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PRINCIPLES of APPLIED MATHEMATICS Transformation and Approximation Revised Edition PRINCIPLES OF To Kristine, Sammy, and Justin, APPLIED Still my best friends after all these years. MATHEMATICS Transformation and Approximation Revised Edition JAMES P. KEENER University of Utah Salt Lake City, Utah ADVANCED BOOK PROGRAM I ewI I I ~ Member of the Perseus Books Group Contents 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 Many of the designations used by manufacturers and sellers to distinguish their 42 products are claimed as trademarks. Where those designations appear in this 1.6 Applications of Eigenvalues and Eigenfunctions book and Westview Press was aware of a trademark claim, the designations have 1.6.1 Exponentiation of Matrices . 42 been printed in initial capital letters. 1.6.2 The Power Method and Positive Matrices 43 45 Library of Congress Catalog Card Number: 99-067545 1.6.3 Iteration Methods Further Reading . 47 ISBN: 0-7382-0129-4 Problems for Chapter 1 49 Copyright© 2000 by James P. Keener 2 Function Spaces 59 Westview Press books are available at special discounts for bulk purchases in the 2.1 Complete Vector Spaces .... 59 U.S. by corporations, institutions, and other organizations. For more informa­ 2.1.1 Sobolev Spaces ..... 65 tion, please contact the Special Markets Department at the Perseus Books Group, 2.2 Approximation in Hilbert Spaces 67 11 Cambridge Center, Cambridge, MA 02142, or call (617) 252-5298. 2.2.1 Fourier Series and Completeness 67 All rights reserved. No part of this publication may be reproduced, stored in a 2.2.2 Orthogonal Polynomials . 69 retrieval system, or transmitted, in any form or by any means, electric, mechani­ 2.2.3 Trigonometric Series . 73 cal, photocopying, recording, or otherwise, without the prior written permission 76 of the publisher. Printed in the United States of America. 2.2.4 Discrete Fourier Transforms . 2.2.5 Sine Functions 78 Westview Press is a member of the Perseus Books Group 2.2.6 Wavelets .... 79 2.2. 7 Finite Elements . 88 First printing, December 1999 Further Reading . 92 Problems for Chapter 2 . 93 Find us on the World Wide Web at http:/ /www.westviewpress.com v CONTENTS vii vi CONTENTS 3 Integral Equations 101 6 Complex Variable Theory 209 3.1 Introduction .............................. 101 6.1 Complex Valued Functions ..................... 209 3.2 Bounded Linear Operators in Hilbert Space . 105 6.2 The Calculus of Complex Functions . 214 3.3 Compact Operators ......................... 111 6.2.1 Differentiation-Analytic Functions ............. 214 3.4 Spectral Theory for Compact Operators .............. 114 6.2.2 Integration . 217 ? 3.5 Resolvent and Pseudo-Resolvent Kernels . 118 6.2.3 Cauchy Integral Formula . 220 3.6 Approximate Solutions . 121 6.2.4 Taylor and Laurent Series . 224 3. 7 Singular Integral Equations . 125 6.3 Fluid Flow and Conformal Mappings . 228 I~ Further R.eading . 127 6.3.1 Laplace's Equation . 228 Problems for Chapter 3 . · . 128 I 6.3.2 Conformal Mappings . 236 6.3.3 Free Boundary Problems . 243 4 Differential Operators 133 6.4 Contour Integration . 248 I 6.5 Special Functions . 259 4.1 Distributions and the Delta Function . 133 I 4.2 Green's Functions ......................... 144 ii! 6.5.1 The Gamma Function . 259 4.3 Differential Operators . 151 I 6.5.2 Bessel Functions . 262 4.3.1 Domain of an Operator . 151 " 6.5.3 Legendre Functions . 268 4.3.2 Adjoint of an Operator .................. 152 6.5.4 Sine Functions . 270 4.3.3 Inhomogeneous Boundary Data . 154 I Further Reading . 273 J Problems for Chapter 6 . ~ . 27 4 4.3.4 The Fredholm Alternative . 155 ~ 4.4 Least Squares Solutions . ~ 157 7 Transform and Spectral Theory 283 4.5 Eigenfunction Expansions . -~i 161 7.1 Spectrum of an Operator ...................... 283 4.5.1 'Irigonometric Functions . 164 I 7.2 Fourier Transforms ........ , ................. 284 4.5.2 Orthogonal Polynomials . 167 ~ 7.2.1 Transform Pairs .................... , .. 284 4.5.3 Special Functions . ~ 169 § 7.2.2 Completeness of Hermite and Laguerre Polynomials ... 297 4.5.4 Discretized Operators . 169 7.2.3 Sine Functions ........................ 299 Further R.eading . · · · · · · · · · · · · · 171 I Problems for Chapter 4 . 171 s" 7.2.4 Windowed Fourier Transforms ............... 300 I 7.2.5 Wavelets . 301 5 Calculus of V~iations 177 7.3 Related Integral Transforms ..................... 307 i 7.3.1 Laplace Transform ...................... 307 5.1 The Euler-Lagrange Equations ................... 177 ~ 5.1.1 Constrained Problems .................... 180 • 7.3.2 Mellin Transform ....................... 308 5.1.2 Several Unknown Functions . 181 I 7.3.3 Hankel Transform ...................... 309 5.1.3 Higher Order Derivatives . 184 I 7.4 Z Transforms ............................. 310 li 7.5 Scattering Theory .......................... 312 5.1.4 Variable Endpoints ...................... 184 ~ 5.1.5 Several Independent Variables . 185 7.5.1 Scattering Examples ..................... 318 5.2 Hamilton's Principle ......................... 186 7.5.2 Spectral Representations ................... 325 5.2.1 The Swinging Pendulum . 188 Further Reading . 327 I Problems for Chapter 7 . , . 328 5.2.2 The Vibrating String . 189 I 5.2.3 The Vibrating Rod . 189 ! Appendix: Fourier Transform Pairs . 335 5.2.4 Nonlinear Deformations of a Thin Beam . 193 i J 8 Partial Differential Equations 337 5.2.5 A Vibrating Membrane . 194 iii 5.3 Approximate Methods ........................ 195 ~ 8.1 Poisson's Equation .......................... 339 5.4 Eigenvalue Problems . 198 8.1.1 Fundamental Solutions .................... 339 5.4.1 Optimal Design of Structures . 201 8.1.2 The Method of Images .................... 343 Further R.eading . .. 202 8.1.3 Transform Methods . 344 Problems for Chapter 5 . 203 8.1.4 Hilbert Transforms ...................... 355 viii CONTENTS CONTENTS ix 8.1.5 Boundary Integral Equations ................ 357 Problems for Chapter 11 . 498 8.1.6 Eigenfunctions . 359 8.2 The Wave Equation . 365 12 Singular Perturbation Theory 505 8.2.1 Derivations .......................... 365 12.1 Initial Value Problems I . 505 8.2.2 Fundamental Solutions . 368 12.1.1 Van der Pol Equation .................... 508 8.2.3 Vibrations . 373 12.1.2 Adiabatic lnvariance . 510 8.2.4 Diffraction Patterns . 376 12.1.3 Averaging . , . 511 8.3 The a:eat Equation . 380 12.1.4 Homogenization Theory ....... , ........... 514 8.3.1 Derivations .......................... 380 12.2 Initial Value Problems II . , . 520 8.3.2 Fundamental Solutions . 383 12.2.1 Operational Amplifiers . 521 8.3.3 Transform Methods . 385 12.2.2 Enzyme Kinetics . 523 8.4 Differential-Difference Equations .................. 390 12.2.3 Slow Selection in Population Genetics . 526 8.4.1 Transform Methods ..................... 392 .~ 12.3 Boundary Value Problems . 528 8.4.2 Numerical Methods . 395 12.3.1 Matched Asymptotic Expansions . 528 I 12.3.2 Flame Fronts . 539 Further Reading . 400 ~ Problems for Chapter 8 . 402 12.3.3 Relaxation Dynamics . 542 12.3.4 Exponentially Slow Motion . 548 9 Inverse Scattering Transform 411 I Further Reading .............................. 551 9.1 Inverse Scattering .......................... 411 Problems for Chapter 12 . 552 9.2 Isospectral Flows . 417 i i Bibliography 559 9.3 Korteweg-deVries Equation ..................... 421 ~ ~ 9.4 The Toda Lattice . 426 I Selected Hints and Solutions 567 Further Reading . 432 I Problems for Chapter 9 . 433 Index 596 10 Asymptotic Expansions 437 ~ 10.1 Definitions and Properties ..................... 437 I 10.2 Integration by Parts ......................... 440 I 10.3 Laplace's Method . 442 10.4 Method of Steepest Descents . ; 449 I 10.5 Method of Stationary Phase . 456 Further Reading . ~ . 463 'I Problems for Chapter 10 . 463 11 Regular Perturbation Theory 469 ! 11.1 The Implicit Function Theorem .................. 469 I 11.2 Perturbation of Eigenvalues ..................... 475 * 11.3 Nonlinear Eigenvalue Problems ................... 478 I 11.3.1 Lyapunov-Schmidt Method ................. 482 I 11.4 Oscillations and Periodic Solutions ................. 482 ~ 11.4.1 Advance of the Perihelion of Mercury . 483 11.4.2 Vander 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 Preface to First Edition Applied mathematics should read like good mystery, with an intriguing begin­ ning, a clever but systematic middle, and a satisfying resolution at the end. Often, however, the resolution of one mystery opens up a whole new problem, and the process starts all over. For the applied mathematical scientist, there is the goal to explain or predict the behavior of some physical situation. One begins by constructing a mathematical model which captures the essential fea­ tures of the problem without masking its content with overwhelming detail. Then comes the analysis of the model where every possible tool is tried, and some new tools developed, in order to understand the behavior of the model as thoroughly as possible. Finally, one must interpret and compare these results with real world facts. Sometimes this comparison is quite satisfactory, but most often one discovers that important features of the problem are not adequately 1 accounted for, and the process begins again. 1 Although every problem has its own distinctive features, through the years it has become apparent that there are a group of tools that are essential to the analysis of problems in many disciplines. This book is about those tools. But more than being just a grab bag of tools and techniques, the purpose of this book is to show that there is a systematic, even esthetic, explanation of how ~ I-?£ these classical tools work and fit together as a unit. Much of applied mathematical analysis can be summarized by the observa­ tion that we continually attempt to reduce our problems to ones that we already I know how to solve.
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