Ordinary and Partial Differential Equations

416 Pages·2010·2.9 MB·English

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.
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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