Principles Of Applied Mathematics: Transformation and Approximation

624 Pages·2000·31.822 MB·\624

by  James P. Keener (Author)

Principles of
Applied
Mathematics
To Kristine, Sammy, and Justin,
Still my best friends after all these years.
P
rinciples of
A
pplied
M
athematics
Transformation
and Approximation
Revised Edition
James P. Keener
University of Utah
Salt Lake City, Utah
A dvanced Book Program
CRC Press
Taylor &.Francis Group
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an Informa business
A CHAPMAN & HALL BOOK
First published 2000 by Westview Press
Published 2018 by CRC Press
Taylor & Francis Group
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CRC Press is an imprint of the Taylor & Francis Group, an informa business
Copyright © 2000 by James P. Keener
No claim to original U.S. Government works
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Library of Congress Catalog Card Number: 99-067545
ISBN 13: 978-0-7382-0129-0 (hbk)
C ontents
Preface to First Edition xi
Preface to Second Edition xvii
1 Finite Dimensional Vector Spaces 1
1.1 Linear Vector Spaces....................................................................... 1
1.2 Spectral Theory for Matrices........................................................ 9
1.3 Geometrical Significance of Eigenvalues ....................................... 17
1.4 Fredholm Alternative Theorem..................................................... 24
1.5 Least Squares Solutions-Pseudo Inverses....................................... 25
1.5.1 The Problem of Procrustes................................................ 41
1.6 Applications of Eigenvalues and Eigenfunctions............................ 42
1.6.1 Exponentiation of Matrices ................................................ 42
1.6.2 The Power Method and Positive Matrices........................ 43
1.6.3 Iteration Methods.............................................................. 45
Further Reading...................................................................................... 47
Problems for Chapter 1 .......................................................................... 49
2 Function Spaces 59
2.1 Complete Vector Spaces................................................................. 59
2.1.1 Sobolev Spaces.................................................................... 65
2.2 Approximation in Hilbert Spaces................................................. 67
2.2.1 Fourier Series and Completeness....................................... 67
2.2.2 Orthogonal Polynomials..................................................... 69
2.2.3 Trigonometric Series........................................................... 73
2.2.4 Discrete Fourier Transforms................................................ 76
2.2.5 Sine Functions.................................................................... 78
2.2.6 Wavelets................................................................................ 79
2.2.7 Finite Elements.................................................................... 88
Further Reading.................................................................'................... 92
Problems for Chapter 2 .......................................................................... 93
v
Integral Equations
3.1 Introduction......................................................... 101
3.2 Bounded Linear Operators in Hilbert Space . . 105
3.3 Compact Operators .......................................... 111
3.4 Spectral Theory for Compact Operators . . . . 114
3.5 Resolvent and Pseudo-Resolvent Kernels . . . . 118
3.6 Approximate Solutions....................................... 121
3.7 Singular Integral Equations.............................. 125
Further Reading......................................................... 127
Problems for Chapter 3 ............................................. 128
Differential Operators 133
4.1 Distributions and the Delta Function............... 133
4.2 Green’s Functions ............................................. 144
4.3 Differential Operators....................................... 151
4.3.1 Domain of an Operator........................ 151
4.3.2 Adjoint of an Operator ........................ 152
4.3.3 Inhomogeneous Boundary Data............ 154
4.3.4 The Fredholm Alternative..................... 155
4.4 Least Squares Solutions.................................... 157
4.5 Eigenfunction Expansions................................. 161
4.5.1 Trigonometric Functions........................ 164
4.5.2 Orthogonal Polynomials........................ 167
4.5.3 Special Functions.................................... 169
4.5.4 Discretized Operators........................... 169
Further Reading......................................................... 171
Problems for Chapter 4 ............................................. 171
Calculus of Variations 177
5.1 The Euler-Lagrange Equations........................ 177
5.1.1 Constrained Problems........................... 180
5.1.2 Several Unknown Functions.................. 181
5.1.3 Higher Order Derivatives ..................... 184
5.1.4 Variable Endpoints................................. 184
5.1.5 Several Independent Variables............... 185
5.2 Hamilton’s Principle.......................................... 186
5.2.1 The Swinging Pendulum........................ 188
5.2.2 The Vibrating String.............................. 189
5.2.3 The Vibrating Rod................................. 189
5.2.4 Nonlinear Deformations of a Thin Beam 193
5.2.5 A Vibrating Membrane........................ 194
5.3 Approximate Methods....................................... 195
5.4 Eigenvalue Problems.......................................... 198
5.4.1 Optimal Design of Structures............... 201
Further Reading......................................................... 202
Problems for Chapter 5 ............................................. 203
CONTENTS vii
6 Complex Variable Theory 209
6.1 Complex Valued Functions ..............................................................209
6.2 The Calculus of Complex Functions ...............................................214
6.2.1 Differentiation-Analytic Functions......................................214
6.2.2 Integration.............................................................................217
6.2.3 Cauchy Integral Formula .....................................................220
6.2.4 Taylor and Laurent Series.....................................................224
6.3 Fluid Flow and Conformal Mappings...............................................228
6.3.1 Laplace’s Equation.................................................................228
6.3.2 Conformal Mappings..............................................................236
6.3.3 Free Boundary Problems .....................................................243
6.4 Contour Integration ..........................................................................248
6.5 Special Functions................................................................................259
6.5.1 The Gamma Function...........................................................259
6.5.2 Bessel Functions....................................................................262
6.5.3 Legendre Functions.................................................................268
6.5.4 Sine Functions.......................................................................270
Further Reading.........................................................................................273
Problems for Chapter 6 ...................................................................... . 274
7 Transform and Spectral Theory 283
7.1 Spectrum of an Operator .................................................................283
7.2 Fourier Transforms.............................................................................284
7.2.1 Transform Pairs ....................................................................284
7.2.2 Completeness of Hermite and Laguerre Polynomials . . . 297
7.2.3 Sine Functions.......................................................................299
7.2.4 Windowed Fourier Transforms ............................................300
7.2.5 Wavelets...................................................................................301
7.3 Related Integral Transforms..............................................................307
7.3.1 Laplace Transform.................................................................307
7.3.2 Mellin Transform....................................................................308
7.3.3 Hankel Transform ........................................................ 309
7.4 Z Transforms......................................................................................310
7.5 Scattering Theory .............................................................................312
7.5.1 Scattering Examples..............................................................318
7.5.2 Spectral Representations........................................................325
Further Reading.........................................................................................327
Problems for Chapter 7 .............................................................................328
Appendix: Fourier Transform Pairs...........................................................335
8 Partial Differential Equations 337
8.1 Poisson’s Equation.............................................................................339
8.1.1 Fundamental Solutions...........................................................339
8.1.2 The Method of Images...........................................................343
8.1.3 Transform Methods ..............................................................344
8.1.4 Hilbert Transforms.................................................................355
viii CONTENTS
8.1.5 Boundary Integral Equations................................................357
8.1.6 Eigenfunctions.......................................................................359
8.2 The Wave Equation ..........................................................................365
8.2.1 Derivations.............................................................................365
8.2.2 Fundamental Solutions...........................................................368
8.2.3 Vibrations................................................................................373
8.2.4 Diffraction Patterns..............................................................376
8.3 The Heat Equation.............................................................................380
8.3.1 Derivations.............................................................................380
8.3.2 Fundamental Solutions...........................................................383
8.3.3 Transform Methods ..............................................................385
8.4 Differential-Difference Equations.....................................................390
8.4.1 Transform Methods ..............................................................392
8.4.2 Numerical Methods ..............................................................395
Further Reading.........................................................................................400
Problems for Chapter 8 .............................................................................402
9 Inverse Scattering Transform 411
9.1 Inverse Scattering .............................................................................411
9.2 Isospectral Flows................................................................................417
9.3 Korteweg-deVries Equation..............................................................421
9.4 The Toda Lattice................................................................................426
Further Reading.........................................................................................432
Problems for Chapter 9 .............................................................................433
10 Asymptotic Expansions 437
10.1 Definitions and Properties ..............................................................437
10.2 Integration by Parts..........................................................................440
10.3 Laplace’s Method................................................................................442
10.4 Method of Steepest Descents ...........................................................449
10.5 Method of Stationary Phase..............................................................456
Further Reading.........................................................................................463
Problems for Chapter 10.............................................................................463
11 Regular Perturbation Theory 469
11.1 The Implicit Function Theorem .....................................................469
11.2 Perturbation of Eigenvalues..............................................................475
11.3 Nonlinear Eigenvalue Problems........................................................478
11.3.1 Lyapunov-Schmidt Method..................................................482
11.4 Oscillations and Periodic Solutions...................................................482
11.4.1 Advance of the Perihelion of Mercury .................................483
11.4.2 Van der Pol Oscillator............................................................485
11.4.3 Knotted Vortex Filaments......................................................488
11.4.4 The Melnikov Function ........................................................493
11.5 Hopf Bifurcations................................................................................494
Further Reading.........................................................................................496
CONTENTS
IX
Problems for Chapter 11.............................................................................498
12 Singular Perturbation Theory 505
12.1 Initial Value Problems I ....................................................................505
12.1.1 Van der Pol Equation...........................................................508
12.1.2 Adiabatic Invariance..............................................................510
12.1.3 Averaging........................................., ................................511
12.1.4 Homogenization Theory........................................................514
12.2 Initial Value Problems II....................................................................520
12.2.1 Operational Amplifiers...........................................................521
12.2.2 Enzyme Kinetics....................................................................523
12.2.3 Slow Selection in Population Genetics................................526
12.3 Boundary Value Problems.................................................................528
12.3.1 Matched Asymptotic Expansions.........................................528
12.3.2 Flame Fronts..........................................................................539
12.3.3 Relaxation Dynamics ...........................................................542
12.3.4 Exponentially Slow Motion..................................................548
Further Reading........................................................................................551
Problems for Chapter 12.............................................................................552
Bibliography 559
Selected Hints and Solutions 567
Index 596