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Partial Differential Equations: Theory and Completely Solved Problems PDF

694 Pages·2012·85.32 MB·English
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PARTIAL DIFFERENTIAL EQUATIONS PARTIAL DIFFERENTIAL EQUATIONS Theory and Completely Solved Problems T. HILLEN I. E. LEONARD H. VAN ROESSEL Department of Mathematical and Statistical Sciences University of Alberta ~WILEY A JOHN WILEY & SONS, INC., PUBLICATION Cover art: Water Wave courtesyo f Brocken Inaglory: Coronal Mass Ejection courtesyo fNASNSDO and AlA, EVE, and HMI sciencet eams. Copyright © 2012 by JohnW iley & Sons,I nc. Publishedb y JohnW iley & Sons,I nc., Hoboken,N ew Jersey.A ll rights reserved. Publisheds imultaneouslyin Canada. No part of this publicationm ay be reproduceds, toredi n a retrieval systemo r transmittedi n any form or by any means,e lectronic,m echanical,p hotocopying,r ecording,s canningo r otherwise,e xcepta s permittedu nder Section1 07 or 108 of the 1976 United StatesC opyrightA ct, without either the prior written permissiono f the Publisher,o r authorizationt hroughp aymento f the appropriatep er-copyf ee to the Copyright ClearanceC enter,I nc., 222 RosewoodD rive, Danvers,M A 01923, (978) 750-8400,f ax (978) 750-4470,o r on the web at www.copyright.com.R equeststo the Publisherf or permissions hould be addressedto the PermissionsD epartment,J ohnW iley & Sons,I nc., 111 River Street,H oboken,N J 07030, (201) 748-6011,f ax (201) 748-6008,o r online at http://www.wiley.comlgo/permission. Limit of LiabilitylDisclaimer of Warranty: While the publishera nd authorh ave usedt heir beste fforts in preparingt his book, they maken o representationo r warrantiesw ith respectt o the accuracyo r completenesos f the contentso f this book and specifically disclaim any implied warrantieso f merchantabilityo r fitness for a particularp urpose.N o warrantym ay be createdo r extendedb y sales representativeos r written salesm aterials.T he advice and strategiesc ontainedh ereinm ay not be suitable for your situation.Y ou shouldc onsultw ith a professionalw herea ppropriate.N eithert he publishern or authors hall be liable for any loss of profit or any other commerciald amagesi,n cluding but not limited to special,i ncidental,c onsequentialo, r other damages. For generali nformation on our otherp roductsa nd servicesp leasec ontacto ur CustomerC are Departmentw ithin the United Statesa t (800) 762-2974,o utsidet he United Statesa t (317) 572-3993o r fax (317) 572-4002. Wiley also publishesi ts booksi n a variety of electronicf ormats. Somec ontentt hat appearsin print, however,m ay not be availablei n electronicf ormats. For more information aboutW iley products,v isit our web site at www.wiley.com. Library ofC ongressC ataloging-in-Publication Data: Hillen, Thomas,1 966- Partial differential equations:t heory and completelys olvedp roblemsI ThomasH illen, I. Ed Leonard, Henry van Roessel. p.cm. Includesb ibliographicalr eferencesa nd index. ISBN 978-1-118-06330-9(h ardback) 1. Differential equations,P artial. I. Leonard,I . Ed., 1938-II. Van Roessel,H enry, 1956 III. Title. QA377.H55 2012 515'.353-dc23 2012017382 Printed in the United Stateso f America. 10 9 8 7 6 5 4 3 2 1 CONTENTS Preface Xl PART I THEORY 1 Introduction 3 1.1 Partial Differential Equations 4 1.2 Classificationo f Second-OrdeLr inear PDEs 7 1.3 Side Conditions 10 1.3.1 BoundaryC onditionso n an Interval 12 1.4 LinearPDEs 12 1.4.1 Principle of Superposition 14 1.5 Steady-Statea nd Equilibrium Solutions 16 1.6 First Examplef or Separationo f Variables 19 1.7 Derivation of the Diffusion Equation 24 1.7.1 BoundaryC onditions 25 1.8 Derivation of the HeatE quation 26 1.9 Derivation of the Wave Equation 29 1.10 Exampleso f Laplace'sE quation 33 1.11 Summary 37 1.11.1 Problemsa nd Notes 38 v vi CONTENTS 2 Fourier Series 39 2.1 PiecewiseC ontinuousF unctions 39 2.2 Even, Odd, andP eriodicF unctions 41 2.3 OrthogonalF unctions 43 2.4 FourierS eries 48 2.4.1 FourierS ine and CosineS eries 53 2.5 Convergenceo f FourierS eries 56 2.5.1 Gibbs' Phenomenon 60 2.6 Operationso n Fourier Series 63 2.7 Mean SquareE rror 74 2.8 ComplexF ourierS eries 78 2.9 Summary 81 2.9.1 Problemsa ndN otes 82 3 Separation of Variables 83 3.1 HomogeneousE quations 83 3.1.1 GeneralL inear HomogeneousE quations 89 3.1.2 Limitations of the Method of Separationo f Variables 93 3.2 NonhomogeneouEsq uations 95 3.2.1 Method of EigenfunctionE xpansions 100 3.3 Summary 111 3.3.1 Problemsa nd Notes 113 4 Sturm-Liouville Theory 115 4.1 Formulation 115 4.2 Propertieso f Sturm-Liouville Problems 119 4.3 EigenfunctionE xpansions 127 4.4 RayleighQ uotient 134 4.5 Summary 141 4.5.1 Problemsa nd Notes 143 5 Heat, Wave, and Laplace Equations 145 5.1 One-DimensionaHl eatE quation 145 5.2 Two-DimensionalH eatE quation 150 5.3 One-DimensionaWl ave Equation 153 5.3.1 d' Alembert'sS olution 157 5.4 Laplace'sE quation 163 5.4.1 Potentiali n a Rectangle 163 5.5 Maximum Principle 167 CONTENTS vii 5.6 Two-DimensionalW ave Equation 168 5.7 Eigenfunctionsin Two Dimensions 172 5.8 Summary 176 5.8.1 Problemsa nd Notes 177 6 Polar Coordinates 179 6.1 Interior Dirichlet Problemf or a Disk 179 6.1.1 PoissonI ntegralF ormula 186 6.2 Vibrating Circular Membrane 188 6.3 Bessel'sE quation 191 6.3.1 SeriesS olutionso f ODEs 191 6.4 BesselF unctions 195 6.4.1 Propertieso f BesselF unctions 201 6.4.2 IntegralR epresentationo f BesselF unctions 204 6.5 Fourier-BesseSl eries 210 6.6 Solutiont o the Vibrating MembraneP roblem 214 6.7 Summary 218 6.7.1 Problemsa ndN otes 220 7 Spherical Coordinates 221 7.1 SphericalC oordinates 221 7.1.1 Derivationo f the Laplacian 222 7.2 Legendre'sE quation 224 7.3 LegendreF unctions 227 7.3.1 LegendreP olynomials 228 7.3.2 Fourier-LegendreS eries 245 7.3.3 LegendreF unctionso f the SecondK ind 248 7.3.4 AssociatedL egendreF unctions 249 7.4 SphericalB esselF unctions 252 7.5 Interior Dirichlet Problemf or a Sphere 253 7.6 Summary 257 7.6.1 Problemsa ndN otes 259 8 Fourier Transforms 261 8.1 FourierI ntegrals 261 8.1.1 FourierI ntegralR epresentation 261 8.1.2 Examples 264 8.1.3 FourierS ine and CosineI ntegralR epresentations 268 8.1.4 Proofo f Fourier'sT heorem 271 viii CONTENTS 8.2 FourierT ransforms 277 8.2.1 OperationalP ropertieso f the FourierT ransform 281 8.2.2 FourierS ine and CosineT ransforms 284 8.2.3 OperationalP ropertieso f the Fourier Sine and Cosine Transforms 288 8.2.4 FourierT ransformsa nd Convolutions 289 8.2.5 FourierT ransformo f a GaussianF unction 294 8.3 Summary 297 8.3.1 Problemsa nd Notes 298 9 Fourier Transform Methods in POEs 299 9.1 The Wave Equation 300 9.1.1 d' Alembert'sS olution to the One-DimensionaWl ave Equation 300 9.2 The Heat Equation 305 9.2.1 Heat Flow in an Infinite Rod 305 9.2.2 FundamentaSl olution to the HeatE quation 306 9.2.3 Error Function 308 9.2.4 HeatF low in a Semi-infinite Rod: Dirichlet Condition 311 9.2.5 HeatF low in a Semi-infinite Rod: NeumannC ondition 317 9.3 Laplace'sE quation 319 9.3.1 Laplace'sE quationi n a Half-Plane 319 9.3.2 Laplace'sE quationi n a Semi-infinite Strip 324 9.4 Summary 328 9.4.1 Problemsa nd Notes 329 10 Method of Characteristics 331 10.1 Introductiont o the Method of Characteristics 331 10.2 GeometricI nterpretation 335 10.3 d' Alembert'sS olution 344 10.4 Extensiont o QuasilinearE quations 348 10.5 Summary 350 10.5.1 Problemsa nd Notes 351

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