Principles of Partial Differential Equations

ISBN-13: 9781441910950 / Angielski / Twarda / 2009 / 161 str.

Andrew Komech; Alexander Komech

Opis

ThisbookisintendedtogivethereaderanopportunitytomastersolvingPDEpr- lems. Ourmaingoalwastohaveaconcisetextthatwouldcovertheclassicaltools ofPDEtheorythatareusedintoday'sscienceandengineering, suchaschar- teristics, thewavepropagation, theFouriermethod, distributions, Sobolevspaces, fundamentalsolutions, andGreen'sfunctions. WhileintroductoryFouriermethod -basedPDEbooksdonotgiveanadequatedescriptionoftheseareas, themore advancedPDEbooksarequitetheoreticalandrequireahighlevelofmathematical backgroundfromareader. Thisbookwaswrittenspeci?callyto?llthisgap, sat- fyingthedemandofthewiderangeofenduserswhoneedtheknowledgeofhow tosolvethePDEproblemsandatthesametimearenotgoingtospecializeinthis areaofmathematics. Arguably, thisistheshortestPDEcourse, whichstretchesfar beyondcommon, Fouriermethod-basedPDEtexts. Forexample, Hab03], which isacommonthoroughtextbookonpartialdifferentialequations, teachesasimilar setoftoolswhilebeingabout?vetimeslonger. Thebookisproblem-oriented. Thetheoreticalpartisrigorousyetshort. So- timeswereferthereadertotextbooksthatgivewidercoverageofthetheory. Yet, - portanttheoreticaldetailsarepresentedwithcare, whilethehintsgivethereaderan opportunitytorestoretheargumentstothefullrigor. Manyexamplesfromphysics areintendedtokeepthebookintuitiveforthereaderandtoillustratetheapplied natureofthesubject. Thebookwillbeusefulforanyhigher-levelundergraduatecourseandforse- studyforbothgraduateandhigher-levelundergraduatestudents, andforanys- cialtyinsciences. ItsRussianversionhasbeenastandardproblem-solvingmanual atMoscowStateUniversitysince1988, andisalsousedbystudentsofSt. Pete- burgUniversityandNovosibirskUniversities. ItsSpanishversionisusedatMorelia UniversityinMexico, whiletheEnglishdrafthasalreadybeenusedinViennaU- versityandatTexasA&MUniversity. Forfurtherreadingwerecommend Str92], Eva98], and EKS99]. Mu]nchen, AlexanderKomech August2007 AndrewKomech v Acknowledgements The?rstauthorisindebtedtoMargaritaKorotkinaforthefortunatesuggestionto writethisbook, toA. F. Filippov, A. S. Kalashnikov, M. A. Shubin, T. D. Ventzel, andM. I. Vishikforcheckingthe?rstversionofthemanuscriptandfortheadvice. BothauthorsaregratefultoH. Spohn(TechnischeUniversita]t, Mu]nchen)andto E. Zeidler(Max-PlanckInstituteforMathematics, Leipzig)fortheirhospitalityand supportduringtheworkonthebook. BothauthorsweresupportedbyInstituteforInformationTransmissionProblems (RussianAcademyofSciences). The?rstauthorwassupportedbytheDepartment ofMechanicsandMathematicsofMoscowStateUniversity, bytheAlexandervon HumboldtResearchAward, FWFGrantP19138-N13, andtheGrantsofRFBR. The secondauthorwassupportedbyTexasA&MUniversityandbytheNationalScience FoundationunderGrantsDMS-0621257andDMS-0600863. vii Contents 1 Hyperbolicequations. Methodofcharacteristics. . . . . . . . . . . . . . . . . . . 1 1 Derivationofthed'Alembertequation. . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Thed'Alembertmethodforin?nitestring . . . . . . . . . . . . . . . . . . . . . . 7 3 Analysisofthed'Alembertformula. . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Second-orderhyperbolicequationsintheplane . . . . . . . . . . . . . . . . . 19 5 Semi-in?nitestring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6 Finitestring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7 Waveequationwithmanyindependentvariables . . . . . . . . . . . . . . . . 46 8 Generalhyperbolicequations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2 TheFouriermethod. . . .

Recenzje

From the reviews:

"This book is intended to give the reader an opportunity to master solving problems in partial differential equations. ... This book has been written specifically to satisfy the demand of a wide audience who needs knowledge of how to solve PDE problems ... . The book under review is mainly addressed to those in higher-level undergraduate courses and for self-study for both graduate and higher-level undergraduate students." (Vicentiu Radulescu, Mathematical Reviews, Issue 2010 k)

Spis treści

Hyperbolic equations. Method of characteristics.- The Fourier method.- Distributions and Green’s functions.- Fundamental solutions and Green’s functions in higher dimensions.- Erratum.

Blurb

This concise book covers the classical tools of PDE theory used in today's science and engineering: characteristics, the wave propagation, the Fourier method, distributions, Sobolev spaces, fundamental solutions, and Green's functions.  The approach is problem-oriented, giving the reader an opportunity to master solution techniques.  The theoretical part is rigorous and with important details presented with care.  Hints are provided to help the reader restore the arguments to their full rigor.  Many examples from physics are intended to keep the book intuitive and to illustrate the applied nature of the subject. The book is useful for a higher-level undergraduate course and for self-study.