fourier series, transforms, and boundary value problems

365 Pages·2012·4.98 MB·English

FOURIER SERIES,
TRANSFORMS, AND
BOUNDARY VALUE
PROBLEMS

Second Edition

TRAY HANNA,
Prefeser timerins
Eire of Wyoming
Taranie. soning

JOHN #, ROWLAND

A Wiley interseience Publication
John Wiley & Sons, Ine.
New York / Chichestar? Baisbane ¢ tinwntn Singapore
PURE AND APPLUED MATHEMATICS

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PREFACE

‘The basic philosophy remias the same ws inthe fit edition. ‘The primary
changes consist of the addition of ew material on integrat tension,
‘dscreto and fat Fourier transforms, series solutions, harmonic analy,
sphoriest harmonics, ad a glance at some of the numescal techniques fot
the talution of boundary value problems. "The order of presentation af some
ff the material from the frst edition has been rearranged to provide more

‘exbiig in arranging courses based on this ext
“The boeic contains more than enough marcral fara one semester cours.
thapters relatively

frdinarif be coweied i a course of tis nature, These could be followed by
any combination of Chapters 4,5, 6, and & except chat Sctions 8. [1-818
depend on Chapter 5 and Sects S14 and 815 depend on Chaplet 6
‘Chapter 7 depends somewhat on te booty presented i Chapter an dhe
Flake! aust Lepeude (rasforms depend ou Chapters Sand 6, expoctvey
Chapter 9, taken ia its entity, is dependent ou all of the preceding
chiplels, Howover, iaviwcto's who prefer @ ieterwenve applications wilh
the devolapmven of coos will id tat Is pore t sleet PertiueDtYopies
Iowa Chapters 8 and 9 a the necesetiy Watheratis & developed.

‘A cncwomester eoulse given at the Univerly of Wyoning wets
‘alstantist portions of Chapters 1, 2, 9, 4, 7, ad 8 "This comnse has pp
lass aud graduate students from Fels such as goupsics, physics, en
‘incoting, computer scones, and mathematic.

‘We ate indchusd to Maria Taylor and Bob Hilt for valuable cditoral
sssstanee and to many Studcncs for catching cross and suggesting improse-
meals, Faull, we express uur special appreciation lo Janet Netzel wad
Milei Slepbers for their slit sping

J. Rav Hasina
ure, Cala

Jour H. Rowse
PREFACE TO THE FIRST
EDITION

This book isa result of the development of a Sct of notes tor 2 court ia
boundary valbe problems, using ourier scrice and integrae. Tis primsty
objective Je to acquaint stodents with the solutions of brundary vale
problems asocated with natural phenomena, Ie tnrafore necusary for
fhe reader to undcrtand basic coneepls and manipulations of elementary
talaulse, Although some mathematical teas from advanced caleniue are
beneficial, many of the conecpts ato contained in thie hoak. A mieimal
background in physics will id onc in understanding the snodcling ofa fet
problems conscming heat, wave, and potential theory.

“This book safes to the main proces fr solving boundary value problowt
asthe Fourier mcthod, Ta understand che detail af this procedure, pics of
Urtbosonility, Fouror series, and integrals precede cho diseussion af the
Fourier method. OF necessity Roch topics ah convergence, exitence, and
tuniquecess arc inchulad. Pimphass ie placed clearly on the use of basic
soncepts and techniques cath than the details af developing the theory
‘There ore many completely solved examples. These are flowed by otc.
ives Ibal allow the reuder ample oppurtunity to tet hia*her understanding
OL the material. Most crecciees arc accompanied with) answers. Some
Answers are imped in the problems, wil others are given in an answer
scetion. The sDbceviatons teed we Usted in the index.

‘Canton simi 1 that of tht hovk bas been ured in 4 oouree of three
semestes houte with several doves, La prerequisite of ruinary diferenta!
equations prescited. much of Chapte | aity be omit. I shorten Oe
‘outvechanters on either Bessel functions of Legendre polyols nay be
deleted. Work with uperarors may he reduced, oF oxkor sections may Be
‘mitted widhou sionsy afecting tho continvity Of the come. The rebee-
fence of the instzustor, the Boskyround und interests of the sludens- wa Ene
Intensity of the course sliwald govera the cheice of abject ater

"Numerous colleague and students hive flueneed the form of tls book.
1k is my plewsuce 10 thank everyone who hae offered sngurvtions forse
inmprewement. Tm paticulary indebte! to my department head, Joseph
Mastin, for his foi support of tke projet, and to Daniel Kate «former
‘ident, for his holpfl ideas nd bis solaions for many of the problems. To
Featrice Shabe for valushe edltoral asiiaace, and 10 all of the Wiley
ication snl, Tam deeply getafu, Fnuly, Lexprose a specel apreci-
fiom tor the skilfoltsping of dhe manuscript by Laureds Lol, Pauls
Meleher, und Prt Twitchell

Hambly 1 acknowledge the volumes of Herre, extonding (eons a time
hnefore J, Faucet 10 the peesane, whish hive influenced the composition ot
this bouk, AL woukd be am endless Task tw meation each one, A Tit of
references. by do means exhaustive. i included to wid the reader

‘In spite of cielul proolreadiag, some errors are elusive and not dscor-
cred, Tencouraye readers. infra me of rastakes und to offer suggestions
{or the inprovereal of the book.

A, Ray Hana
CONTENTS

4, Linear Difercfia ations

11. Lineur Operators, 1
12. Ordinary Ditteeetial Equations, 2

5. Homogeneous Linear ODL with Constant Coetiieats, $
4. Eulers ODE. 7

3. Series Solutions, &

‘& Frateniae Method, 13,

7 Numeral Solutions, 19

A. Linear PDEs, 23

8. Classification of a Linsar PDE of Sesond Order, 23

(0. Boundary Value Problems with PLES, 24

1. Second Order Linear PDEs with Constant Coctiiems, 26
12. Sepaation of Variables, 35

Orthogonal §

6 of Puneton

24. Ontbogonality wd Veutus, 40

22. Oalogonsl Fonetous, 2

23. Complex Funstions, 46

2.4. Auditional Conccpts af Orehogonatity, 47

2.3, “The Siore-Liguvlle Boundary Value Problem, 50
25, Uniform Convergenee of Series, 34

2.7. Sosiox of Orthogonal Fonetions, 61

28, Approximation hy Least Squares, 64

28, Completeness of Set, «5

3, Router Series

3.1, Piorwiw Continuous Functions. 64
32 A Gosie Fourier Series, 72
3a. Fea ind Odd Vnctions, 76
44, Fourier Sine and Cosine Serie, 77
AS. Compies Fourier Series, 9

JA. Harmonic Analysis, 42

32. Uniform Consergesce uf Fourier Sevies, #9
38. Differentiation of Fourier Sexes, 92

3.8. Intogration of Fourier Series, 94
3.10. Double Four‘ Series, 97

Fours Integrals 1m

4.1. Unitorm Convergence of Integrals, 312
42, A Generalization of the Fourier Seri, 10?
43. Fourier Sine and Cosine Integels, 10
44, “The Expnental Fourie insgral, 112

esol Fuetloas 1

5.1. “The Gamma Function and the Bost Funetion, 147
5.2. Additional Bere! Functions, 120

53, ‘Differential Equations fafvable with Boscl Funecions, 122
5.4. Spocial Bows! Functions and Tdenities, 124

5.5. An Integral Form tor J,(2), 130

56. Singular SLPs, 135,

5.7. Onthagonality af Bessel Functions, UH

58. Onthogonal Serica of Bestel Functions, 137

59. Beal Functions and Cylindrical Geometry, 10

Legendre Potynomias oy

64, Solutions to the Legendre Equation, 182
6.2, Rodsigaes’ Formula for Legendre Polynomials, 146
63, A Generating Function for 2,2), 199

6:4, ‘The Legendre Polynomial £,{c0s2), 151

65. Onthogonaliy and Norms f P(x), 152

86. Legendre Series, 154

67. Legendre Polynomial and Sphesieal Geometry, 158
8 Spherical Harmonics, 161

69, “The Generalized Legendre Hyustion, 162

Jnleral Treaster 18

3.1. Leplace Transforms. 168
2.2, Faslenie of the Traneforin, 169

13. The Gama Punction vad Laplace Teuzsforms, 170
7. ‘Tracsfons of Desvatives, 72

35. Derivatives uf Trvmfcne, 172

6, The Inverse Laplace ‘Tvanstorm, 173
13. Solutions af ODFs aed TPs, 173,
TS. Purtal Fractions, 74

29. ‘Ihe Unie Step Fonetion, (75

TAD. Shiting Propecia, 136

TAL. The Dire Dts Function, 177

TAZ. Convolution, 190

TS. Laplace Tearsten Method for PDEs. 16

TAL Finite Fourier Teansorms, 189

TS. Fourier Transforms, 191

TG. The Discrete Fourier Teansonm, 19?

FAT. The Fase Fourier Transform, 203

TB. Fourier Transforms of Euncivns of Two Variable, 28

3.19. Hanket Transforms, 200

320. Legendre Transform, 214

321. Mellin Trowsform. 215,

Applicaton af AEs 29

ALL The Vibrating Stcing, 219
42. Vorication and Uniqusnoss of the Solution ofthe
Vibrating String Probiem, 725
8.3, The Vibrvig Sting with Two Nonhomogeneous
Coudiinas, 228
‘Rd. Langitudinal Vibrations slong an Elastic Rod. 230
4S. Heat Conduction, 234
[RG Numerical Solution of the Host Equation, 241
[R7._Verifeation and Uniqueness of dhe Solution for che
Hest Prabism, 282
88 Gravitatonat Potential, 246
89, Laplace's Lquation, 247
Gib, Numerical Solution of the Laplace Equation, 251
RIL, “Temperature in u Circular Disk with fsulited Faves, 254
R12. Stead State Temperature in a Right Senicicular Coier, 256
8.13. Mirmonic loterior of a Right Cieular Cylinder. 260
RA. Steady State Temperature Distribution in a Sphere, 264
BAS. Potential for Sphere. 267

Aadhionsl Applleutons m

9.1. Mechunical and ectrical Osis, 270

9.3. The Ving Metbeane, 273

43, Vibrations of Circular Memtrune Dependent on Distance
trom Cepter. 250

9.4. The Vibrating String with an Excerl Fores, 283

9.3, Nonhomogencous End Temperatures it «Rod, 289

98. A Rod with rsalaed Ends 291