istailoredtodevelopanalyticskillsand"workingknowledge"inbothclassicalandLiesmethodsforsolvinglinearandnonlinearequations.Thisapproachhelpstomakecoursesndifferentialequations，mathematicalmodelling，distributionsandfundamentalsolution，etc.easytofollowandinterestingforstudents.ThebookisbasedontheauthorsextensiveteachingexperienceatNovosibirskandMoscowuniversitiesinRussia，CollegedeFrance，GeorgiaTechandStanfordUniversityintheUnitedStates，universitiesinSouthAfrica，Cyprus，Turkey，andBlekingeInstituteofTechnology(BTH)inSweden.Thenewcurriculumpreparesstudentsforsolvingmodernnonlinearproblemsandwillessentiallybemoreappealingtostudentscomparedtothetraditionalwayofteachingmathematics.Thebookcanbeusedasamaintextbookbyundergraduateandgraduatestudentsanduniversitylecturersinappliedmathematics，physicsandengineering. Preface
1Selectedtopicsfromanalysis
1.1Elementarymathematics
1.1.1Numbers,variablesandelementaryfunctions
1.1.2Quadraticandcubicequations
1.1.3Areasofsimilarfigures.Ellipseasanexample
1.1.4Algebraiccurvesoftheseconddegree
1.2Differentialandintegralcalculus
1.2.1Rulesfordifferentiation
1.2.2Themeanvaluetheorem
1.2.3Invarianceofthedifferential
1.2.4Rulesforintegration
1.2.5TheTaylorseries
1.2.6Complexvariables
1.2.7Approximaterepresentationoffunctions
1.2.8Jacobian.Functionalindependence.Changeofvariablesinmultipleintegrals
1.2.9Linearindependenceoffunctions.Wronskian
1.2.10Integrationbyquadrature
1.2.11Differentialequationsforfamiliesofcurves
1.3Vectoranalysis
1.3.1Vectoralgebra
1.3.2Vectorfunctions
1.3.3Vectorfields
1.3.4Threeclassicalintegraltheorems
1.3.5TheLaplaceequation
1.3.6Differentiationofdeterminants
1.4Notationofdifferentialalgebra
1.4.1Differentialvariables.Totaldifferentiation
1.4.2Higherderivativesoftheproductandofcompositefunctions
1.4.3Differentialfunctionswithseveralvariables
1.4.4Theframeofdifferentialequations
1.4.5Transformationofderivatives
1.5Variationalcalculus
1.5.1Principleofleastaction
1.5.2Euler-Lagrangeequationswithseveralvariables
ProblemstoChapter1
2Mathematicalmodels
2.1Introduction
2.2Naturalphenomena
2.2.1Populationmodels
2.2.2Ecology:Radioactivewasteproducts
2.2.3Keplerslaws.Newtonsgravitationlaw
2.2.4Freefallofabodyneartheearth
2.2.5Meteoroid
2.2.6Amodelofrainfall
2.3Physicsandengineeringsciences
2.3.1Newtonsmodelofcooling
2.3.2Mechanicalvibrations.Pendulum
2.3.3Collapseofdrivingshafts
2.3.4ThevanderPolequation
2.3.5Telegraphequation
2.3.6Electrodynamics
2.3.7TheDiracequation
2.3.8Fluiddynamics
2.3.9TheNavier-Stokesequations
2.3.10Amodelofanirrigationsystem
2.3.11Magnetohydrodynamics
2.4Diffusionphenomena
2.4.1Linearheatequation
2.4.2Nonlinearheatequation
2.4.3TheBurgersandKorteweg-deVriesequations.
2.4.4Mathematicalmodellinginfinance
2.5Biomathematics
2.5.1Smartmushrooms
2.5.2Atumourgrowthmodel
2.6Wavephenomena
2.6.1Smallvibrationsofastring
2.6.2Vibratingmembrane
2.6.3Minimalsurfaces
2.6.4Vibratingslenderrodsandplates
2.6.5Nonlinearwaves
2.6.6TheChaplyginandTricomiequations
ProblemstoChapter2
3Ordinarydifferentialequations:Traditionalapproach
3.1Introductionandelementarymethods
3.1.1Differentialequations.Initialvalueproblem
3.1.2Integrationoftheequationy(n)=f(x)
3.1.3Homogeneousequations
3.1.4Differenttypesofhomogeneity
3.1.5Reductionoforder
3.1.6Linearizationthroughdifferentiation
3.2First-orderequations
3.2.1Separableequations
3.2.2Exactequations
3.2.3Integratingfactor(A.Clairaut,1739)
3.2.4TheRiccatiequation
3.2.5TheBernoulliequation
3.2.6Homogeneouslinearequations
3.2.7Non-homogeneouslinearequations.Variationoftheparameter
3.3Second-orderlinearequations
3.3.1Homogeneousequation:Superposition
3.3.2Homogeneousequation:Equivalenceproperties
3.3.3Homogeneousequation:Constantcoefficients
3.3.4Non-homogeneousequation:Variationofparameters
3.3.5BesselsequationandtheBesselfunctions
3.3.6Hypergeometricequation
3.4Higher-orderlinearequations
3.4.1Homogeneousequations.Fundamentalsystem
3.4.2Non-homogeneousequations.Variationofparameters
3.4.3Equationswithconstantcoefficients
3.4.4Eulersequation
3.5Systemsoffirst-orderequations
3.5.1Generalpropertiesofsystems
3.5.2Firstintegrals
3.5.3Linearsystemswithconstantcoefficients
3.5.4Variationofparametersforsystems
ProblemstoChapter3
4First-orderpartialdifferentialequations
4.1Introduction
4.2Homogeneouslinearequation
4.3Particularsolutionsofnon-homogeneousequations
4.4Quasi-linearequations
4.5Systemsofhomogeneousequations
ProblemstoChapter4
5Linearpartialdifferentialequationsofthesecondorder
5.1Equationswithseveralvariables
5.1.1Classificationatafixedpoint
5.1.2Adjointlineardifferentialoperators
5.2Classificationofequationsintwoindependentvariables
5.2.1Characteristics.Threetypesofequations
5.2.2Thestandardformofthehyperbolicequations
5.2.3Thestandardformoftheparabolicequations
5.2.4Thestandardformoftheellipticequations
5.2.5Equationsofamixedtype
5.2.6Thetypeofnonlinearequations
5.3Integrationofhyperbolicequationsintwovariables
5.3.1dAlembertssolution
5.3.2Equationsreducibletothewaveequation
5.3.3Eulersmethod
5.3.4Laplacescascademethod
5.4Theinitialvalueproblem
5.4.1Thewaveequation
5.4.2Non-homogeneouswaveequation
5.5Mixedproblem.Separationofvariables
5.5.1Vibrationofastringtiedatitsends
5.5.2Mixedproblemfortheheatequation
ProblemstoChapter5
6Nonlinearordinarydifferentialequations
6.1Introduction
6.2Transformationgroups
6.2.1One-parametergroupsontheplane
6.2.2GroupgeneratorandtheLieequations
6.2.3Exponentialmap
6.2.4Invariantsandinvariantequations
6.2.5Canonicalvariables
6.3Symmetriesoffirst-orderequations
6.3.1Firstprolongationofgroupgenerators
6.3.2Symmetrygroup:definitionandmainproperty
6.3.3Equationswithagivensymmetry
6.4Integrationoffirst-orderequationsusingsymmetries
6.4.1Liesintegratingfactor
6.4.2Integrationusingcanonicalvariables
6.4.3Invariantsolutions
6.4.4Generalsolutionprovidedbyinvariantsolutions
6.5Second-orderequations
6.5.1SecondprolongationofgroupgeneratorsCalculationofsymmetries
6.5.2Liealgebras
6.5.3Standardformsoftwo-dimensionalLiealgebras
6.5.4Liesintegrationmethod
6.5.5Integrationoflinearequationswithaknownparticularsolution
6.5.6Lieslinearizationtest
6.6Higher-orderequations
6.6.1Invariantsolutions.DerivationofEulersansatz
6.6.2Integratingfactor(N.H.Ibragimov,2006)
6.6.3Linearizationofthird-orderequations
6.7Nonlinearsuperposition
6.7.1Introduction
6.7.2Maintheoremonnonlinearsuperposition
6.7.3Examplesofnonlinearsuperposition
6.7.4Integrationofsystemsusingnonlinearsuperposition
ProblemstoChapter6
7Nonlinearpartialdifferentialequations
7.1Symmetries
7.1.1Definitionandcalculationofsymmetrygroups
7.1.2Grouptransformationsofsolutions
7.2Groupinvariantsolutions
7.2.1Introduction
7.2.2TheBurgersequation
7.2.3Anonlinearboundary-valueproblem
7.2.4Invariantsolutionsforanirrigationsystem
7.2.5Invariantsolutionsforatumourgrowthmodel
7.2.6Anexamplefromnonlinearoptics
7.3Invarianceandconservationlaws
7.3.1Introduction
7.3.2Preliminaries
7.3.3Noetherstheorem
7.3.4Higher-orderLagrangians
7.3.5ConservationtheoremsforODEs
7.3.6GeneralizationofNoetherstheorem
7.3.7Examplesfromclassicalmechanics
7.3.8DerivationofEinsteinsformulaforenergy
7.3.9ConservationlawsfortheDiracequations
ProblemstoChapter7
8Generalizedfunctionsordistributions
8.1Introductionofgeneralizedfunctions
8.1.1Heuristicconsiderations
8.1.2Definitionandexamplesofdistributions
8.1.3Representationsoftheδ-functionasalimit
8.2Operationswithdistributions
8.2.1Multiplicationbyafunction
8.2.2Differentiation
8.2.3Directproductofdistributions
8.2.4Convolution
8.3Thedistribution△(r2-n)
8.3.1Themeanvalueoverthesphere
8.3.2SolutionoftheLaplaceequation△v(r)=0
8.3.3Evaluationofthedistribution△(r2-n)
8.4Transformationsofdistributions
8.4.1Motivationbylineartransformations
8.4.2Changeofvariablesinthed-function
8.4.3Arbitrarygrouptransformations
8.4.4Infinitesimaltransformationofdistributions
ProblemstoChapter8
9Invarianceprincipleandfundamentalsolutions
9.1Introduction
9.2Theinvarianceprinciple
9.2.1Formulationoftheinvarianceprinciple
9.2.2Fundamentalsolutionoflinearequationswithconstantcoefficients
9.2.3ApplicationtotheLaplaceequation
9.2.4Applicationtotheheatequation
9.3Cauchysproblemfortheheatequation
9.3.1FundamentalsolutionfortheCauchyproblem
9.3.2DerivationofthefundamentalsolutionfortheCauchyproblemfromtheinvarianceprinciple
9.3.3SolutionoftheCauchyproblem
9.4Waveequation
9.4.1Preliminariesondifferentialforms
9.4.2Auxiliaryequationswithdistributions
9.4.3Symmetriesanddefinitionoffundamentalsolutionsforthewaveequation
9.4.4Derivationofthefundamentalsolution
9.4.5SolutionoftheCauchyproblem
9.5Equationswithvariablecoefficients
ProblemstoChapter9
Answers
Bibliography
Index
内容简介:
　　APracticalCourseinDifferentialEquationsandMathematicalModellingisauniqueblendofthetraditionalmethodsofordinaryandpartialdifferentialequationswithLiegroupanalysisenrichedbytheauthorsowntheoreticaldevelopments.Thebook——whichaimstopresentnewmathematicalcurriculabasedonsymmetryandinvarianceprinciples——istailoredtodevelopanalyticskillsand"workingknowledge"inbothclassicalandLiesmethodsforsolvinglinearandnonlinearequations.Thisapproachhelpstomakecoursesndifferentialequations，mathematicalmodelling，distributionsandfundamentalsolution，etc.easytofollowandinterestingforstudents.ThebookisbasedontheauthorsextensiveteachingexperienceatNovosibirskandMoscowuniversitiesinRussia，CollegedeFrance，GeorgiaTechandStanfordUniversityintheUnitedStates，universitiesinSouthAfrica，Cyprus，Turkey，andBlekingeInstituteofTechnology(BTH)inSweden.Thenewcurriculumpreparesstudentsforsolvingmodernnonlinearproblemsandwillessentiallybemoreappealingtostudentscomparedtothetraditionalwayofteachingmathematics.Thebookcanbeusedasamaintextbookbyundergraduateandgraduatestudentsanduniversitylecturersinappliedmathematics，physicsandengineering.
目录:
Preface
1Selectedtopicsfromanalysis
1.1Elementarymathematics
1.1.1Numbers,variablesandelementaryfunctions
1.1.2Quadraticandcubicequations
1.1.3Areasofsimilarfigures.Ellipseasanexample
1.1.4Algebraiccurvesoftheseconddegree
1.2Differentialandintegralcalculus
1.2.1Rulesfordifferentiation
1.2.2Themeanvaluetheorem
1.2.3Invarianceofthedifferential
1.2.4Rulesforintegration
1.2.5TheTaylorseries
1.2.6Complexvariables
1.2.7Approximaterepresentationoffunctions
1.2.8Jacobian.Functionalindependence.Changeofvariablesinmultipleintegrals
1.2.9Linearindependenceoffunctions.Wronskian
1.2.10Integrationbyquadrature
1.2.11Differentialequationsforfamiliesofcurves
1.3Vectoranalysis
1.3.1Vectoralgebra
1.3.2Vectorfunctions
1.3.3Vectorfields
1.3.4Threeclassicalintegraltheorems
1.3.5TheLaplaceequation
1.3.6Differentiationofdeterminants
1.4Notationofdifferentialalgebra
1.4.1Differentialvariables.Totaldifferentiation
1.4.2Higherderivativesoftheproductandofcompositefunctions
1.4.3Differentialfunctionswithseveralvariables
1.4.4Theframeofdifferentialequations
1.4.5Transformationofderivatives
1.5Variationalcalculus
1.5.1Principleofleastaction
1.5.2Euler-Lagrangeequationswithseveralvariables
ProblemstoChapter1
2Mathematicalmodels
2.1Introduction
2.2Naturalphenomena
2.2.1Populationmodels
2.2.2Ecology:Radioactivewasteproducts
2.2.3Keplerslaws.Newtonsgravitationlaw
2.2.4Freefallofabodyneartheearth
2.2.5Meteoroid
2.2.6Amodelofrainfall
2.3Physicsandengineeringsciences
2.3.1Newtonsmodelofcooling
2.3.2Mechanicalvibrations.Pendulum
2.3.3Collapseofdrivingshafts
2.3.4ThevanderPolequation
2.3.5Telegraphequation
2.3.6Electrodynamics
2.3.7TheDiracequation
2.3.8Fluiddynamics
2.3.9TheNavier-Stokesequations
2.3.10Amodelofanirrigationsystem
2.3.11Magnetohydrodynamics
2.4Diffusionphenomena
2.4.1Linearheatequation
2.4.2Nonlinearheatequation
2.4.3TheBurgersandKorteweg-deVriesequations.
2.4.4Mathematicalmodellinginfinance
2.5Biomathematics
2.5.1Smartmushrooms
2.5.2Atumourgrowthmodel
2.6Wavephenomena
2.6.1Smallvibrationsofastring
2.6.2Vibratingmembrane
2.6.3Minimalsurfaces
2.6.4Vibratingslenderrodsandplates
2.6.5Nonlinearwaves
2.6.6TheChaplyginandTricomiequations
ProblemstoChapter2
3Ordinarydifferentialequations:Traditionalapproach
3.1Introductionandelementarymethods
3.1.1Differentialequations.Initialvalueproblem
3.1.2Integrationoftheequationy(n)=f(x)
3.1.3Homogeneousequations
3.1.4Differenttypesofhomogeneity
3.1.5Reductionoforder
3.1.6Linearizationthroughdifferentiation
3.2First-orderequations
3.2.1Separableequations
3.2.2Exactequations
3.2.3Integratingfactor(A.Clairaut,1739)
3.2.4TheRiccatiequation
3.2.5TheBernoulliequation
3.2.6Homogeneouslinearequations
3.2.7Non-homogeneouslinearequations.Variationoftheparameter
3.3Second-orderlinearequations
3.3.1Homogeneousequation:Superposition
3.3.2Homogeneousequation:Equivalenceproperties
3.3.3Homogeneousequation:Constantcoefficients
3.3.4Non-homogeneousequation:Variationofparameters
3.3.5BesselsequationandtheBesselfunctions
3.3.6Hypergeometricequation
3.4Higher-orderlinearequations
3.4.1Homogeneousequations.Fundamentalsystem
3.4.2Non-homogeneousequations.Variationofparameters
3.4.3Equationswithconstantcoefficients
3.4.4Eulersequation
3.5Systemsoffirst-orderequations
3.5.1Generalpropertiesofsystems
3.5.2Firstintegrals
3.5.3Linearsystemswithconstantcoefficients
3.5.4Variationofparametersforsystems
ProblemstoChapter3
4First-orderpartialdifferentialequations
4.1Introduction
4.2Homogeneouslinearequation
4.3Particularsolutionsofnon-homogeneousequations
4.4Quasi-linearequations
4.5Systemsofhomogeneousequations
ProblemstoChapter4
5Linearpartialdifferentialequationsofthesecondorder
5.1Equationswithseveralvariables
5.1.1Classificationatafixedpoint
5.1.2Adjointlineardifferentialoperators
5.2Classificationofequationsintwoindependentvariables
5.2.1Characteristics.Threetypesofequations
5.2.2Thestandardformofthehyperbolicequations
5.2.3Thestandardformoftheparabolicequations
5.2.4Thestandardformoftheellipticequations
5.2.5Equationsofamixedtype
5.2.6Thetypeofnonlinearequations
5.3Integrationofhyperbolicequationsintwovariables
5.3.1dAlembertssolution
5.3.2Equationsreducibletothewaveequation
5.3.3Eulersmethod
5.3.4Laplacescascademethod
5.4Theinitialvalueproblem
5.4.1Thewaveequation
5.4.2Non-homogeneouswaveequation
5.5Mixedproblem.Separationofvariables
5.5.1Vibrationofastringtiedatitsends
5.5.2Mixedproblemfortheheatequation
ProblemstoChapter5
6Nonlinearordinarydifferentialequations
6.1Introduction
6.2Transformationgroups
6.2.1One-parametergroupsontheplane
6.2.2GroupgeneratorandtheLieequations
6.2.3Exponentialmap
6.2.4Invariantsandinvariantequations
6.2.5Canonicalvariables
6.3Symmetriesoffirst-orderequations
6.3.1Firstprolongationofgroupgenerators
6.3.2Symmetrygroup:definitionandmainproperty
6.3.3Equationswithagivensymmetry
6.4Integrationoffirst-orderequationsusingsymmetries
6.4.1Liesintegratingfactor
6.4.2Integrationusingcanonicalvariables
6.4.3Invariantsolutions
6.4.4Generalsolutionprovidedbyinvariantsolutions
6.5Second-orderequations
6.5.1SecondprolongationofgroupgeneratorsCalculationofsymmetries
6.5.2Liealgebras
6.5.3Standardformsoftwo-dimensionalLiealgebras
6.5.4Liesintegrationmethod
6.5.5Integrationoflinearequationswithaknownparticularsolution
6.5.6Lieslinearizationtest
6.6Higher-orderequations
6.6.1Invariantsolutions.DerivationofEulersansatz
6.6.2Integratingfactor(N.H.Ibragimov,2006)
6.6.3Linearizationofthird-orderequations
6.7Nonlinearsuperposition
6.7.1Introduction
6.7.2Maintheoremonnonlinearsuperposition
6.7.3Examplesofnonlinearsuperposition
6.7.4Integrationofsystemsusingnonlinearsuperposition
ProblemstoChapter6
7Nonlinearpartialdifferentialequations
7.1Symmetries
7.1.1Definitionandcalculationofsymmetrygroups
7.1.2Grouptransformationsofsolutions
7.2Groupinvariantsolutions
7.2.1Introduction
7.2.2TheBurgersequation
7.2.3Anonlinearboundary-valueproblem
7.2.4Invariantsolutionsforanirrigationsystem
7.2.5Invariantsolutionsforatumourgrowthmodel
7.2.6Anexamplefromnonlinearoptics
7.3Invarianceandconservationlaws
7.3.1Introduction
7.3.2Preliminaries
7.3.3Noetherstheorem
7.3.4Higher-orderLagrangians
7.3.5ConservationtheoremsforODEs
7.3.6GeneralizationofNoetherstheorem
7.3.7Examplesfromclassicalmechanics
7.3.8DerivationofEinsteinsformulaforenergy
7.3.9ConservationlawsfortheDiracequations
ProblemstoChapter7
8Generalizedfunctionsordistributions
8.1Introductionofgeneralizedfunctions
8.1.1Heuristicconsiderations
8.1.2Definitionandexamplesofdistributions
8.1.3Representationsoftheδ-functionasalimit
8.2Operationswithdistributions
8.2.1Multiplicationbyafunction
8.2.2Differentiation
8.2.3Directproductofdistributions
8.2.4Convolution
8.3Thedistribution△(r2-n)
8.3.1Themeanvalueoverthesphere
8.3.2SolutionoftheLaplaceequation△v(r)=0
8.3.3Evaluationofthedistribution△(r2-n)
8.4Transformationsofdistributions
8.4.1Motivationbylineartransformations
8.4.2Changeofvariablesinthed-function
8.4.3Arbitrarygrouptransformations
8.4.4Infinitesimaltransformationofdistributions
ProblemstoChapter8
9Invarianceprincipleandfundamentalsolutions
9.1Introduction
9.2Theinvarianceprinciple
9.2.1Formulationoftheinvarianceprinciple
9.2.2Fundamentalsolutionoflinearequationswithconstantcoefficients
9.2.3ApplicationtotheLaplaceequation
9.2.4Applicationtotheheatequation
9.3Cauchysproblemfortheheatequation
9.3.1FundamentalsolutionfortheCauchyproblem
9.3.2DerivationofthefundamentalsolutionfortheCauchyproblemfromtheinvarianceprinciple
9.3.3SolutionoftheCauchyproblem
9.4Waveequation
9.4.1Preliminariesondifferentialforms
9.4.2Auxiliaryequationswithdistributions
9.4.3Symmetriesanddefinitionoffundamentalsolutionsforthewaveequation
9.4.4Derivationofthefundamentalsolution
9.4.5SolutionoftheCauchyproblem
9.5Equationswithvariablecoefficients
ProblemstoChapter9
Answers
Bibliography
Index

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