Partial Di(cid:11)erential Equations Analytical Methods and Applications Textbooks in Mathematics Series editors: Al Boggess and Ken Rosen CRYPTOGRAPHY: THEORY AND PRACTICE, FOURTH EDITION Douglas R. S!nson and Maura B. Paterson GRAPH THEORY AND ITS APPLICATIONS, THIRD EDITION Jonathan L. Gross, Jay Yellen and Mark Anderson COMPLEX VARIABLES: A PHYSICAL APPROACH WITH APPLICATIONS, SECOND EDITION Steven G. Krantz GAME THEORY: A MODELING APPROACH Richard Alan Gillman and David Housman FORMAL METHODS IN COMPUTER SCIENCE Jiacun Wang and William Tepfenhart AN ELEMENTARY TRANSITION TO ABSTRACT MATHEMATICS Gove Effinger and Gary L. Mullen ORDINARY DIFFERENTIAL EQUATIONS: AN INTRODUCTION TO THE FUNDAMENTALS, SECOND EDITION Kenneth B. Howell SPHERICAL GEOMETRY AND ITS APPLICATIONS Marshall A. Whi#lesey COMPUTATIONAL PARTIAL DIFFERENTIAL PARTIAL EQUATIONS USING MATLAB®, SECOND EDITION Jichun Li and Yi-Tung Chen AN INTRODUCTION TO MATHEMATICAL PROOFS Nicholas A. Loehr DIFFERENTIAL GEOMETRY OFMANIFOLDS, SECOND EDITION Stephen T. Love# MATHEMATICAL MODELING WITH EXCEL Brian Albright and William P. Fox THE SHAPE OF SPACE, THIRD EDITION Jeffrey R. Weeks CHROMATIC GRAPH THEORY, SECOND EDITION Gary Chartrand and Ping Zhang PARTIAL DIFFERENTIAL EQUATIONS: ANALYTICAL METHODS AND APPLICATIONS Victor Henner, Tatyana Belozerova, and Alexander Nepomnyashchy https://www.crcpress.com/Textbooks-in-Mathematics/book-series/CANDHTEXBOOMTH Partial Di(cid:11)erential Equations Analytical Methods and Applications Victor Henner Tatyana Belozerova Alexander Nepomnyashchy CRCPress Taylor&FrancisGroup 6000BrokenSoundParkwayNW,Suite300 BocaRaton,FL33487-2742 (cid:13)c 2020byTaylor&FrancisGroup,LLC CRCPressisanimprintofTaylor&FrancisGroup,anInformabusiness NoclaimtooriginalU.S.Governmentworks Printedonacid-freepaper InternationalStandardBookNumber-13:978-1-138-33983-5(Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable e(cid:11)orts have been made to publish reliable data and information, but the author and publisher cannot assume responsibilityforthevalidityofallmaterialsortheconsequencesoftheiruse.Theauthorsandpublishers haveattemptedtotracethecopyrightholdersofallmaterialreproducedinthispublicationandapologize tocopyrightholdersifpermissiontopublishinthisformhasnotbeenobtained.Ifanycopyrightmaterial hasnotbeenacknowledgedpleasewriteandletusknowsowemayrectifyinanyfuturereprint. ExceptaspermittedunderU.S.CopyrightLaw,nopartofthisbookmaybereprinted,reproduced,trans- mitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, micro(cid:12)lming, and recording, or in any information storage or retrieval system,withoutwrittenpermissionfromthepublishers. Forpermissiontophotocopyorusematerialelectronicallyfromthiswork,pleaseaccesswww.copyright.com (http://www.copyright.com/)orcontacttheCopyrightClearanceCenter,Inc.(CCC),222RosewoodDrive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-pro(cid:12)t organization that provides licenses and regis- trationforavarietyofusers.FororganizationsthathavebeengrantedaphotocopylicensebytheCCC,a separatesystemofpaymenthasbeenarranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are usedonlyforidenti(cid:12)cationandexplanationwithoutintenttoinfringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com In memory of the wonderful scientist and friend Sergei Shklyaev Contents Preface xi 1 Introduction 1 1.1 Basic De(cid:12)nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 First-Order Equations 7 2.1 Linear First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Quasilinear First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Characteristic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Second-Order Equations 21 3.1 Classi(cid:12)cation of Second-Order Equations . . . . . . . . . . . . . . . . . . . 21 3.2 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.1 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.2 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.3 Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 The Sturm-Liouville Problem 29 4.1 General Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Examples of Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . 34 5 One-Dimensional Hyperbolic Equations 43 5.1 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 Longitudinal Vibrations of a Rod and Electrical Oscillations . . . . . . . . 48 5.3.1 Rod Oscillations: Equations and Boundary Conditions . . . . . . . . 48 5.3.2 Electrical Oscillations in a Circuit . . . . . . . . . . . . . . . . . . . 50 5.4 Traveling Waves: D’Alembert Method . . . . . . . . . . . . . . . . . . . . . 52 5.5 Cauchy Problem for Nonhomogeneous Wave Equation . . . . . . . . . . . . 57 5.5.1 D’Alembert’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.5.2 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.5.3 Well-Posedness of the Cauchy Problem. . . . . . . . . . . . . . . . . 59 5.6 Finite Intervals: The Fourier Method for Homogeneous Equations . . . . . 60 5.7 The Fourier Method for Nonhomogeneous Equations . . . . . . . . . . . . . 71 5.8 The Laplace Transform Method: Simple Cases . . . . . . . . . . . . . . . . 76 5.9 Equations with Nonhomogeneous Boundary Conditions . . . . . . . . . . . 78 5.10 The Consistency Conditions and Generalized Solutions . . . . . . . . . . . 83 5.11 Energy in the Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 vii viii Contents 5.12 Dispersion of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.12.1 Cauchy Problem in an In(cid:12)nite Region . . . . . . . . . . . . . . . . . 88 5.12.2 Propagation of a Wave Train . . . . . . . . . . . . . . . . . . . . . . 91 5.13 Wave Propagation on an Inclined Bottom: Tsunami E(cid:11)ect . . . . . . . . . 93 6 One-Dimensional Parabolic Equations 99 6.1 Heat Conduction and Di(cid:11)usion: Boundary Value Problems . . . . . . . . . 99 6.1.1 Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.1.2 Di(cid:11)usion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.1.3 One-dimensional Parabolic Equations and Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2 The Fourier Method for Homogeneous Equations . . . . . . . . . . . . . . . 103 6.3 Nonhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.4 Green’s Function and Duhamel’s Principle . . . . . . . . . . . . . . . . . . 114 6.5 The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . 118 6.6 Large Time Behavior of Solutions . . . . . . . . . . . . . . . . . . . . . . . 126 6.7 Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.8 The Heat Equation in an In(cid:12)nite Region . . . . . . . . . . . . . . . . . . . 131 7 Elliptic Equations 139 7.1 Elliptic Di(cid:11)erential Equations and Related Physical Problems . . . . . . . 139 7.2 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.3.1 Example of an Ill-posed Problem . . . . . . . . . . . . . . . . . . . . 142 7.3.2 Well-posed Boundary Value Problems . . . . . . . . . . . . . . . . . 143 7.3.3 Maximum Principle and its Consequences . . . . . . . . . . . . . . . 144 7.4 Laplace Equation in Polar Coordinates . . . . . . . . . . . . . . . . . . . . 146 7.5 Laplace Equation and Interior BVP for Circular Domain . . . . . . . . . . 147 7.6 Laplace Equation and Exterior BVP for Circular Domain . . . . . . . . . . 151 7.7 Poisson Equation: General Notes and a Simple Case . . . . . . . . . . . . . 151 7.8 Poisson Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.9 Application of Bessel Functions for the Solution of Poisson Equations in a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.10 Three-dimensional Laplace Equation for a Cylinder . . . . . . . . . . . . . 160 7.11 Three-dimensional Laplace Equation for a Ball . . . . . . . . . . . . . . . . 164 7.11.1 Axisymmetric Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.11.2 Non-axisymmetric Case . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.12 BVP for Laplace Equation in a Rectangular Domain . . . . . . . . . . . . . 167 7.13 The Poisson Equation with Homogeneous Boundary Conditions . . . . . . 169 7.14 Green’s Function for Poisson Equations . . . . . . . . . . . . . . . . . . . . 171 7.14.1 Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . 171 7.14.2 Nonhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . 175 7.15 Some Other Important Equations . . . . . . . . . . . . . . . . . . . . . . . 176 7.15.1 Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.15.2 Schr(cid:127)odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8 Two-Dimensional Hyperbolic Equations 187 8.1 Derivation of the Equations of Motion . . . . . . . . . . . . . . . . . . . . . 187 8.1.1 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . 189 8.2 Oscillations of a Rectangular Membrane . . . . . . . . . . . . . . . . . . . . 191 Contents ix 8.2.1 The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . 192 8.2.2 The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . 199 8.2.3 The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . 203 8.3 Small Transverse Oscillations of a Circular Membrane . . . . . . . . . . . . 205 8.3.1 The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . 206 8.3.2 Axisymmetric Oscillations of a Membrane . . . . . . . . . . . . . . . 209 8.3.3 The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . 214 8.3.4 Forced Axisymmetric Oscillations. . . . . . . . . . . . . . . . . . . . 216 8.3.5 The Fourier Method for Equations with Nonhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 218 9 Two-Dimensional Parabolic Equations 227 9.1 Heat Conduction within a Finite Rectangular Domain . . . . . . . . . . . . 227 9.1.1 The Fourier Method for the Homogeneous Heat Equation (Free Heat Exchange) . . . . . . . . . . . . . . . . . . . . . . . . . . 230 9.1.2 The Fourier Method for Nonhomogeneous Heat Equation with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . 233 9.2 Heat Conduction within a Circular Domain . . . . . . . . . . . . . . . . . . 237 9.2.1 The Fourier Method for the Homogeneous Heat Equation . . . . . . 238 9.2.2 The Fourier Method for the Nonhomogeneous Heat Equation . . . . 241 9.2.3 The Fourier Method for the Nonhomogeneous Heat Equation with Nonhomogeneous Boundary Conditions . . . . . . . . . . . . . 246 9.3 Heat Conduction in an In(cid:12)nite Medium . . . . . . . . . . . . . . . . . . . . 248 9.4 Heat Conduction in a Semi-In(cid:12)nite Medium . . . . . . . . . . . . . . . . . 250 10 Nonlinear Equations 261 10.1 Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 10.1.1 Kink Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 10.1.2 Symmetries of the Burger’s Equation . . . . . . . . . . . . . . . . . . 262 10.2 General Solution of the Cauchy Problem . . . . . . . . . . . . . . . . . . . 264 10.2.1 Interaction of Kinks . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.3 Korteweg-de Vries Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 267 10.3.1 Symmetry Properties of the KdV Equation . . . . . . . . . . . . . . 267 10.3.2 Cnoidal Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 10.3.3 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 10.3.4 Bilinear Formulation of the KdV Equation . . . . . . . . . . . . . . 271 10.3.5 Hirota’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 10.3.6 Multisoliton Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 274 10.4 Nonlinear Schr(cid:127)odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 277 10.4.1 Symmetry Properties of NSE . . . . . . . . . . . . . . . . . . . . . . 277 10.4.2 Solitary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 A Fourier Series, Fourier and Laplace Transforms 283 A.1 Periodic Processes and Periodic Functions . . . . . . . . . . . . . . . . . . 283 A.2 Fourier Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 A.3 Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 286