Essentials of applied mathematics for scientists and engineers

180 Pages·2007·1.189 MB·English

by  Robert Watts

MOBK070-FM MOBKXXX-Sample.cls March22,2007 13:6
Essentials of Applied Mathematics
for Scientists and Engineers
MOBK070-FM MOBKXXX-Sample.cls March22,2007 13:6
Copyright© 2007byMorgan&Claypool
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedin
anyformorbyanymeans—electronic,mechanical,photocopy,recording,oranyotherexceptforbriefquotations
inprintedreviews,withoutthepriorpermissionofthepublisher.
EssentialsofAppliedMathematicsforScientistsandEngineers
RobertG.Watts
www.morganclaypool.com
ISBN:1598291866 paperback
ISBN:9781598291865 paperback
ISBN:1598291874 ebook
ISBN:9781598291872 ebook
DOI10.2200/S00082ED1V01Y200612ENG003
APublicationintheMorgan&ClaypoolPublishersseries
SYNTHESISLECTURESONENGINEERINGSEQUENCEINSERIES#3
Lecture#3
SeriesISSN: 1559-811X print
SeriesISSN: 1559-8128 electronic
FirstEdition
10987654321
MOBK070-FM MOBKXXX-Sample.cls March22,2007 13:6
Essentials of Applied Mathematics
for Scientists and Engineers
RobertG.Watts
TulaneUniversity
SYNTHESISLECTURESONENGINEERINGSEQUENCEINSERIES#3
M
&C &
M o r g a n C l a y p o o l P u b l i s h e r s
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iv
ABSTRACT
This is a book about linear partial differential equations that are common in engineering and
the physical sciences. It will be useful to graduate students and advanced undergraduates in
all engineering fields as well as students of physics, chemistry, geophysics and other physical
sciences and professional engineers who wish to learn about how advanced mathematics can
be used in their professions. The reader will learn about applications to heat transfer, fluid
flow, mechanical vibrations. The book is written in such a way that solution methods and
applicationtophysicalproblemsareemphasized.Therearemanyexamplespresentedindetail
and fully explained in their relation to the real world. References to suggested further reading
areincluded.Thetopicsthatarecoveredincludeclassicalseparationofvariablesandorthogonal
functions,Laplacetransforms,complexvariablesandSturm-Liouvilletransforms.
KEYWORDS
Engineering mathematics, separation of variables, orthogonal functions, Laplace transforms,
complexvariablesandSturm-Liouvilletransforms,differentialequations.
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Contents
1. PartialDifferentialEquationsinEngineering...................................1
1.1 IntroductoryComments .................................................. 1
1.2 FundamentalConcepts ................................................... 1
Problems......................................................3
1.3 TheHeatConduction(orDiffusion)Equation..............................3
1.3.1 RectangularCartesianCoordinates.................................3
1.3.2 CylindricalCoordinates ........................................... 5
1.3.3 SphericalCoordinates.............................................6
TheLaplacianOperator........................................6
1.3.4 BoundaryConditions..............................................7
1.4 TheVibratingString ..................................................... 7
1.4.1 BoundaryConditions..............................................8
1.5 VibratingMembrane..................................................... 8
1.6 LongitudinalDisplacementsofanElasticBar...............................9
FurtherReading..........................................................9
2. TheFourierMethod:SeparationofVariables ..................................11
2.1 HeatConduction ....................................................... 12
2.1.1 ScalesandDimensionlessVariables................................12
2.1.2 SeparationofVariables...........................................13
2.1.3 Superposition ................................................... 14
2.1.4 Orthogonality...................................................15
2.1.5 Lessons.........................................................15
Problems .................................................... 16
2.1.6 ScalesandDimensionlessVariables................................16
2.1.7 SeparationofVariables...........................................17
2.1.8 ChoosingtheSignoftheSeparationConstant......................17
2.1.9 Superposition ................................................... 19
2.1.10 Orthogonality...................................................19
2.1.11 Lessons.........................................................20
2.1.12 ScalesandDimensionlessVariables................................20
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vi ESSENTIALSOFAPPLIEDMATHEMATICSFORSCIENTISTSANDENGINEERS
2.1.13 GettingtoOneNonhomogeneousCondition ...................... 20
2.1.14 SeparationofVariables...........................................21
2.1.15 ChoosingtheSignoftheSeparationConstant......................21
2.1.16 Superposition ................................................... 22
2.1.17 Orthogonality...................................................22
2.1.18 Lessons.........................................................23
2.1.19 ScalesandDimensionlessVariables................................23
2.1.20 RelocatingtheNonhomogeneity..................................24
2.1.21 SeparatingVariables ............................................. 25
2.1.22 Superposition ................................................... 25
2.1.23 Orthogonality...................................................25
2.1.24 Lessons.........................................................26
Problems .................................................... 26
2.2 Vibrations..............................................................26
2.2.1 ScalesandDimensionlessVariables................................27
2.2.2 SeparationofVariables...........................................27
2.2.3 Orthogonality...................................................28
2.2.4 Lessons.........................................................29
Problems .................................................... 29
FurtherReading ........................................................ 29
3. OrthogonalSetsofFunctions.................................................31
3.1 Vectors.................................................................31
3.1.1 OrthogonalityofVectors.........................................31
3.1.2 OrthonormalSetsofVectors......................................32
3.2 Functions .............................................................. 32
3.2.1 OrthonormalSetsofFunctionsandFourierSeries..................32
3.2.2 BestApproximation..............................................34
3.2.3 ConvergenceofFourierSeries.....................................35
3.2.4 ExamplesofFourierSeries........................................36
Problems .................................................... 38
3.3 Sturm–LiouvilleProblems:OrthogonalFunctions..........................39
3.3.1 OrthogonalityofEigenfunctions..................................40
Problems .................................................... 42
FurtherReading ........................................................ 43
4. SeriesSolutionsofOrdinaryDifferentialEquations............................45
4.1 GeneralSeriesSolutions.................................................45
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4.1.1 Definitions......................................................45
4.1.2 OrdinaryPointsandSeriesSolutions..............................46
4.1.3 Lessons:FindingSeriesSolutionsforDifferentialEquations
withOrdinaryPoints.............................................48
Problems .................................................... 48
4.1.4 RegularSingularPointsandtheMethodofFrobenius...............49
4.1.5 Lessons:FindingSeriesSolutionforDifferentialEquationswith
RegularSingularPoints .......................................... 54
4.1.6 LogarithmsandSecondSolutions.................................55
Problems .................................................... 57
4.2 BesselFunctions........................................................58
4.2.1 SolutionsofBessel’sEquation.....................................58
HerearetheRules............................................61
4.2.2 Fourier–BesselSeries.............................................64
Problems .................................................... 68
4.3 LegendreFunctions.....................................................69
4.4 AssociatedLegendreFunctions...........................................72
Problems .................................................... 73
FurtherReading ........................................................ 74
5. SolutionsUsingFourierSeriesandIntegrals...................................75
5.1 Conduction(orDiffusion)Problems......................................75
5.1.1 Time-DependentBoundaryConditions............................80
5.2 VibrationsProblems.....................................................83
Problems .................................................... 88
5.3 FourierIntegrals........................................................89
Problem.....................................................93
FurtherReading ........................................................ 93
6. IntegralTransforms:TheLaplaceTransform..................................95
6.1 TheLaplaceTransform..................................................95
6.2 SomeImportantTransforms.............................................96
6.2.1 Exponentials....................................................96
6.2.2 Shiftinginthes-domain ......................................... 96
6.2.3 ShiftingintheTimeDomain.....................................96
6.2.4 SineandCosine.................................................97
6.2.5 HyperbolicFunctions............................................97
6.2.6 Powersoft:tm .................................................. 97
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viii ESSENTIALSOFAPPLIEDMATHEMATICSFORSCIENTISTSANDENGINEERS
6.2.7 HeavisideStep .................................................. 99
6.2.8 TheDiracDeltaFunction.......................................100
6.2.9 TransformsofDerivatives.......................................100
6.2.10 LaplaceTransformsofIntegrals..................................101
6.2.11 DerivativesofTransforms.......................................101
6.3 LinearOrdinaryDifferentialEquationswithConstantCoefficients.........102
6.4 SomeImportantTheorems ............................................. 103
6.4.1 InitialValueTheorem...........................................103
6.4.2 FinalValueTheorem ........................................... 103
6.4.3 Convolution ................................................... 103
6.5 PartialFractions ....................................................... 104
6.5.1 NonrepeatingRoots ............................................ 104
6.5.2 RepeatedRoots.................................................107
6.5.3 QuadraticFactors:ComplexRoots...............................108
Problems ................................................... 109
FurtherReading.......................................................110
7. ComplexVariablesandtheLaplaceInversionIntegral.........................111
7.1 BasicProperties........................................................111
7.1.1 LimitsandDifferentiationofComplexVariables:
AnalyticFunctions..............................................115
Integrals....................................................117
7.1.2 TheCauchyIntegralFormula....................................118
Problems ................................................... 120
8. SolutionswithLaplaceTransforms...........................................121
8.1 MechanicalVibrations..................................................121
Problems ................................................... 125
8.2 DiffusionorConductionProblems ...................................... 125
Problems ................................................... 134
8.3 Duhamel’sTheorem....................................................135
Problems ................................................... 138
FurtherReading.......................................................139
9. Sturm–LiouvilleTransforms.................................................141
9.1 APreliminaryExample:FourierSineTransform..........................141
9.2 Generalization:TheSturm–LiouvilleTransform:Theory..................143
9.3 TheInverseTransform.................................................146
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CONTENTS ix
Problems ................................................... 151
FurtherReading.......................................................151
10. IntroductiontoPerturbationMethods........................................153
10.1 ExamplesfromAlgebra.................................................153
10.1.1 RegularPerturbation............................................153
10.1.2 SingularPerturbation...........................................155
AppendixA:TheRootsofCertainTranscendentalEquations..................159
AppendixB:................................................................165
AuthorBiography...........................................................169
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