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Ordinary and Partial Differential Equations PDF

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Ordinary and Partial Differential Equations AnIntroductiontoDynamicalSystems John W. Cain, Ph.D. and Angela M. Reynolds, Ph.D. MathematicsTextbookSeries.Editor:LonMitchell 1. BookofProof byRichardHammack 2. LinearAlgebrabyJimHefferon 3. AbstractAlgebra:TheoryandApplicationsbyThomasJudson 4. OrdinaryandPartialDifferentialEquationsbyJohnW.CainandAngelaM.Reynolds DepartmentofMathematics&AppliedMathematics VirginiaCommonwealthUniversity Richmond,Virginia,23284 PublicationofthiseditionsupportedbytheCenterforTeachingExcellenceatvcu OrdinaryandPartialDifferentialEquations:AnIntroductiontoDynamicalSystems Edition1.0 ©2010byJohnW.CainandAngelaReynolds This work is licensed under the Creative Commons Attribution-NonCommercial-No Derivative Works3.0Licenseandispublishedwiththeexpresspermissionoftheauthors. Typesetin10ptPalladioLwithPazoMathfontsusingPDFLATEX Acknowledgements John W. Cain expresses profound gratitude to his advisor, Dr. David G. Scha- effer, James B. Duke Professor of Mathematics at Duke University. The first fivechaptersarebasedinpartuponProfessorSchaeffer’sintroductorygradu- ate course on ordinary differential equations. The material has been adapted to accommodate upper-level undergraduate students, essentially by omitting technicalproofsofthemajortheoremsandincludingadditionalexamples. Other majorinfluencesonthisbookincludetheexcellenttextsofPerko[8],Strauss[10], andStrogatz[11]. Inparticular,thematerialpresentedinthelastfivechapters (includingtheorderingofthetopics)isbasedheavilyonStrauss’book. Onthe other hand, our exposition, examples, and exercises are more “user-friendly”, makingourtextmoreaccessibletoreaderswithlessbackgroundinmathematics. Dr.Reynoldsdedicatesherportionofthistextbooktohermother,fatherand sisters,shethanksthemforalltheirsupportandlove. Finally,Dr.CaindedicateshisportionofthistextbooktohisparentsJeanette andHarry,whohelovesmorethanwordscanexpress. iii Contents Acknowledgements iii Contents iv 1 Introduction 1 1.1 InitialandBoundaryValueProblems . . . . . . . . . . . . . . . . . 4 2 Linear,Constant-CoefficientSystems 8 2.1 HomogeneousSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 DiagonalizableMatrices. . . . . . . . . . . . . . . . . . . . . 12 2.1.2 AlgebraicandGeometricMultiplicitiesofEigenvalues. . . . 21 2.1.3 ComplexEigenvalues. . . . . . . . . . . . . . . . . . . . . . . 29 2.1.4 RepeatedEigenvaluesandNon-DiagonalizableMatrices. . 37 2.2 PhasePortraitsandPlanarSystems . . . . . . . . . . . . . . . . . . 45 2.3 Stable,Unstable,andCenterSubspaces . . . . . . . . . . . . . . . . 57 2.4 TraceandDeterminant . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.5 InhomogeneousSystems . . . . . . . . . . . . . . . . . . . . . . . . . 67 3 NonlinearSystems: LocalTheory 78 3.1 LinearApproximationsofFunctionsofSeveralVariables . . . . . . 81 3.2 FundamentalExistenceandUniquenessTheorem . . . . . . . . . . 84 3.3 GlobalExistence,DependenceonInitialConditions . . . . . . . . . 86 3.4 EquilibriaandLinearization . . . . . . . . . . . . . . . . . . . . . . . 94 3.5 TheHartman-GrobmanTheorem . . . . . . . . . . . . . . . . . . . . 98 3.6 TheStableManifoldTheorem . . . . . . . . . . . . . . . . . . . . . . 100 3.7 Non-HyperbolicEquilibriaandLyapunovFunctions . . . . . . . . 105 4 Periodic,Heteroclinic,andHomoclinicOrbits 122 4.1 PeriodicOrbitsandthePoincaré-BendixonTheorem . . . . . . . . 122 iv contents v 4.2 HeteroclinicandHomoclinicOrbits . . . . . . . . . . . . . . . . . . 130 5 Bifurcations 140 5.1 ThreeBasicBifurcations . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.2 DependenceofSolutionsonParameters . . . . . . . . . . . . . . . . 148 5.3 Andronov-HopfBifurcations . . . . . . . . . . . . . . . . . . . . . . 151 6 IntroductiontoDelayDifferentialEquations 166 6.1 InitialValueProblems . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.2 SolvingConstant-CoefficientDelayDifferentialEquations . . . . . 169 6.3 CharacteristicEquations . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.4 TheHutchinson-WrightEquation . . . . . . . . . . . . . . . . . . . . 172 7 IntroductiontoDifferenceEquations 180 7.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.2 Linear,Constant-CoefficientDifferenceEquations . . . . . . . . . . 181 7.3 First-OrderNonlinearEquationsandStability . . . . . . . . . . . . 191 7.4 SystemsofNonlinearEquationsandStability . . . . . . . . . . . . 195 7.5 Period-DoublingBifurcations . . . . . . . . . . . . . . . . . . . . . . 200 7.6 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 7.7 HowtoControlChaos . . . . . . . . . . . . . . . . . . . . . . . . . . 208 8 IntroductiontoPartialDifferentialEquations 218 8.1 BasicClassificationofPartialDifferentialEquations . . . . . . . . . 221 8.2 SolutionsofPartialDifferentialEquations . . . . . . . . . . . . . . . 227 8.3 InitialConditionsandBoundaryConditions . . . . . . . . . . . . . 228 8.4 VisualizingSolutionsofPartialDifferentialEquations . . . . . . . . 233 9 Linear,First-OrderPartialDifferentialEquations 236 9.1 DerivationandSolutionoftheTransportEquation . . . . . . . . . 239 9.2 MethodofCharacteristics: MoreExamples . . . . . . . . . . . . . . 241 10 TheHeatandWaveEquationsonanUnboundedDomain 250 10.1 DerivationoftheHeatandWaveEquations . . . . . . . . . . . . . . 250 10.2 CauchyProblemfortheWaveEquation . . . . . . . . . . . . . . . . 255 10.3 CauchyProblemfortheHeatEquation . . . . . . . . . . . . . . . . 265 10.4 Well-PosednessandtheHeatEquation . . . . . . . . . . . . . . . . 276 10.5 InhomogeneousEquationsandDuhamel’sPrinciple . . . . . . . . 284 vi 11 Initial-BoundaryValueProblems 297 11.1 HeatandWaveEquationsonaHalf-Line . . . . . . . . . . . . . . . 297 11.2 SeparationofVariables . . . . . . . . . . . . . . . . . . . . . . . . . . 306 11.2.1 WaveEquation,DirichletProblem. . . . . . . . . . . . . . . . 307 11.2.2 HeatEquation,DirichletProblem. . . . . . . . . . . . . . . . 313 11.2.3 WaveEquation,NeumannProblem. . . . . . . . . . . . . . . 318 11.2.4 HeatEquation,NeumannProblem. . . . . . . . . . . . . . . 324 11.2.5 MixedBoundaryConditions: AnExample. . . . . . . . . . . 324 12 IntroductiontoFourierSeries 330 12.1 Fourierseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 12.1.1 Fouriersineseries. . . . . . . . . . . . . . . . . . . . . . . . . 332 12.1.2 Fouriercosineseries. . . . . . . . . . . . . . . . . . . . . . . . 337 12.1.3 Fourierseries. . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 12.2 ConvergenceofFourierSeries . . . . . . . . . . . . . . . . . . . . . . 344 12.2.1 Norms,distances,innerproducts,andconvergence. . . . . 347 12.2.2 Convergencetheorems. . . . . . . . . . . . . . . . . . . . . . 359 13 TheLaplaceandPoissonEquations 367 13.1 DirchletandNeumannProblems . . . . . . . . . . . . . . . . . . . . 370 13.2 Well-posednessandtheMaximumPrinciple . . . . . . . . . . . . . 372 13.3 TranslationandRotationInvariance . . . . . . . . . . . . . . . . . . 375 13.4 Laplace’sEquationonBoundedDomains . . . . . . . . . . . . . . . 383 13.4.1 Dirichletproblemonarectangle.. . . . . . . . . . . . . . . . 383 13.4.2 Dirichletproblemonadisc. . . . . . . . . . . . . . . . . . . . 390 GuidetoCommonlyUsedNotation 404 References 406 Index 407 CHAPTER 1 Introduction Themathematicalsub-disciplineofdifferentialequationsanddynamicalsystems isfoundationalinthestudyofappliedmathematics. Differentialequations arise in a variety of contexts, some purely theoretical and some of practical interest. As you read this textbook, you will find that the qualitative and quantitativestudyofdifferentialequationsincorporatesanelegantblendoflinear algebraandadvancedcalculus. Forthisreason,itisexpectedthatthereaderhas alreadycompletedcoursesin(i)linearalgebra;(ii)multivariablecalculus;and (iii)introductorydifferentialequations. Familiaritywiththefollowingtopicsis especiallydesirable: (cid:43) Frombasicdifferentialequations: separabledifferentialequationsandsepara- tionof variables; andsolvinglinear, constant-coefficient differentialequations usingcharacteristicequations. (cid:43) From linear algebra: solving systems of m algebraic equations with n un- knowns;matrixinversion;linearindependence;andeigenvalues/eigenvectors. (cid:43) From multivariable calculus: parametrized curves; partial derivatives and gradients;andapproximatingasurfaceusingatangentplane. Someofthesetopicswillbereviewedasweencounterthemlater—inthis chapter,wewillrecallafewbasicnotionsfromanintroductorycoursein differentialequations. Readersareencouragedtosupplementthisbookwiththe excellenttextbooksofHubbardandWest[5],Meiss[7],Perko[8],Strauss[10], andStrogatz[11]. Question: Whystudydifferentialequations? 1 2 Answer: When scientists attempt to mathematically model various natural phenomena,theyofteninvokephysical“laws”orbiological“principles”which governtheratesofchangeofcertainquantitiesofinterest. Hence,theequations in mathematical models tend to include derivatives. For example, suppose thatahotcupofcoffeeisplacedinaroomofconstantambienttemperature α. Newton’sLawofCoolingstatesthattherateofchangeofthecoffeetemperature T(t)isproportionaltothedifferencebetweenthecoffee’stemperatureandthe room temperature. Mathematically, this can be expressed as dT = k(T−α), dt where k isaproportionalityconstant. Solutiontechniquesfordifferentialequations(des)dependinpartuponhow manyindependentvariablesanddependentvariablesthesystemhas. Example 1.0.1. One independent variable and one independent variable. In writingtheequation d2y +cos(xy) = 3, dx2 it is understood that y is the dependent variable and x is the independent variable. Whenadifferentialequationinvolvesasingleindependentvariable,werefer totheequationasanordinarydifferentialequation(ode). Example1.0.2. Ifthereareseveraldependentvariablesandasingleindependent variable,wemighthaveequationssuchas dy dz = x2y−xy2+z, = z−ycosx. dx dx This is a system of two odes, and it is understood that x is the independent variable. Example1.0.3. Onedependentvariable,severalindependentvariables. Consider thede ∂u ∂2u ∂2u = + . ∂t ∂x2 ∂y2 Thisequationinvolvesthreeindependentvariables(x, y,and t)andonedepen- dentvariable, u. Thisisanexampleofapartialdifferentialequation(pde). Ifthere areseveralindependentvariablesandseveraldependentvariables,onemayhave systemsofpdes. introduction 3 Althoughtheseconceptsareprobablyfamiliartothereader,wegiveamore exactdefinitionforwhatwemeanbyode. Supposethatxandyareindependent anddependentvariables,respectively,andlet y(k)(x) denotethe kthderivative of y withrespectto x. (If k ≤3,wewilluseprimes.) Definition 1.0.4. Any equation of the form F(x,y,y(cid:48),y(cid:48)(cid:48),...,y(n)) = 0 is called anordinarydifferentialequation. If y(n) isthehighestderivativeappearinginthe equation,wesaythattheodeisofordern. Example1.0.5. (cid:18)d3y(cid:19)2 dy d2y −(cosx) = y dx3 dx dx2 canbewrittenas(y(cid:48)(cid:48)(cid:48))2−yy(cid:48)(cid:48)−(cosx)y(cid:48) =0,sousingthenotationintheabove Definition,wewouldhave F(x,y,y(cid:48),y(cid:48)(cid:48),y(cid:48)(cid:48)(cid:48)) = (y(cid:48)(cid:48)(cid:48))2−yy(cid:48)(cid:48)−(cosx)y(cid:48). Thisisa third-orderode. Definition1.0.6. Asolutionoftheode F(x,y,y(cid:48),y(cid:48)(cid:48),...,y(n)) =0onaninterval I isanyfunction y(x) whichis n-timesdifferentiableandsatisfiestheequation on I. Example1.0.7. Foranychoiceofconstant A,thefunction Aex y(x) = 1+Aex isasolutionofthefirst-orderode y(cid:48) = y−y2 forallreal x. Toseewhy,weuse thequotientruletocalculate Aex(1+Aex)−(Aex)2 Aex y(cid:48) = = . (1+Aex)2 (1+Aex)2 Bycomparison,wecalculatethat Aex (Aex)2 Aex y−y2 = − = . (1+Aex) (1+Aex)2 (1+Aex)2 Therefore, y(cid:48) = y−y2,asclaimed. The definition of a solution of an ode is easily extended to systems of odes (seebelow). Inwhatfollows,wewillfocussolelyonsystemsoffirst-orderodes. Thismayseemoverlyrestrictive,untilwemakethefollowingobservation. 4 initial and boundary value problems Observation. Any nth-order ode can be written as a system of n first-order odes. Theprocessofdoingsoisstraightforward,asillustratedinthefollowing example: Example1.0.8. Considerthesecond-orderodey(cid:48)(cid:48)+(cosx)y(cid:48)+y2 =ex. Toavoid usingsecondderivatives,weintroduceanewdependentvariable z = y(cid:48) sothat z(cid:48) = y(cid:48)(cid:48). Our ode can be re-written as z(cid:48)+(cosx)z+y2 = ex. Thus, we have obtainedasystemoftwofirst-orderodes: dy dz = z, = −(cosx)z−y2+ex. dx dx A solution of the above system of odes on an open interval I is any vector of differentiable functions [y(x),z(x)] which simultaneously satisfy both odes when x ∈ I. Example1.0.9. Considerthesystem dy dz = z, = −y. dt dt WeclaimthatforanychoicesofconstantsC andC , 1 2 (cid:34) (cid:35) (cid:34) (cid:35) y(t) C cost+C sint = 1 2 z(t) −C sint+C cost 1 2 is a solution of the system. To verify this, assume that y and z have this form. Differentiationrevealsthat y(cid:48) = −C sint+C cost and z(cid:48) = −C cost−C sint. 1 2 1 2 Thus, y(cid:48) = z and z(cid:48) = −y,asrequired. 1.1. InitialandBoundaryValueProblems Inthepreviousexample,thesolutionofthesystemof odescontainsarbitrary constants C and C . Therefore, the system has infinitely many solutions. In 1 2 practice, one often has additional information about the underlying system, allowing us to select a particular solution of practical interest. For example, supposethatacupofcoffeeiscoolingoffandobeysNewton’sLawofCooling. Inordertopredictthecoffee’stemperatureatfuturetimes,wewouldneedto specifythetemperatureofthecoffeeatsomereferencetime(usuallyconsidered tobethe“initial”time). Byspecifyingauxiliaryconditionsthatsolutionsofan

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