Advanced Mathematics for Engineers and Physicists

833 Pages·2023·10 MB·English

by  Sever Angel Popescu, Marilena Jianu

Sever Angel Popescu
Marilena Jianu
Advanced
Mathematics
for Engineers
and Physicists
Advanced Mathematics for Engineers
and Physicists
Sever Angel Popescu • Marilena Jianu
Advanced Mathematics
for Engineers and Physicists
SeverAngelPopescu MarilenaJianu
DepartmentofMathematicsandComputer DepartmentofMathematicsandComputer
Science Science
TechnicalUniversityofCivilEngineering TechnicalUniversityofCivilEngineering
ofBucharest ofBucharest
Bucharest,Romania Bucharest,Romania
ISBN978-3-031-21501-8 ISBN978-3-031-21502-5 (eBook)
https://doi.org/10.1007/978-3-031-21502-5
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Preface
This book is designed to be an introductory course to some important chapters
of Advanced Mathematics for Engineering and Physics students: Differential
Equations,FourierSeries,LaplaceandFourierTransforms,CalculusofVariations,
and Probability Theory. It requires a complete course in Mathematical Analysis
and in Linear Algebra. We mostly tried to use the engineering intuition instead
of insisting on mathematical tricks. Even if the book was first of all intended for
engineeringstudents, it may be successfully used by other students and scientists
as an introduction to Applied Mathematics. The main feature of the material
presented here is its clarity, motivation, and the genuine desire of the authors to
makeextremelytransparentthe“mysterious”mathematicaltoolswhichareusedto
describeandorganizethegreatvarietyofimpressionsofoursearchingMindinthe
infinitehiddencornersofNature.Wehavealwayshadinmindthecommonstudent
who really tries to understand the process of mathematical modeling of various
physical phenomena, because we are fully convinced that mathematics education
doesnotmeanlearning(lotsof)formulasbyheart,buttheconsciousunderstanding
of the meaning of these formulas. We believe that this course is also useful for
mathematicsstudentsbecause the presentationis elementary,but rigorousenough
tosatisfythetasteofamathematician.
In the first three chapters, we offer a detailed study of ordinary differential
equations.InChap.3, we beginby presentingthegeneralframework,focusingon
first-orderordinarydifferentialequations.Weshowbyseveralexampleshowdiffer-
entialequationscandescribe (model)physicalphenomenain mathematicalterms.
Themostcommontypesoffirst-orderdifferentialequationsarepresented,together
withthetechniquesforobtainingtheirsolutions(eitherinimplicitorexplicitform).
The last section is dedicated to the theorem of existence and uniqueness of the
solution of the Cauchy problem for first-order differentialequations. Chapter 2 is
devotedtohigher-orderdifferentialequations,mainlylineardifferentialequations.
Weintroducethevectorspaceofthesolutionsofahomogeneouslineardifferential
equationandtheaffinespaceofthesolutionsforthenon-homogeneouscase.Linear
equations with constant coefficients are placed in the foreground, because in this
case we have a closed-form solution for homogeneousequations as well as for a
v
vi Preface
wideclassofnon-homogeneousequations.WealsopresenttheEuler-typeequations
whichhavevariablecoefficientsbutcan betransformedintolinear equationswith
constantcoefficients.ThesystemsofdifferentialequationsarepresentedinChap.3.
We state andprovethe existenceanduniquenesstheoremforthe Cauchyproblem
in this case. Particular attention is paid to linear systems of differentialequations
and to the relation between a linear system of n equations and an n-th order
linearequation.Autonomoussystemsarealsostudied,andsincefirst-orderpartial
differentialequationsarecloselyrelatedtoautonomoussystems,wehavechosento
presenttheminthischapter.
In Chap.4 (Fourier series), we discuss the expansion of periodic functions in
infinite series of sines and cosines and introduce the general orthogonal systems
of functions. Fourier series will be used in Chap.7 to solve partial differential
equations such as the wave equation, the heat equation, or the Laplace equation.
Fourier series are used to represent periodic functions (or functions defined on
a finite interval, which can be extended by periodicity to the entire axis). The
Fouriertransformgeneralizesthisideatotheintegralrepresentationofnonperiodic
functionsdefinedonthewholesetofrealnumbers.WepresenttheFouriertransform
and some of its applications in Chap.5. The discrete Fourier transform, which
can be used when a function is given only in terms of values at a finite number
of points, is also introduced. Another important mathematical tool is the Laplace
transform,presentedinChap.6.AsinthecaseoftheFouriertransform,theLaplace
transform simplifies the solution of linear differential equations by transforming
themintoalgebraicequations.Itisalsoappliedforthesolutionofpartialdifferential
equations—in Chap.7, we use the Laplace transformfor the solution of the finite
vibrating string equation. By using the Heaviside step function or the Dirac delta
function,theLaplacetransformcanbeappliedinproblemswherethefreetermhas
some discontinuities or represents short impulses. Chapter 7 is an introduction to
the field of the “Equations of Mathematical Physics,” the most important partial
differential equations used in Physics and Engineering. We begin by classifying
theseequationsandintroducethecharacteristicsmethodtofindthecanonicalform
of a quasilinear second-order partial differential equation. Then we present the
solutionoftheone-dimensionalandtwo-dimensionalwaveequation,theheatflow
equation,and the Dirichlet problemfor Laplace’s equation.Althoughthis chapter
is mainly devoted to second-order PDE, we have decided to include here also a
fourth-orderpartialdifferentialequation—thesimplysupportedbeamequation.
Chapter 8 containsthe basic theoryofCalculus ofVariationsappliedto funda-
mentaltypesofvariationalproblemswithapplicationsinphysicsandengineering.
We begin by stating several classical problems (such as the brachistochrone
problem,theminimalsurfaceofrevolution,Hamilton’sprincipleoftheleastaction,
andDido’sproblem).Thenweintroducethegeneralframeofcalculusofvariation,
focusing on necessary conditions of extremum of a functional. We deduce the
basicdifferentialequationsofcalculusofvariationsandapplythemtosolvesome
classicalvariationalproblems.
InChap.9,wemakeanelementaryintroductiontoprobabilitytheory.Wepresent
Laplace’s and Kolmogorov’sdefinitions of probabilityand show how they can be
Preface vii
applied to solve practical problems. We introduce the conditional probability and
the Bayes formula, providing also numerous applications. We define the discrete
randomvariablesandthecontinuousrandomvariables,emphasizingthemostused
distributionsofdiscretetype(Bernoulli,binomial,Poisson),andofcontinuoustype
(normal, gamma, chi-squared, student). We also present the most important limit
theorems:the weak law and the strong law of large numbersand the central limit
theorem,highlightingtheroleplayedbythenormaldistribution.Eachchapterhasa
finalsectionwithexercises.Thecompletesolutionsoftheseproblemsarepresented
stepbystepinChap.10.
Considering the great diversity of topics covered in this book, we have tried
to make it as self-contained and unitary as possible. This is why we have added
a supplementary chapter—Chap.11—which contains the basic theory in some
areas of Linear Algebra, Calculus, and Complex Analysis, necessary for a deep
understandingof the materialpresentedin the book.Thus, the first section of this
chaptercontainselementaryresultsonmetric,normed,Banach,andHilbertspaces,
while the secondsectionprovidesa briefintroductionto complexanalysis, with a
specialfocusonthecalculusofresidues(usedinthecalculationoftheFourierand
Laplacetransforms).
Itisnotpossibletoexpressourgratitudetoallthosewhocontributedindirectly
to the writing of this book. A special role in the formationof the first author as a
mathematicianwasplayedbyhisformerProfessorofAlgebraandNumberTheory,
hisPhDsupervisor,andhisspiritualmaster,Dr.doc.NicolaePopescu.
WewouldliketoexpressoursinceregratefulnesstoProfessorDr.OctavOlteanu,
Professor Dr. Gavriil Pa˘ltineanu, Professor Dr. Ghiocel Groza, and the reviewers
fortheir considerableattentionandpatiencein readingthe manuscriptand forthe
valuablesuggestionsandcommentsthatgreatlyimprovedtheinitialversionofthis
book.
Our familieswere so patientduringthe complicatedandlengthyconceptionof
thiswriting.Wethankthemallfortheirlovingsupport.
We shallbeverygratefultoallthereaderswhowillletusknowaboutpossible
mistakes or some of their particular opinions related to the topics of this book,
becauseimprovementisthemostimportanthumanactivity.
Bucharest,Romania SeverAngelPopescu
September2022 MarilenaJianu
Contents
1 First-OrderDifferentialEquations ...................................... 1
1.1 IntroductiontoOrdinaryDifferentialEquations.................... 1
1.2 SeparableEquations.................................................. 7
1.3 HomogeneousEquations............................................. 13
1.4 First-OrderLinearDifferentialEquations........................... 16
1.5 BernoulliEquations.................................................. 20
1.6 RiccatiEquations..................................................... 22
1.7 ExactDifferentialEquations......................................... 24
1.8 LagrangeEquationsandClairautEquations........................ 29
1.9 ExistenceandUniquenessofSolutionoftheCauchyProblem.... 34
1.10 Exercises.............................................................. 42
2 Higher-OrderDifferentialEquations .................................... 45
2.1 Introduction........................................................... 45
2.2 HomogeneousLinearDifferentialEquationsofOrdern........... 57
2.3 Non-HomogeneousLinearDifferentialEquationsof
Ordern................................................................ 64
2.4 HomogeneousLinearEquationswithConstantCoefficients ...... 68
2.5 NonhomogeneousLinearEquationswithConstant
Coefficients........................................................... 82
2.6 EulerEquations....................................................... 93
2.7 Exercises.............................................................. 97
3 SystemsofDifferentialEquations ........................................ 101
3.1 Introduction........................................................... 101
3.2 First-OrderSystemsandDifferentialEquationsofOrdern........ 109
3.3 LinearSystemsofDifferentialEquations........................... 114
3.4 LinearSystemswithConstantCoefficients......................... 125
3.4.1 TheHomogeneousCase(theAlgebraicMethod)......... 125
3.4.2 TheNon-HomogeneousCase(theMethodof
UndeterminedCoefficients)................................ 138
ix
x Contents
3.4.3 MatrixExponentialandLinearSystemswith
ConstantCoefficients....................................... 148
3.4.4 EliminationMethodforLinearSystemswith
ConstantCoefficients....................................... 164
3.5 AutonomousSystemsofDifferentialEquations.................... 170
3.6 First-OrderPartialDifferentialEquations........................... 178
3.6.1 LinearHomogeneousFirst-OrderPDE.................... 179
3.6.2 QuasilinearFirst-OrderPartialDifferentialEquations.... 183
3.7 Exercises.............................................................. 186
4 FourierSeries............................................................... 191
4.1 Introduction:Periodic,PiecewiseSmoothFunctions............... 191
4.1.1 PeriodicFunctions.......................................... 192
4.1.2 PiecewiseContinuousandPiecewiseSmooth
Functions.................................................... 195
4.2 FourierSeriesExpansions ........................................... 198
4.2.1 SeriesofFunctions ......................................... 199
4.2.2 ABasicTrigonometricSystem ............................ 201
4.2.3 FourierCoefficients......................................... 203
4.3 OrthogonalSystemsofFunctions................................... 205
4.3.1 InnerProduct................................................ 205
4.3.2 BestApproximationintheMean:Bessel’sInequality.... 208
4.4 TheConvergenceofFourierSeries.................................. 212
4.5 DifferentiationandIntegrationoftheFourierSeries............... 225
4.6 TheConvergenceintheMean:CompleteSystems................. 227
4.7 ExamplesofFourierExpansions.................................... 235
4.8 TheComplexformoftheFourierSeries............................ 241
4.9 Exercises.............................................................. 246
5 FourierTransform.......................................................... 249
5.1 ImproperIntegrals.................................................... 249
5.2 TheFourierIntegralFormula........................................ 258
5.3 TheFourierTransform............................................... 265
5.4 SolvingLinearDifferentialEquations .............................. 284
5.5 MomentsTheorems.................................................. 288
5.6 SamplingTheorem................................................... 297
5.7 DiscreteFourierTransform.......................................... 298
5.8 Exercises.............................................................. 302
6 LaplaceTransform ......................................................... 305
6.1 Introduction........................................................... 305
6.2 PropertiesoftheLaplaceTransform ................................ 312
6.3 InverseLaplaceTransform........................................... 332
6.4 SolvingLinearDifferentialEquations .............................. 344
6.5 TheDiracDeltaFunction............................................ 352
6.6 Exercises.............................................................. 358
Contents xi
7 Second-OrderPartialDifferentialEquations ........................... 359
7.1 Classification:CanonicalForm...................................... 359
7.2 TheWaveEquation................................................... 378
7.2.1 InfiniteVibratingString:D’AlembertFormula........... 381
7.2.2 FiniteVibratingString:FourierMethod................... 385
7.2.3 LaplaceTransformMethodfortheVibratingString...... 398
7.2.4 Vibrations of a Rectangular Membrane:
Two-DimensionalWaveEquation ......................... 400
7.3 VibrationsofaSimplySupportedBeam:FourierMethod......... 404
7.4 TheHeatEquation ................................................... 409
7.4.1 ModelingtheHeatFlowfromaBodyinSpace........... 409
7.4.2 HeatFlowinaFiniteRod:FourierMethod............... 411
7.4.3 HeatFlowinanInfiniteRod............................... 415
7.4.4 HeatFlowinaRectangularPlate.......................... 417
7.5 TheLaplace’sEquation.............................................. 422
7.5.1 DirichletProblemforaRectangle......................... 423
7.5.2 DirichletProblemforaDisk............................... 425
7.6 Exercises.............................................................. 432
8 IntroductiontotheCalculusofVariations............................... 435
8.1 ClassicalVariationalProblems ...................................... 435
8.2 GeneralFrameofC(cid:2)alculusofVariations............................ 443
8.3 TheCaseF[y]= bF(x,y,y(cid:2))dx................................. 448
(cid:2)a
8.4 TheCaseF[y]= bF(x,y,y(cid:2),...,y(n))dx....................... 456
a (cid:2)
8.5 TheCaseF[y1,..(cid:2).(cid:2),yn(cid:3)]= abF(x,y1(cid:4),...,yn,y1(cid:2),...,yn(cid:2))dx ... 460
8.6 TheCaseF[z]= F x,y,z, ∂z, ∂z dxdy...................... 467
∂x ∂y
D
8.7 IsoperimetricProblemsandGeodesicProblems.................... 472
8.7.1 IsoperimetricProblems..................................... 472
8.7.2 GeodesicProblems......................................... 477
8.8 Exercises.............................................................. 482
9 ElementsofProbabilityTheory........................................... 485
9.1 SampleSpace:EventSpace.......................................... 485
9.2 ProbabilitySpace..................................................... 492
9.3 ConditionalProbability:BayesFormula............................ 506
9.4 DiscreteRandomVariables.......................................... 511
9.4.1 RandomVariables .......................................... 511
9.4.2 ExpectedValue;Moments ................................. 518
9.4.3 Variance..................................................... 529
9.4.4 DiscreteUniformDistribution............................. 535
9.4.5 BernoulliDistribution...................................... 536
9.4.6 BinomialDistribution ...................................... 536
9.4.7 PoissonDistribution........................................ 540
9.4.8 GeometricDistribution..................................... 543