Mathematical Methods for Engineers and Scientists

440 Pages·2007·3.127 MB·English

by  Kwong-Tin Tang

K.T. Tang
Mathematical Methods
for Engineers and Scientists
3
Fourier Analysis, Partial Differential Equations
and Variational Methods
With 79 Figures and 4 Tables
123
ProfessorDr.Kwong-Tin Tang
PacificLutheranUniversity
DepartmentofPhysics
Tacoma,WA98447,USA
E-mail:[email protected]
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ISBN-103-540-44695-8SpringerBerlinHeidelbergNewYork
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Preface
Forsome30years,Ihavetaughttwo“MathematicalPhysics”courses.Oneof
them was previously named “Engineering Analysis.” There are several text-
books of unquestionable merit for such courses, but I could not find one that
fitted our needs. It seemed to me that students might have an easier time if
some changes were made in these books. I ended up using class notes. Ac-
tually, I felt the same about my own notes, so they got changed again and
again. Throughout the years, many students and colleagues have urged me
to publish them. I resisted until now, because the topics were not new and
I was not sure that my way of presenting them was really much better than
others. In recent years, some former students came back to tell me that they
still found my notes useful and looked at them from time to time. The fact
thattheyalwayssingledoutthesecourses,amongmanyothersIhavetaught,
made me think that besides being kind, they might even mean it. Perhaps, it
is worthwhile to share these notes with a wider audience.
It took far more work than expected to transcribe the lecture notes into
printed pages. The notes were written in an abbreviated way without much
explanationbetweenanytwoequations,becauseIwassupposedtosupplythe
missing links in person. How much detail I would go into depended on the
reaction of the students. Now without them in front of me, I had to decide
the appropriate amount of derivation to be included. I chose to err on the
side of too much detail rather than too little. As a result, the derivation does
not look very elegant, but I also hope it does not leave any gap in students’
comprehension.
Precisely stated and elegantly proved theorems looked great to me when
I was a young faculty member. But in the later years, I found that elegance
in the eyes of the teacher might be stumbling blocks for students. Now I
am convinced that before a student can use a mathematical theorem with
confidence,heorshemustfirstdevelopanintuitivefeeling.Themosteffective
way to do that is to follow a sufficient number of examples.
This book is written for students who want to learn but need a firm
hand-holding. I hope they will find the book readable and easy to learn from.
VI Preface
Learning, as always, has to be done by the student herself or himself. No one
can acquire mathematical skill without doing problems, the more the better.
However, realistically students have a finite amount of time. They will be
overwhelmed if problems are too numerous, and frustrated if problems are
too difficult. A common practice in textbooks is to list a large number of
problems and let the instructor to choose a few for assignments. It seems to
me that is not a confidence building strategy. A self-learning person would
notknowwhattochoose.Thereforeamoderatenumberofnotoverlydifficult
problems, with answers, are selected at the end of each chapter. Hopefully
after the student has successfully solved all of them, he will be encouraged
to seek more challenging ones. There are plenty of problems in other books.
Ofcourse,aninstructorcanalwaysassignmoreproblemsatlevelssuitableto
the class.
Professor I.I. Rabi used to say “All textbooks are written with the prin-
ciple of least astonishment.” Well, there is a good reason for that. After all,
textbooksaresupposedtoexplainawaythemysteriesandmaketheprofound
obvious. This book is no exception. Nevertheless, I still hope the reader will
find something in this book exciting.
ThissetofbooksiswritteninthespiritofwhatSommerfeldcalled“phys-
ical mathematics.” For example, instead of studying the properties of hyper-
bolic,parabolic,andellipticpartialdifferentialequations,materialsonpartial
differential equations are organized around wave, diffusion and Laplace equa-
tions. Physical problems are used as the framework for various mathematical
techniques to hang together, rather than as just examples for mathematical
theories.Inordernottosacrificetheunderlyingmathematicalconcepts,these
materials are preceded by a chapter on Sturm–Livouville theory in infinite
dimensional vector space. It is author’s experience that this approach not
only stimulates students’ intuitive thinking but also increase their confidence
in using mathematical tools.
These books are dedicated to my students. I want to thank my A and B
students, their diligence and enthusiasm have made teaching enjoyable and
worthwhile. I want to thank my C and D students, their difficulties and mis-
takes made me search for better explanations.
I want to thank Brad Oraw for drawing many figures in this book and
Mathew Hacker for helping me to typeset the manuscript.
I want to express my deepest gratitude to Professor S.H. Patil, Indian
Institute of Technology, Bombay. He has read the entire manuscript and pro-
vided many excellent suggestions. He has also checked the equations and the
problems and corrected numerous errors.
The responsibility for remaining errors is, of course, entirely mine. I will
greatly appreciate if they are brought to my attention.
Tacoma, Washington K.T. Tang
June 2006
Contents
Part I Fourier Analysis
1 Fourier Series ............................................. 3
1.1 Fourier Series of Functions with Periodicity 2π .............. 3
1.1.1 Orthogonality of Trigonotric Functions............... 3
1.1.2 The Fourier Coefficients............................ 5
1.1.3 Expansion of Functions in Fourier Series ............. 6
1.2 Convergence of Fourier Series ............................. 9
1.2.1 Dirichlet Conditions ............................... 9
1.2.2 Fourier Series and Delta Function ................... 10
1.3 Fourier Series of Functions of any Period ................... 13
1.3.1 Change of Interval................................. 13
1.3.2 Fourier Series of Even and Odd Functions ............ 21
1.4 Fourier Series of Nonperiodic Functions in Limited Range .... 24
1.5 Complex Fourier Series................................... 29
1.6 The Method of Jumps ................................... 32
1.7 Properties of Fourier Series ............................... 37
1.7.1 Parseval’s Theorem................................ 37
1.7.2 Sums of Reciprocal Powers of Integers ............... 39
1.7.3 Integration of Fourier Series ........................ 42
1.7.4 Differentiation of Fourier Series ..................... 43
1.8 Fourier Series and Differential Equations ................... 45
1.8.1 Differential Equation with Boundary Conditions....... 45
1.8.2 Periodically Driven Oscillator ....................... 49
Exercises ................................................... 52
2 Fourier Transforms ....................................... 61
2.1 Fourier Integral as a Limit of a Fourier Series ............... 61
2.1.1 Fourier Cosine and Sine Integrals.................... 65
2.1.2 Fourier Cosine and Sine Transforms ................. 67
2.2 Tables of Transforms..................................... 72
VIII Contents
2.3 The Fourier Transform ................................... 72
2.4 Fourier Transform and Delta Function ..................... 79
2.4.1 Orthogonality..................................... 79
2.4.2 Fourier Transforms Involving Delta Functions......... 80
2.4.3 Three-Dimensional Fourier Transform Pair ........... 81
2.5 Some Important Transform Pairs .......................... 85
2.5.1 Rectangular Pulse Function......................... 85
2.5.2 Gaussian Function................................. 85
2.5.3 Exponentially Decaying Function.................... 87
2.6 Properties of Fourier Transform ........................... 88
2.6.1 Symmetry Property ............................... 88
2.6.2 Linearity, Shifting, Scaling.......................... 89
2.6.3 Transform of Derivatives ........................... 91
2.6.4 Transform of Integral .............................. 92
2.6.5 Parseval’s Theorem................................ 92
2.7 Convolution ............................................ 94
2.7.1 Mathematical Operation of Convolution.............. 94
2.7.2 Convolution Theorems ............................. 96
2.8 Fourier Transform and Differential Equations ............... 99
2.9 The Uncertainty of Waves ................................103
Exercises ...................................................105
Part II Sturm–Liouville Theory and Special Functions
3 Orthogonal Functions and Sturm–Liouville Problems .....111
3.1 Functions as Vectors in Infinite Dimensional Vector Space ....111
3.1.1 Vector Space......................................111
3.1.2 Inner Product and Orthogonality....................113
3.1.3 Orthogonal Functions..............................116
3.2 Generalized Fourier Series ................................121
3.3 Hermitian Operators.....................................123
3.3.1 Adjoint and Self-adjoint (Hermitian) Operators .......123
3.3.2 Properties of Hermitian Operators...................125
3.4 Sturm–Liouville Theory ..................................130
3.4.1 Sturm–Liouville Equations .........................130
3.4.2 Boundary Conditions of Sturm–Liouville Problems ....132
3.4.3 Regular Sturm–Liouville Problems...................133
3.4.4 Periodic Sturm–Liouville Problems ..................141
3.4.5 Singular Sturm–Liouville Problems ..................142
3.5 Green’s Function ........................................149
3.5.1 Green’s Function and Inhomogeneous Differential
Equation .........................................149
3.5.2 Green’s Function and Delta Function ................150
Exercises ...................................................157
Contents IX
4 Bessel and Legendre Functions ............................163
4.1 Frobenius Method of Differential Equations .................164
4.1.1 Power Series Solution of Differential Equation.........164
4.1.2 Classifying Singular Points .........................166
4.1.3 Frobenius Series...................................167
4.2 Bessel Functions.........................................171
4.2.1 Bessel Functions J (x) of Integer Order ..............172
n
4.2.2 Zeros of the Bessel Functions .......................174
4.2.3 Gamma Function..................................175
4.2.4 Bessel Function of Noninteger Order .................177
4.2.5 Bessel Function of Negative Order...................179
4.2.6 Neumann Functions and Hankel Functions............179
4.3 Properties of Bessel Function .............................182
4.3.1 Recurrence Relations ..............................182
4.3.2 Generating Function of Bessel Functions .............185
4.3.3 Integral Representation ............................186
4.4 Bessel Functions as Eigenfunctions of Sturm–Liouville
Problems...............................................187
4.4.1 Boundary Conditions of Bessel’s Equation ............187
4.4.2 Orthogonality of Bessel Functions ...................188
4.4.3 Normalization of Bessel Functions ...................189
4.5 Other Kinds of Bessel Functions...........................191
4.5.1 Modified Bessel Functions ..........................191
4.5.2 Spherical Bessel Functions..........................192
4.6 Legendre Functions ......................................196
4.6.1 Series Solution of Legendre Equation ................196
4.6.2 Legendre Polynomials..............................200
4.6.3 Legendre Functions of the Second Kind ..............202
4.7 Properties of Legendre Polynomials .......................204
4.7.1 Rodrigues’ Formula................................204
4.7.2 Generating Function of Legendre Polynomials.........206
4.7.3 Recurrence Relations ..............................208
4.7.4 Orthogonality and Normalization of Legendre
Polynomials ......................................211
4.8 Associated Legendre Functions and Spherical Harmonics .....212
4.8.1 Associated Legendre Polynomials....................212
4.8.2 Orthogonality and Normalization of Associated
Legendre Functions................................214
4.8.3 Spherical Harmonics ...............................217
4.9 Resources on Special Functions............................218
Exercises ...................................................219
X Contents
Part III Partial Differential Equations
5 Partial Differential Equations in Cartesian Coordinates....229
5.1 One-Dimensional Wave Equations .........................230
5.1.1 The Governing Equation of a Vibrating String ........230
5.1.2 Separation of Variables.............................232
5.1.3 Standing Wave....................................238
5.1.4 Traveling Wave ...................................242
5.1.5 Nonhomogeneous Wave Equations ...................248
5.1.6 D’Alembert’s Solution of Wave Equations ............252
5.2 Two-Dimensional Wave Equations .........................261
5.2.1 The Governing Equation of a Vibrating Membrane ....261
5.2.2 Vibration of a Rectangular Membrane ...............262
5.3 Three-Dimensional Wave Equations........................267
5.3.1 Plane Wave.......................................268
5.3.2 Particle Wave in a Rectangular Box .................270
5.4 Equation of Heat Conduction .............................272
5.5 One-Dimensional Diffusion Equations ......................274
5.5.1 Temperature Distributions with Specified Values
at the Boundaries .................................275
5.5.2 Problems Involving Insulated Boundaries.............278
5.5.3 Heat Exchange at the Boundary.....................280
5.6 Two-Dimensional Diffusion Equations: Heat
Transfer in a Rectangular Plate ...........................284
5.7 Laplace’s Equations......................................286
5.7.1 Two-Dimensional Laplace’s Equation: Steady-State
Temperature in a Rectangular Plate .................287
5.7.2 Three-Dimensional Laplace’s Equation: Steady-State
Temperature in a Rectangular Parallelepiped .........289
5.8 Helmholtz’s Equations ...................................291
Exercises ...................................................292
6 Partial Differential Equations with Curved Boundaries ....301
6.1 The Laplacian ..........................................302
6.2 Two-Dimensional Laplace’s Equations......................304
6.2.1 Laplace’s Equation in Polar Coordinates .............304
6.2.2 Poisson’s Integral Formula..........................312
6.3 Two-Dimensional Helmholtz’s Equations in Polar Coordinates.315
6.3.1 Vibration of a Drumhead: Two Dimensional Wave
Equation in Polar Coordinates ......................316
6.3.2 Heat Conduction in a Disk: Two Dimensional
Diffusion Equation in Polar Coordinates..............322
6.3.3 Laplace’s Equations in Cylindrical Coordinates........326
6.3.4 Helmholtz’s Equations in Cylindrical Coordinates .....331
Contents XI
6.4 Three-Dimensional Laplacian in Spherical Coordinates .......334
6.4.1 Laplace’s Equations in Spherical Coordinates .........334
6.4.2 Helmholtz’s Equations in Spherical Coordinates .......345
6.4.3 Wave Equations in Spherical Coordinates.............346
6.5 Poisson’s Equations......................................349
6.5.1 Poisson’s Equation and Green’s Function.............351
6.5.2 Green’s Function for Boundary Value Problems .......355
Exercises ...................................................359
Part IV Variational Methods
7 Calculus of Variation ......................................367
7.1 The Euler–Lagrange Equation.............................368
7.1.1 Stationary Value of a Functional ....................368
7.1.2 Fundamental Theorem of Variational Calculus ........370
7.1.3 Variational Notation...............................372
7.1.4 Special Cases .....................................373
7.2 Constrained Variation....................................377
7.3 Solutions to Some Famous Problems .......................380
7.3.1 The Brachistochrone Problem.......................380
7.3.2 Isoperimetric Problems.............................384
7.3.3 The Catenary.....................................386
7.3.4 Minimum Surface of Revolution .....................391
7.3.5 Fermat’s Principle.................................394
7.4 Some Extensions ........................................397
7.4.1 Functionals with Higher Derivatives .................397
7.4.2 Several Dependent Variables ........................399
7.4.3 Several Independent Variables ......................401
7.5 Sturm–Liouville Problems and Variational Principles.........403
7.5.1 Variational Formulation of Sturm–Liouville Problems ..403
7.5.2 Variational Calculations of Eigenvalues and
Eigenfunctions ....................................405
7.6 Rayleigh–Ritz Methods for Partial Differential Equations .....410
7.6.1 Laplace’s Equation ................................411
7.6.2 Poisson’s Equation ................................415
7.6.3 Helmholtz’s Equation ..............................417
7.7 Hamilton’s Principle .....................................420
Exercises ...................................................425
References.....................................................431
Index..........................................................433