Methods of Applied Mathematics for Engineers and Scientists

897 Pages·2013·12.12 MB·English

by  Tomas B. Co

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METHODS OF APPLIED MATHEMATICS FOR
ENGINEERS AND SCIENTISTS
Basedoncoursenotesfrommorethantwentyyearsofteachingengi-
neering and physical sciences at Michigan Technological University,
TomasCo’sengineeringmathematicstextbookisrichwithexamples,
applications, and exercises. Professor Co uses analytical approaches
tosolvesmallerproblemstoprovidemathematicalinsightandunder-
standing,andnumericalmethodsforlargeandcomplexproblems.The
book emphasizes applying matrices with strong attention to matrix
structure and computational issues such as sparsity and efficiency.
Chapters on vector calculus and integral theorems are used to build
coordinate-freephysicalmodels,withspecialemphasisonorthogonal
coordinates. Chapters on ordinary differential equations and partial
differentialequationscoverbothanalyticalandnumericalapproaches.
Topics on analytical solutions include similarity transform methods,
direct formulas for series solutions, bifurcation analysis, Lagrange-
Charpit formulas, shocks/rarefaction, and others. Topics on numeri-
calmethodsincludestabilityanalysis,differentialalgebraicequations,
high-order finite-difference formulas, Delaunay meshes, and others.
MATLAB implementations of the methods and concepts are fully
integrated.
Tomas Co is an associate professor of chemical engineering at
MichiganTechnologicalUniversity.AftercompletinghisPhDinchem-
icalengineeringattheUniversityofMassachusettsatAmherst,hewas
a postdoctoral researcher at Lehigh University, a visiting researcher
at Honeywell Corp., and a visiting professor at Korea University.
Hehasbeenteachingappliedmathematicstograduateandadvanced
undergraduate students at Michigan Tech for more than twenty
years. His research areas include advanced process control, includ-
ing plantwide control, nonlinear control, and fuzzy logic. His journal
publications span broad areas in such journals as IEEE Transactions
in Automatic Control, Automatica, AIChE Journal, Computers in
Chemical Engineering, and Chemical Engineering Progress. He has
been nominated twice for the Distinguished Teaching Awards at
Michigan Tech and is a member of the Michigan Technological
UniversityAcademyofTeachingExcellence.
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CUUS1935-FM cuus1935/Co 9781107004122 February14,2013 22:13
Methods of Applied Mathematics for
Engineers and Scientists
Tomas B. Co
MichiganTechnologicalUniversity
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CUUS1935-FM cuus1935/Co 9781107004122 February14,2013 22:13
cambridgeuniversitypress
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,MexicoCity
CambridgeUniversityPress
32AvenueoftheAmericas,NewYork,NY10013-2473,USA
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107004122
©TomasB.Co2013
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2013
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationData
Co,TomasB.,1959–
Methodsofappliedmathematicsforengineersandscientists:analyticaland
numericalapproaches/TomasB.Co.,MichiganTechnologicalUniversity.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-1-107-00412-2(hardback)
1.Matrices. 2.Differentialequations–Numericalsolutions. I.Title.
QA188.C63 2013
512.9(cid:2)434–dc23 2012043979
ISBN978-1-107-00412-2Hardback
Additionalresourcesforthispublicationat[insertURLhere].
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyInternetWebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchWebsitesis,orwillremain,accurate
orappropriate.
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Contents
Preface pagexi
I MATRIXTHEORY
1 MatrixAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 DefinitionsandNotations 4
1.2 FundamentalMatrixOperations 6
1.3 PropertiesofMatrixOperations 18
1.4 BlockMatrixOperations 30
1.5 MatrixCalculus 31
1.6 SparseMatrices 39
1.7 Exercises 41
2 SolutionofMultipleEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.1 Gauss-JordanElimination 55
2.2 LUDecomposition 59
2.3 DirectMatrixSplitting 65
2.4 IterativeSolutionMethods 66
2.5 Least-SquaresSolution 71
2.6 QRDecomposition 77
2.7 ConjugateGradientMethod 78
2.8 GMRES 79
2.9 Newton’sMethod 80
2.10 EnhancedNewtonMethodsviaLineSearch 82
2.11 Exercises 86
3 MatrixAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.1 MatrixOperators 100
3.2 EigenvaluesandEigenvectors 107
3.3 PropertiesofEigenvaluesandEigenvectors 113
3.4 SchurTriangularizationandNormalMatrices 116
3.5 Diagonalization 117
3.6 JordanCanonicalForm 118
3.7 FunctionsofSquareMatrices 120
v
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vi Contents
3.8 StabilityofMatrixOperators 124
3.9 SingularValueDecomposition 127
3.10 PolarDecomposition 132
3.11 MatrixNorms 135
3.12 Exercises 138
II VECTORSANDTENSORS
4 VectorandTensorAlgebraandCalculus. . . . . . . . . . . . . . . . . . . .149
4.1 NotationsandFundamentalOperations 150
4.2 VectorAlgebraBasedonOrthonormalBasisVectors 154
4.3 TensorAlgebra 157
4.4 MatrixRepresentationofVectorsandTensors 162
4.5 DifferentialOperationsforVectorFunctionsofOneVariable 164
4.6 ApplicationtoPositionVectors 165
4.7 DifferentialOperationsforVectorFields 169
4.8 CurvilinearCoordinateSystem:CylindricalandSpherical 184
4.9 OrthogonalCurvilinearCoordinates 189
4.10 Exercises 196
5 VectorIntegralTheorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .204
5.1 Green’sLemma 205
5.2 DivergenceTheorem 208
5.3 Stokes’TheoremandPathIndependence 210
5.4 Applications 215
5.5 LeibnitzDerivativeFormula 224
5.6 Exercises 225
III ORDINARYDIFFERENTIALEQUATIONS
6 AnalyticalSolutionsofOrdinaryDifferentialEquations . . . . . . . . . . 235
6.1 First-OrderOrdinaryDifferentialEquations 236
6.2 SeparableFormsviaSimilarityTransformations 238
6.3 ExactDifferentialEquationsviaIntegratingFactors 242
6.4 Second-OrderOrdinaryDifferentialEquations 245
6.5 MultipleDifferentialEquations 250
6.6 DecoupledSystemDescriptionsviaDiagonalization 258
6.7 LaplaceTransformMethods 262
6.8 Exercises 263
7 NumericalSolutionofInitialandBoundaryValueProblems . . . . . . . 273
7.1 EulerMethods 274
7.2 RungeKuttaMethods 276
7.3 MultistepMethods 282
7.4 DifferenceEquationsandStability 291
7.5 BoundaryValueProblems 299
7.6 DifferentialAlgebraicEquations 303
7.7 Exercises 305
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Contents vii
8 QualitativeAnalysisofOrdinaryDifferentialEquations. . . . . . . . . .311
8.1 ExistenceandUniqueness 312
8.2 AutonomousSystemsandEquilibriumPoints 313
8.3 IntegralCurves,PhaseSpace,Flows,andTrajectories 314
8.4 LyapunovandAsymptoticStability 317
8.5 Phase-PlaneAnalysisofLinearSecond-Order
AutonomousSystems 321
8.6 LinearizationAroundEquilibriumPoints 327
8.7 MethodofLyapunovFunctions 330
8.8 LimitCycles 332
8.9 BifurcationAnalysis 340
8.10 Exercises 340
9 SeriesSolutionsofLinearOrdinaryDifferentialEquations . . . . . . . . 347
9.1 PowerSeriesSolutions 347
9.2 LegendreEquations 358
9.3 BesselEquations 363
9.4 PropertiesandIdentitiesofBesselFunctionsand
ModifiedBesselFunctions 369
9.5 Exercises 371
IV PARTIALDIFFERENTIALEQUATIONS
10 First-OrderPartialDifferentialEquationsandtheMethodof
Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
10.1 TheMethodofCharacteristics 380
10.2 AlternateFormsandGeneralSolutions 387
10.3 TheLagrange-CharpitMethod 389
10.4 ClassificationBasedonPrincipalParts 393
10.5 HyperbolicSystemsofEquations 397
10.6 Exercises 399
11 LinearPartialDifferentialEquations . . . . . . . . . . . . . . . . . . . . . . 405
11.1 LinearPartialDifferentialOperator 406
11.2 ReducibleLinearPartialDifferentialEquations 408
11.3 MethodofSeparationofVariables 411
11.4 NonhomogeneousPartialDifferentialEquations 431
11.5 SimilarityTransformations 439
11.6 Exercises 443
12 IntegralTransformMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
12.1 GeneralIntegralTransforms 451
12.2 FourierTransforms 452
12.3 SolutionofPDEsUsingFourierTransforms 459
12.4 LaplaceTransforms 464
12.5 SolutionofPDEsUsingLaplaceTransforms 474
12.6 MethodofImages 476
12.7 Exercises 477
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viii Contents
13 FiniteDifferenceMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
13.1 FiniteDifferenceApproximations 484
13.2 Time-IndependentEquations 491
13.3 Time-DependentEquations 504
13.4 StabilityAnalysis 512
13.5 Exercises 519
14 MethodofFiniteElements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
14.1 TheWeakForm 524
14.2 TriangularFiniteElements 527
14.3 AssemblyofFiniteElements 533
14.4 MeshGeneration 539
14.5 SummaryofFiniteElementMethod 541
14.6 AxisymmetricCase 546
14.7 Time-DependentSystems 547
14.8 Exercises 552
Bibliography B-1
Index I-1
A AdditionalDetailsandFortificationforChapter1 . . . . . . . . . . . . . . 561
A.1 MatrixClassesandSpecialMatrices 561
A.2 MotivationforMatrixOperationsfromSolutionofEquations 568
A.3 TaylorSeriesExpansion 572
A.4 ProofsforLemmaandTheoremsofChapter1 576
A.5 PositiveDefiniteMatrices 586
B AdditionalDetailsandFortificationforChapter2 . . . . . . . . . . . . . 589
B.1 GaussJordanEliminationAlgorithm 589
B.2 SVDtoDetermineGauss-JordanMatricesQandW 594
B.3 BooleanMatricesandReducibleMatrices 595
B.4 ReductionofMatrixBandwidth 600
B.5 BlockLUDecomposition 602
B.6 MatrixSplitting:DiakopticMethodandSchur
ComplementMethod 605
B.7 LinearVectorAlgebra:FundamentalConcepts 611
B.8 DeterminationofLinearIndependenceofFunctions 614
B.9 Gram-SchmidtOrthogonalization 616
B.10ProofsforLemmaandTheoremsinChapter2 617
B.11ConjugateGradientAlgorithm 620
B.12GMRESalgorithm 629
B.13Enhanced-NewtonUsingDouble-DoglegMethod 635
B.14NonlinearLeastSquaresviaLevenberg-Marquardt 639
C AdditionalDetailsandFortificationforChapter3 . . . . . . . . . . . . . . 644
C.1 ProofsofLemmasandTheoremsofChapter3 644
C.2 QRMethodforEigenvalueCalculations 649
C.3 CalculationsfortheJordanDecomposition 655
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Contents ix
C.4 SchurTriangularizationandSVD 658
C.5 Sylvester’sMatrixTheorem 659
C.6 DanilevskiiMethodforCharacteristicPolynomial 660
D AdditionalDetailsandFortificationforChapter4 . . . . . . . . . . . . . . 664
D.1 ProofsofIdentitiesofDifferentialOperators 664
D.2 DerivationofFormulasinCylindricalCoordinates 666
D.3 DerivationofFormulasinSphericalCoordinates 669
E AdditionalDetailsandFortificationforChapter5 . . . . . . . . . . . . . . 673
E.1 LineIntegrals 673
E.2 SurfaceIntegrals 678
E.3 VolumeIntegrals 684
E.4 Gauss-LegendreQuadrature 687
E.5 ProofsofIntegralTheorems 691
F AdditionalDetailsandFortificationforChapter6 . . . . . . . . . . . . . . 700
F.1 SupplementalMethodsforSolvingFirst-OrderODEs 700
F.2 SingularSolutions 703
F.3 FiniteSeriesSolutionofdx/dt =Ax+b(t) 705
F.4 ProofforLemmasandTheoremsinChapter6 708
G AdditionalDetailsandFortificationforChapter7 . . . . . . . . . . . . . . 715
G.1 DifferentialEquationSolversinMATLAB 715
G.2 DerivationofFourth-OrderRungeKuttaMethod 718
G.3 Adams-BashforthParameters 722
G.4 VariableStepSizesforBDF 723
G.5 ErrorControlbyVaryingStepSize 724
G.6 ProofofSolutionofDifferenceEquation,Theorem7.1 730
G.7 NonlinearBoundaryValueProblems 731
G.8 RicattiEquationMethod 734
H AdditionalDetailsandFortificationforChapter8 . . . . . . . . . . . . . . 738
H.1 BifurcationAnalysis 738
I AdditionalDetailsandFortificationforChapter9 . . . . . . . . . . . . . . 745
I.1 DetailsonSeriesSolutionofSecond-OrderSystems 745
I.2 MethodofOrderReduction 748
I.3 ExamplesofSolutionofRegularSingularPoints 750
I.4 SeriesSolutionofLegendreEquations 753
I.5 SeriesSolutionofBesselEquations 757
I.6 ProofsforLemmasandTheoremsinChapter9 761
J AdditionalDetailsandFortificationforChapter10 . . . . . . . . . . . . . 771
J.1 ShocksandRarefaction 771
J.2 ClassificationofSecond-OrderSemilinearEquations:n >2 781
J.3 ClassificationofHigh-OrderSemilinearEquations 784