微分方程及其应用

　　Mathematics is playing an ever more important role in the physical and biological sciences， provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest， both in research and teaching， has led to the establishment of the series： Texts in Applied Mathematics （TAM）. The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques， such as numerical and symbolic computer systems， dynamical systems， and chaos， mix with and reinforce the traditional methods of applied mathematics. Thus， the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses.

暂缺《微分方程及其应用》作者简介

Chapter1
First-orderdifferentialequations
1.1Introduction
1.2First-orderlineardifferentialequations
1.3TheVanMeegerenartforgeries
1.4Separableequations
1.5Populationmodels
1.6Thespreadoftechnologicalinnovations
1.7Anatomicwastedisposalproblem
1.8Thedynamicsoftumorgrowth,mixingproblems,and
orthogonaltrajectories
1.9Exactequations,andwhywecannotsolveverymany
differentialequations
1.10Theexistence-uniquenesstheorem;Picarditeration
1.11Findingrootsofequationsbyiteration
1.11.1Newton'smethod
1.12Differenceequations,andhowtocomputetheinterest
dueonyourstudentloans
1.13Numericalapproximations;Euler'smethod
1.13.1ErroranalysisforEuler'smethod
1.14ThethreetermTaylorseriesmethod
1.15AnimprovedEulermethod
1.16TheRnnge-Kuttamethod
1.17Whattodoinpractice
Chapter2
Second-orderlineardifferentialequations
2.1Algebraicpropertiesofsolutions
2.2Linearequationswithconstantcoefficients
2.2.1Complexroots
2.2.2Equalroots;reductionoforder
2.3Thenonhomogeneousequation
2.4Themethodofvariationofparameters
2.5Themethodofjudiciousguessing
2.6Mechanicalvibrations
2.6.1TheTacomaBridgedisaster
2.6.2Electricalnetworks
2.7Amodelforthedetectionofdiabetes
2.8Seriessolutions
2.8.1Singularpoints,Eulerequations
2.8.2Regularsingularpoints,themethodofFrobenius
2.8.3Equalroots,androotsdifferingbyaninteger
2.9ThemethodofLaplacetransforms
2.10SomeusefulpropertiesofLaplacetransforms
2.11Differentialequationswithdiscontinuousright-handsides
2.12TheDiracdeltafunction
2.13Theconvolutionintegral
2.14Themethodofeliminationforsystems
2.15Higher-orderequations
Chapter3
Systemsofdifferentialequations
3.1Algebraicpropertiesofsolutionsoflinearsystems
3.2Vectorspaces
3.3Dimensionofavectorspace
3.4Applicationsoflinearalgebratodifferentialequations
3.5Thetheoryofdeterminants
3.6Solutionsofsimultaneouslinearequations
3.7Lineartransformations
3.8Theeigenvalue-eigenvectormethodoffindingsolutions
3.9Complexroots
3.10Equalroots
3.11Fundamentalmatrixsolutions;eA'
3.12Thenonhomogeneousequation;variationofparameters
3.13SolvingsystemsbyLaplacetransforms
Chapter4
Qualitativetheoryofdifferentialequations
4.1Introduction
4.2Stabilityoflinearsystems
4.3Stabilityofequilibriumsolutions
4.4Thephase-plane
4.5Mathematicaltheoriesofwar
4.5.1L.F.Richardson'stheoryofconflict
4.5.2Lanchester'scombatmodelsandthebattleoflwoJima
4.6Qualitativepropertiesoforbits
4.7Phaseportraitsoflinearsystems
4.8Longtimebehaviorofsolutions;thePoincare-BendixsonTheorem
4.9Introductiontobifurcationtheory
4.10Predator-preyproblems;orwhy
thepercentageofsharkscaughtintheMediterranean
SearosedramaticallyduringWorldWarI
4.11Theprincipleofcompetitiveexclusioninpopulationbiology
4.12TheThresholdTheoremofepidemiology
4.13Amodelforthespreadofgonorrhea
Chapter5
SeparationofvariablesandFourierseries
5.1Twopointboundary-valueproblems
5.2Introductiontopartialdifferentialequations
5.3Theheatequation;separationofvariables
5.4Fourierseries
5.5Evenandoddfunctions
5.6Returntotheheatequation
5.7Thewaveequation
5.8Laplace'sequation
Chapter6
Sturm-Liouvilleboundaryvalueproblems
6.1Introduction
6.2Innerproductspaces
6.3Orthogonalbases,Hermitianoperators
6.4Sturm-Liouvilletheory
AppendixA
Somesimplefactsconcerningfunctions
ofseveralvariables
AppendixB
Sequencesandseries
AppendixC
CPrograms
Answerstoodd-numberedexercises
Index