Boundary value problems and partial differential equations

515 Pages·2005·2.367 MB·English

by  David L. Powers

B O U N D A R Y
VALUE PROBLEMS
FIFTH EDITION
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B O U N D A R Y
VALUE PROBLEMS
ANDPARTIALDIFFERENTIALEQUATIONS
DAVID L. POWERS
C l a r k s o n U n i v e r s i t y
FIFTH EDITION
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Contents
Preface ix
CHAPTER0 OrdinaryDifferentialEquations 1
0.1 HomogeneousLinearEquations 1
0.2 NonhomogeneousLinearEquations 14
0.3 BoundaryValueProblems 26
0.4 SingularBoundaryValueProblems 38
0.5 Green’sFunctions 43
ChapterReview 51
MiscellaneousExercises 51
CHAPTER1 FourierSeriesandIntegrals 59
1.1 PeriodicFunctionsandFourierSeries 59
1.2 ArbitraryPeriodandHalf-RangeExpansions 64
1.3 ConvergenceofFourierSeries 73
1.4 UniformConvergence 79
1.5 OperationsonFourierSeries 85
1.6 MeanErrorandConvergenceinMean 90
1.7 ProofofConvergence 95
1.8 NumericalDeterminationofFourierCoefficients 100
1.9 FourierIntegral 106
1.10 ComplexMethods 113
1.11 ApplicationsofFourierSeriesandIntegrals 117
1.12 CommentsandReferences 124
ChapterReview 125
MiscellaneousExercises 125
v
vi Contents
CHAPTER2 TheHeatEquation 135
2.1 DerivationandBoundaryConditions 135
2.2 Steady-StateTemperatures 143
2.3 Example:FixedEndTemperatures 149
2.4 Example:InsulatedBar 157
2.5 Example:DifferentBoundaryConditions 163
2.6 Example:Convection 170
2.7 Sturm–LiouvilleProblems 175
2.8 ExpansioninSeriesofEigenfunctions 181
2.9 GeneralitiesontheHeatConductionProblem 184
2.10 Semi-InfiniteRod 188
2.11 InfiniteRod 193
2.12 TheErrorFunction 199
2.13 CommentsandReferences 204
ChapterReview 206
MiscellaneousExercises 206
CHAPTER3 TheWaveEquation 215
3.1 TheVibratingString 215
3.2 SolutionoftheVibratingStringProblem 218
3.3 d’Alembert’sSolution 227
3.4 One-DimensionalWaveEquation:Generalities 233
3.5 EstimationofEigenvalues 236
3.6 WaveEquationinUnboundedRegions 239
3.7 CommentsandReferences 246
ChapterReview 247
MiscellaneousExercises 247
CHAPTER4 ThePotentialEquation 255
4.1 PotentialEquation 255
4.2 PotentialinaRectangle 259
4.3 FurtherExamplesforaRectangle 264
4.4 PotentialinUnboundedRegions 270
4.5 PotentialinaDisk 275
4.6 ClassificationandLimitations 280
4.7 CommentsandReferences 283
ChapterReview 285
MiscellaneousExercises 285
CHAPTER5 HigherDimensionsandOtherCoordinates 295
5.1 Two-DimensionalWaveEquation:Derivation 295
5.2 Three-DimensionalHeatEquation 298
5.3 Two-DimensionalHeatEquation:Solution 303
Contents vii
5.4 ProblemsinPolarCoordinates 308
5.5 Bessel’sEquation 311
5.6 TemperatureinaCylinder 316
5.7 VibrationsofaCircularMembrane 321
5.8 SomeApplicationsofBesselFunctions 329
5.9 SphericalCoordinates;LegendrePolynomials 335
5.10 SomeApplicationsofLegendrePolynomials 345
5.11 CommentsandReferences 353
ChapterReview 354
MiscellaneousExercises 354
CHAPTER6 LaplaceTransform 363
6.1 DefinitionandElementaryProperties 363
6.2 PartialFractionsandConvolutions 369
6.3 PartialDifferentialEquations 376
6.4 MoreDifficultExamples 383
6.5 CommentsandReferences 389
MiscellaneousExercises 389
CHAPTER7 NumericalMethods 397
7.1 BoundaryValueProblems 397
7.2 HeatProblems 403
7.3 WaveEquation 408
7.4 PotentialEquation 414
7.5 Two-DimensionalProblems 420
7.6 CommentsandReferences 428
MiscellaneousExercises 428
Bibliography 433
Appendix:MathematicalReferences 435
AnswerstoOdd-NumberedExercises 441
Index 495
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Preface
This textis designed for a one-semester or two-quarter course in partial dif-
ferentialequationsgiventothird-andfourth-yearstudentsofengineeringand
science.Itcanalsobeusedasthebasisforanintroductorycourseforgraduate
students.Mathematicalprerequisiteshavebeenkepttoaminimum—calculus
anddifferentialequations.Vectorcalculusisusedforonlyonederivation,and
necessarylinearalgebraislimitedtodeterminantsofordertwo.Areaderneeds
enoughbackgroundinphysicstofollowthederivationsoftheheatandwave
equations.
The principal objective of the book is solving boundary value problems
involving partial differential equations. Separation of variables receives the
greatestattentionbecauseitiswidelyusedinapplicationsandbecauseitpro-
vides a uniform method for solving important cases of the heat, wave, and
potentialequations.Onetechniqueisnotenough,ofcourse.D’Alembert’sso-
lution of the wave equation is developed in parallel with the series solution,
and the distributed-source solution is constructed for the heat equation. In
addition,therearechaptersonLaplacetransformtechniquesandonnumeri-
calmethods.
Thesecond objectiveistotietogetherthemathematicsdevelopedandthe
student’sphysicalintuition.Thisisaccomplishedbyderivingthemathemati-
calmodelinanumberofcases,byusingphysicalreasoninginthemathemat-
icaldevelopment,byinterpretingmathematicalresultsinphysicalterms,and
bystudyingtheheat,wave,andpotentialequationsseparately.
Intheserviceofbothobjectives,therearemanyfullyworkedexamplesand
nowabout900exercises,includingmiscellaneousexercisesattheendofeach
chapter. The level of difficulty ranges from drill and verification of details
to development of new material. Answers to odd-numbered exercises are in
ix