Table Of Content

Atul Kumar Razdan V. Ravichandran Fundamentals of Partial Differential Equations Fundamentals of Partial Differential Equations · Atul Kumar Razdan V. Ravichandran Fundamentals of Partial Differential Equations AtulKumarRazdan V.Ravichandran DepartmentofAppliedSciences DepartmentofMathematics Engineering NationalInstituteofTechnology MIET Tiruchirappalli Meerut,UttarPradesh,India Tiruchirappalli,TamilNadu,India ISBN978-981-16-9864-4 ISBN978-981-16-9865-1 (eBook) https://doi.org/10.1007/978-981-16-9865-1 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SingaporePteLtd.2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore TomywifeAnjuandchildren,Anujaand Aman —AtulKumarRazdan; ToProf.Dato’InderaRosihanM.Aliandmy wifeKalaiselvi —V.Ravichandran Preface Differentialequationsprovideaunifyingthemetostudyphysicalsystems.Asingle differentialequationinvolvesthederivativesofascalarfunctionofoneormoreinde- pendent variables that, in practical terms, is an approximation of a law governing naturalprocessesofthesystem.Weobtaina(coupled)systemofdifferentialequa- tionswhiledealingwithaphenomenondefinedintermsofavectorfunctionofone ormoreindependentvariables.Ingeneral,differentialequationsarefundamentalto modelmanydifferenttypesofphenomena.Togetherwithsomedeeperconceptsin analysisandgeometry,theyhelpsolveimportantpracticalproblemsinscienceand engineering[1–3]. Astudyofunivariatephenomenasuchasrelatedtoplanetarymotionandgravita- tionalforcesamonglargebodiesconductedintheseventeenthcenturyledNewtonto applyordinarydifferentialequationsasanimportanttooltoanalyserelatedpractical problems. In 1747, d’Alembert was first to formulate differential equation model foramultivariatephenomenonsuchasvibratingstring,whichisknownasthewave equationindimensionone.Subsequently,severaleminentmathematiciansdeveloped differentialequationmodelsforvariousothertypesofmultivariatephenomenasuch asconcerningfluidflow(Euler,1755),minimalsurfaces(Lagrange,1760),hydrody- namics(DanielBernoulli,1762),potentialtheory(Laplace,1780),heatconduction (Fourier, 1805), three-body problems (Poincáre, 1892), continuous symmetry (Lie, 1888),andmanymore. Applications of differential equations in diverse scientific investigations, and certainrelateddevelopments,haveprovedtobeaseffectivetothephysicalsciences as it has been to the evolution of modern mathematics [4]. The discovery of some mathematical theories ensued in the twentieth century facilitated wide spectrum applicationsofdifferentialequationstodynamicalsystemsinalmostalldisciplines of science and engineering [5]. The present book is mainly about basic aspects of some important partial differential equations that find applications to problems in physics and engineering such as concerning mechanical vibrations, wave motion, heatconduction,fluidflow,andelectrodynamics.Weshallbedealingwithdifferen- tial equations involving two or three independent variables. Throughout the book, wehaveattemptedtomaintainaproperbalancebetweenthemathematicalconcepts vii viii Preface introduced and their applications to the physical world. The reader interested in abstracttheoryofpartialdifferentialequationsmayliketoreadexcellenttextssuch as[6]or[7]or[8]. Theworkonthisbookstartedsometimeagowithclassnotesofthesecondauthor onthetopicrealanalysisandpartialdifferentialequationsthathetaughttounder- graduateengineeringstudentsattheNITTiruchirappalli.Thefirstpartofthebook developsmathematicaltoolsrelatedtoclassicalvectoranalysisandordinarydiffer- ential equations. In Chap. 1, we review some relevant concepts of multivariable analysis,basictheoryofsurfacesandcurves,andalsoclassicalvectorcalculus.In Chap.2,wediscussthetheoryofordinarydifferentialequationsinasmuchasneeded in the subsequent chapters. The former is used in Chap. 3 to derive fundamental differentialequationmodelsofsomemultivariatephenomenaconcerningtransport of physical quantities such as the mass, momentum, energy, and the charge. The treatmentofmosttopicscoveredinthefirstpartofthebookismainlyillustrative, withanoccasionaldigressiontorelatedtheory. In Chap. 4, we introduce basic concepts such as order and linearity type of a general partial differential equation. The focus of the chapter is to discuss mathe- maticalissuesrelatedtoaclassicalsolutionofinitial-boundaryvalueproblems.In Chap. 5, we discuss Lagrange–Charpit method of finding the complete integral of general first-order partial differential equations. In Chap. 6, we apply the method of characteristics to solve Cauchy problems for some hyperbolic type differential equations. The main part of the book discusses mathematical concepts relevant to some standardanalyticalsolutionmethods suchasseparationofvariables,Fourierseries, eigenfunctionsexpansion,andalsotransformtechniquesduetoFourierandLaplace. Weapplyaforementionedmethodstosolvesomeinterestinginitial-boundaryvalue problemsforprototypicalsecondorderlinearpartialdifferentialequationssuchas wave, heat, and Laplace equations. The book is largely self-contained excepting the details related to some important theorems of analysis that are used in various chapters.Thereadermayrefercompanionvolume“FundamentalsofAnalysiswith Applications”[9],oranyotherstandardtextsuchas[10],forprerequisitesrelatedto realanalysis. Themainaudienceforthepresentbookarestudentspursuinganadvancedlevel undergraduatecourseinphysicsandengineering.However,anappropriateselection oftopicsmayalsobeusefultothebeginner-levelgraduatestudentsinmathematics. For example, a suitable amplification of topics presented in the first four chapters may be offered as an introductory course on the subject to the graduate students interestedincertainspecificapplicationsofdifferentialequations. Preface ix We are thankful to two reviewers for their comments. We are thankful to Dr. ShamimAhmad,PublishingEditor,Mathematics,SpringerIndia,forhiscontinued support;hehasinsistedustowritetwoseparatebooksinsteadofouroriginalplan ofasinglebookonRealAnalysisandPDE!Theauthorsarethankfultotheteamat SpringerandMr.T.ParimelAzhaganandMr.NareshKumarManifortheircontin- uoussupportduringtheproductionprocess.Wewillbehappytoreceivecomments forimprovementsincludingerrors/misprints. Meerut,India AtulKumarRazdan Tiruchirappalli,India V.Ravichandran December2021 References 1. Keshet,L.E.,MathematicalModelinginBiology,ClassicsinAppliedMathematics,Vol.46, SIAM,2005. 2. Murray,J.D.,MathematicalBiology–AnIntroduction,Springer,NY,2002. 3. Perthame,B.,TransportEquationsinBiology,Birkhäuser,2007. 4. Kline, M., Mathematical Thought from Ancient to Modern Times, Vol. 1, 2, 3, Oxford UniversityPress,London,1972. 5. Brezis, H. and Browder, F., Partial Differential Equations in the 20th Century, Adv. in Mathematics,135,76-144,1998. 6. Arnold,V.I.,LecturersonPartialDifferentialEquations,Springer,2004. 7. Evans,L.C.,PartialDifferentialEquations,AMSGTS,Vol.19,2010. 8. Jost,J.,PartialDifferentialEquations,GTM-214,Springer,NewYork,2002. 9. Razdan,A.andRavichandran,V.,FundamentalsofAnalysiswithApplications,Springer,to appear. 10. Rudin,W.,PrinciplesofMathematicalAnalysis,thirdedition,McGraw-HillBookCo.,New York,1976. Contents 1 Introduction ................................................. 1 References .................................................... 12 2 ClassicalVectorAnalysis ....................................... 13 2.1 MultivariableCalculus .................................... 13 2.2 ClassicalTheoryofSurfacesandCurves ..................... 36 2.3 VectorCalculus .......................................... 49 References .................................................... 76 3 OrdinaryDifferentialEquations ................................ 77 3.1 Introduction ............................................. 78 3.2 FirstOrderDifferentialEquations .......................... 82 3.2.1 IntegrableForms ................................... 84 3.2.2 Picard–LindelöfTheorem ........................... 95 3.3 HigherOrderLinearDifferentialEquations .................. 102 3.3.1 TheCaseofConstantCoefficients .................... 109 3.3.2 PowerSeriesSolution .............................. 116 3.4 BoundaryValueProblems ................................. 128 3.4.1 Green’sFunctionsandNonhomogeneousProblems ..... 131 3.4.2 Sturm–LiouvilleTheory ............................ 140 3.4.3 EigenfunctionsExpansions .......................... 146 3.5 FirstOrderSystemofDifferentialEquations ................. 154 3.5.1 ExistenceandUniquenessTheorem .................. 157 3.5.2 LinearSystems .................................... 160 References .................................................... 172 4 PartialDifferentialEquationModels ............................ 175 4.1 MathematicalModelling .................................. 176 4.2 ThreePrototypicalEquations .............................. 179 4.3 ModelsforTransportPhenomena ........................... 193 References .................................................... 201 xi xii Contents 5 PartialDifferentialEquations .................................. 203 5.1 Preliminaries ............................................ 204 5.2 ClassificationandCanonicalForms ......................... 220 5.3 ClassicalSolution ........................................ 227 5.4 Initial-BoundaryValueProblems ........................... 233 5.5 UniquenessTheoremsandStabilityIssues ................... 242 References .................................................... 249 6 GeneralSolutionandCompleteIntegral ......................... 251 6.1 CharacteristicsCoordinates ................................ 251 6.2 Lagrange’sMethod ....................................... 270 6.3 LinearEquationswithConstantCoefficients ................. 281 6.4 Lagrange–CharpitMethod ................................. 294 7 MethodofCharacteristics ...................................... 305 7.1 LinearandSemilinearEquations ........................... 306 7.2 QuasilinearEquations ..................................... 318 7.3 FullyNonlinearEquation .................................. 324 References .................................................... 334 8 SeparationofVariables ........................................ 335 8.1 VibratingStringControversy ............................... 336 8.2 FourierSeries ............................................ 339 8.3 SeparationofVariables .................................... 379 References .................................................... 402 9 MethodofEigenfunctionsExpansion ........................... 403 9.1 GeneralisedFourierSeries ................................. 404 9.2 NonhomogeneousBoundaryValueProblems ................. 406 9.3 PoissonEquations ........................................ 411 References .................................................... 416 10 FourierTransforms ........................................... 417 10.1 Introduction ............................................. 418 10.2 Fourier’sTransformPair .................................. 426 10.3 TransformsofGeneralisedFunctions ........................ 440 10.4 FundamentalProperties ................................... 444 10.5 ApplicationstoPartialDifferentialEquations ................. 457 References .................................................... 478 11 LaplaceTransform ............................................ 479 11.1 BasicTheoremsandExamples ............................. 480 11.2 PropertiesofLaplaceTransform ............................ 485 11.3 InverseLaplaceTransform ................................. 499 11.4 ApplicationstoDifferentialEquations ....................... 506 Reference ..................................................... 521