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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. . . ."
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. . . ."
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