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Applied Mathematics
Applied Mathematics
Third Edition
J. David Logan
Willa Cather Professor
Department of Mathematics
University of Nebraska at Lincoln
WILEY
[M’ERSCIENCE
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright C 2006 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data.
Logan, J. David (John David)
Applied mathematics / J. David Logan.-3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN-13 978-0-471-74662-1 (acid-free paper)
ISBN-10 0-471-74662-2 (acid-free paper)
1. Mathematics-Textbooks. I. Title.
QA37.3.L64 2006
510-dc22
Printed in the United States of America.
10 9 8 7 6 5 4 3
2005058223
Dedicated to
my mother, Dorothy Elizabeth Logan, and
my father, George E. Logan (1919-1996)
Contents
Preface ………………………………………………… xiii
1.
Dimensional Analysis, Scaling, and Differential Equations ..
1.1
……………………………….
…………….
………………………….
1
Dimensional Analysis
1.1.1 The Program of Applied Mathematics
1.1.2 Dimensional Methods
2
2
5
1.1.3
The Pi Theorem ……………………………..
8
1.1.4
13
Scaling ………………………………………….. 19
1.2.1 Characteristic Scales
19
1.2.2 A Chemical Reactor Problem …………………… 22
1.2.3
25
Proof of the Pi Theorem ……………………….
1.2
1.3
…………………………..
The Projectile Problem ………………………..
Differential Equations ……………………………….
1.3.1
1.3.2
1.4
Stability and Bifurcation ……………………….
Two-Dimensional Problems
1.4.1
2.
Review of Elementary Methods ………………….
…………………………..
35
36
44
54
Phase Plane Phenomena ………………………. 54
1.4.2
Linear Systems ………………………………. 63
1.4.3
Nonlinear Systems ……………………………. 68
1.4.4
Bifurcation …………………………………. 76
Perturbation Methods ……………………………….. 85
2.1
Regular Perturbation ………………………………. 87
2.1.1
2.1.2
2.1.3
Motion in a Resistive Medium
………………….. 88
Nonlinear Oscillations …………………………. 90
The Poincare-Lindstedt Method ………………… 93
Vii
Applied Mathematics, Third Edition
viii
2.1.4
2.2
2.4
2.3.1
Inner and Outer Approximations ………………… 112
2.3.2
Matching …………………………………… 114
2.3.3
Uniform Approximations ………………………. 116
2.3.4
General Procedures …………………………… 119
Initial Layers …………………………………….. 123
2.4.1
2.4.2
3.
………………………….. 104
…………………………. 107
…………………………….. 108
……………………………. 112
Algebraic Equations
Differential Equations
Boundary Layers
Boundary Layer Analysis
2.2.1
2.2.2
2.2.3
2.3
Asymptotic Analysis ………………………….. 95
Singular Perturbation ………………………………. 104
Damped Spring-Mass System …………………… 123
Chemical Reaction Kinetics …………………….. 127
2.5
The WKB Approximation …………………………… 135
2.5.1 The Non-oscillatory Case ………………………. 137
2.5.2 The Oscillatory Case …………………………. 138
2.6
Asymptotic Expansion of Integrals
…………………….. 142
2.6.1
Laplace Integrals …………………………….. 142
2.6.2
Integration by Parts ………………………….. 146
2.6.3
Other Integrals ………………………………. 147
Calculus of Variations ……………………………….. 153
3.1
Variational Problems …………………………………153
3.1.1
3.1.2
3.2
Functionals …………………………………. 153
Examples ………………………………….. 155
Necessary Conditions for Extrema …………………….. 159
Normed Linear Spaces ………………………… 159
Derivatives of Functionals ……………………… 163
Necessary Conditions …………………………. 165
The Simplest Problem ……………………………… 168
3.2.1
3.2.2
3.2.3
3.3
3.3.1
3.3.2
3.3.3
3.4
3.5
The Euler Equation ………………………….. 168
Solved Examples …………………………….. 171
First Integrals ………………………………. 172
Generalizations …………………………………….177
3.4.1
Higher Derivatives ……………………………. 177
3.4.2
Several Functions …………………………….. 179
3.4.3
Natural Boundary Conditions …………………… 181
The Canonical Formalism …………………………… 185
3.5.1 Hamilton’s Principle ………………………….. 185
3.5.2 Hamilton’s Equations …………………………. 191
3.5.3 The Inverse Problem ………………………….. 194
Contents
3.6
4.
ix
Isoperimetric Problems
……………………………… 199
Eigenvalue Problems, Integral Equations, and Green’s Functions ………………………………………………..207
4.1
Orthogonal Expansions ……………………………… 207
4.1.1
4.1.2
4.2
Classical Fourier Series ………………………… 216
Sturm-Liouville Problems …………………………… 220
4.3
Integral Equations …………………………………. 226
4.4
4.3.1
Introduction ………………………………… 226
4.3.2
Volterra Equations …………………………… 230
4.3.3
4.3.4
Fredholm Equations with Degenerate Kernels ………. 236
4.5
Symmetric Kernels …………………………… 239
Green’s Functions …………………………………. 247
4.4.1
4.4.2
4.4.3
5.
Orthogonality ……………………………….. 207
Inverses of Differential Operators ………………… 247
Physical Interpretation ………………………… 250
Green’s Function via Eigenfunctions ……………… 255
Distributions ……………………………………… 258
4.5.1
Test Functions ………………………………. 258
4.5.2
Distributions ……………………………….. 261
4.5.3
Distribution Solutions to Differential Equations …….. 265
Discrete Models …………………………………….. 271
5.1
One-Dimensional Models ……………………………. 272
Linear and Nonlinear Models …………………… 272
Equilibria, Stability, and Chaos …………………. 277
Systems of Difference Equations ………………………. 285
5.1.1
5.1.2
5.2
5.3
5.2.1
Linear Models ………………………………. 285
5.2.2
Nonlinear Interactions ………………………… 296
Stochastic Models …………………………………. 303
5.3.1
5.4
6.
Elementary Probability ……………………….. 303
5.3.2
Stochastic Processes ………………………….. 310
5.3.3
Environmental and Demographic Models …………..
Probability-Based Models ……………………………
314
321
5.4.1
Markov Processes ……………………………. 321
5.4.2
Random Walks ………………………………. 327
5.4.3
The Poisson Process ………………………….. 331
Partial Differential Equations …………………………. 337
6.1
Basic Concepts ……………………………………. 337
6.1.1
6.2
Linearity and Superposition ……………………. 341
Conservation Laws ………………………………… 346
Applied Mathematics, Third Edition
x
6.3
6.4
6.2.1
6.2.2
6.2.3
6.2.4
6.2.5
One Dimension ……………………………… 347
6.3.1
6.3.2
Laplace’s Equation …………………………… 367
Basic Properties ……………………………… 370
Boundary Conditions
Equilibrium Equations
…………………………. 361
……………………………… 367
Eigenfunction Expansions …………………………… 374
6.4.1
6.4.2
6.5
Several Dimensions …………………………… 349
Constitutive Relations ………………………… 354
Probability and Diffusion ………………………. 358
Spectrum of the Laplacian ……………………… 374
Evolution Problems …………………………… 377
Integral Transforms ………………………………… 383
6.5.1
Laplace Transforms …………………………… 383
Fourier Transforms …………………………… 387
Stability of Solutions ……………………………….. 398
6.5.2
6.6
6.6.1
6.6.2
6.7
Elliptic Problems …………………………….. 406
Tempered Distributions ……………………….. 411
Diffusion Problems …………………………… 412
Wave Phenomena …………………………………….419
7.1
7.2
7.3
7.4
Wave Propagation …………………………………. 419
7.1.1
Waves …………………………………….. 419
7.1.2
The Advection Equation ………………………. 425
Nonlinear Waves ………………………………….. 430
7.2.1
7.2.2
7.2.3
Nonlinear Advection ………………………….. 430
7.3.1
Age-Structured Populations …………………….. 449
Traveling Wave Solutions ………………………. 435
Conservation Laws …………………………… 440
Quasi-linear Equations ……………………………… 445
The Wave Equation …………………………………454
7.4.1
7.4.2
7.4.3
8.
Pattern Formation ……………………………. 400
Distributions ……………………………………… 406
6.7.1
6.7.2
6.7.3
7.
Reaction-Diffusion Equations …………………… 398
The Acoustic Approximation …………………… 454
Solutions to the Wave Equation …………………. 458
Scattering and Inverse Problems ………………… 463
Mathematical Models of Continua …………………….. 471
8.1
Kinematics ………………………………………. 472
8.1.1
8.1.2
8.1.3
Mass Conservation …………………………… 477
Momentum Conservation ………………………. 478
Thermodynamics and Energy Conservation ………… 482
Contents
xi
8.1.4
8.2
8.3
Stress Waves in Solids
………………………… 487
Gas Dynamics ……………………………………. 493
8.2.1
Riemann’s Method …………………………… 493
8.2.2
The Rankine-Hugoniot Conditions ………………. 499
Fluid Motions in R3 ……………………………….. 502
8.3.1
8.3.2
Kinematics …………………………………. 502
Dynamics ………………………………….. 508
8.3.3
Energy …………………………………….. 515
Index ………………………………………………….. 525
Preface
This third edition of Applied Mathematics shares the same goals, philosophy,
and writing style as the original edition, which appeared in 1987-namely, to
introduce key ideas about mathematical methods and modeling to seniors and
graduate students in mathematics, engineering, and science. The emphasis is
upon the interdependence of mathematics and the applied and natural sciences.
The prerequisites include elementary, sophomore-level courses in differential
equations and matrices.
There are major changes in this new third edition. Summarizing, the material has been rearranged and basically divided into two parts. Chapters 1
through 4 involve models leading to ordinary differential equations and integral
equations, while Chapters 6 through 8 focus on partial differential equations
and their applications. Motivated by problems in the biological sciences, where
quantitative methods are becoming central, there is a new chapter (Chapter 5)
on discrete-time models, which includes some material on stochastic processes.
Two sections reviewing elementary methods for solving systems of ordinary
differential equations have been added at the end of Chapter 1. Many new exercises appear throughout the book. The Table of Contents details the specific
topics covered.
My colleagues in Lincoln, who have often used the text in our core sequence in applied mathematics, deserve special thanks. Tom Shores provided
me with an extensive errata, and he, Steve Cohn, and Glenn Ledder supplied
several new exercises from graduate qualifying examinations, homework, and
course exams. Bill Woleserisky read some of the manuscript and was often a
sounding board for suggestions. Also, I wish to thank those who used earlier
editions of the book and helped establish it as one of the basic textbooks in
the area; many have given me corrections and suggestions. Any comments, or
AN
xiv
Preface
otherwise, will be greatly appreciated, and they may be forwarded to me at dlogan@math.unl.edu. Finally, my wife Tess has been a constant source of support
for my writing, and I take this opportunity to publicly express my appreciation
for her encouragement and affection.
Suggestions for use of the text. Chapters 1 through 4 can form the
basis of a one-semester course involving differential and integral equations and
the basic core of applied mathematics. Chapter 3, on calculus of variations, is
independent from the others, so it may be deleted or covered in part. Portions of
Chapter 5, on discrete models could be added to this course. A second semester,
involving partial differential equation models, would cover Chapters 6, 7, and
8. Students may take the second semester, as is often done at the University of
Nebraska, without having the first, provided a small portion of Chapter 1 on
scaling is briefly reviewed.
There is much independence among the chapters, enabling an instructor to
design his or her special one-semester course in applied mathematics.
David Logan
Lincoln, Nebraska
1
Dimensional Analysis, Scaling, and
Differential Equations
The techniques of dimensional analysis and scaling are basic in the theory and
practice of mathematical modeling. In every physical setting a good grasp of
the possible relationships and comparative magnitudes among the various dimensioned parameters nearly always leads to a better understanding of the
problem and sometimes points the way toward approximations and solutions.
In this chapter we introduce some of the basic concepts from these two topics.
A statement and proof of the fundamental result in dimensional analysis, the
Pi theorem, is presented, and scaling is discussed in the context of reducing
problems to dimensionless form. The notion of scaling points the way toward a
proper treatment of perturbation methods, especially boundary layer phenomena in singular perturbation theory.
The second half of this chapter, Sections 1.3 and 1.4, contains a review of
differential equations. This material may be perused or skipped if readers are
familiar with the basic concepts and elementary solution methods.
I
2
1.
Dimensional Analysis, Scaling, and Differential Equations
formulation
Problem –
methods
Mathematical
Solution
Model
Comparison
to experiment
revision
Figure 1.1 Schematic of the modeling process.
1.1 Dimensional Analysis
1.1.1 The Program of Applied Mathematics
Applied mathematics is a broad subject area in the mathematical sciences
dealing with those topics, problems, and techniques that have been useful in
analyzing real-world phenomena. In a very limited sense it is a set of methods
that are used to solve the equations that come out of science, engineering, and
other areas. Traditionally, these methods were techniques used to examine and
solve ordinary and partial differential equations, and integral equations. At the
other end of the spectrum, applied mathematics is applied analysis, or the theory that underlies the methods. But, in a broader sense, applied mathematics
is about mathematical modeling and an entire process that intertwines with
the physical reality that underpins its origins.
By a mathematical model we mean an equation, or set of equations, that
describes some physical problem or phenomenon that has its origin in science,
engineering, economics, or some other area. By mathematical modeling we
mean the process by which we formulate and analyze the model. This process includes introducing the important and relevant quantities or variables
involved in the model, making model specific assumptions about those quantities, solving the model equations by some method, and then comparing the
solutions to real data and interpreting the results. Often the solution method
involves computer simulation or approximation. This comparison may lead to
revision and refinement until we are satisfied that the model accurately describes the physical situation and is predictive of other similar observations.
This process is depicted schematically in Fig. 1.1. Therefore the subject of
mathematical modeling involves physical intuition, formulation of equations,
solution methods, and analysis. A good mathematical model is simple, applies
to many situations, and in predictive.
Overall, in mathematical modeling the overarching objective is to make
1.1 Dimensional Analysis
3
sense of the world as we observe it, often by inventing caricatures of reality.
Scientific exactness is sometimes sacrificed for mathematical tractability. Model
predictions depend strongly on the assumptions, and changing the assumptions
changes the model. If some assumptions are less critical than others, we say
the model is robust to those assumptions. They help us clarify verbal descriptions of nature and the mechanisms that make up natural law, and they help
us determine which parameters and processes are important, and which are
unimportant.
Another issue is the level of complexity of a model. With modern computer
technology it is tempting to build complicated models that i …
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