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Item Details
| Title:
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PRINCIPLES OF PARTIAL DIFFERENTIAL EQUATIONS
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| By: |
Andrew Komech, Alexander Komech |
| Format: |
Hardback |
| List price:
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£49.99 |
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further information.
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| ISBN 10: |
1441910956 |
| ISBN 13: |
9781441910950 |
| Publisher: |
SPRINGER-VERLAG NEW YORK INC. |
| Pub. date: |
1 September, 2009 |
| Edition: |
2010 ed. |
| Series: |
Problem Books in Mathematics |
| Pages: |
171 |
| Description: |
This concise book covers the classical tools of Partial Differential Equations Theory in today's science and engineering. The rigorous theoretical presentation includes many hints, and the book contains many illustrative applications from physics. |
| Synopsis: |
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...65 9 Derivationoftheheatequation ...65 10 Mixedproblemfortheheatequation ...67 11 TheSturm-Liouvilleproblem...68 12 Eigenfunctionexpansions...74 13 TheFouriermethodfortheheatequation...78 14 Mixedproblemforthed'Alembertequation ...83 15 TheFouriermethodfornonhomogeneousequations ...86 16 TheFouriermethodfornonhomogeneousboundaryconditions ...93 17 TheFouriermethodfortheLaplaceequation ...95 3 DistributionsandGreen'sfunctions...105 18 Motivation ...105 19 Distributions ...109 20 Operationsondistributions ...110 21 Differentiationofjumpsandtheproductrule ...115 22 Fundamentalsolutionsofordinarydifferentialequations...118 23 Green'sfunctiononaninterval ...121 24 Solvabilityconditionfortheboundaryvalueproblems...125 25 TheSobolevfunctionalspaces...128 26 Well-posednessofthewaveequationintheSobolevspaces ...130 ix x Contents 27 Solutionstothewaveequationinthesenseofdistributions...131 4 FundamentalsolutionsandGreen'sfunctionsinhigherdimensions. . 133 n 28 FundamentalsolutionsoftheLaplaceoperatorinR ...133 29 Potentialsandtheirproperties...137 30 ComputingpotentialsviatheGausstheorem...143 31 Methodofre?ections ...144 32 Green'sfunctionsin2Dviaconformalmappings...149 A Classi?cationofthesecond-orderequations...155 References...159 Index ...161 Chapter1 Hyperbolicequations. Methodofcharacteristics 1Derivationofthed'Alembertequation Thed'Alembertequation,alsocalledtheone-dimensionalwaveequation, 2 2 ? u ? u 2 (x,t)=a +f(x,t), x?[0,l], t>0, (1. 1) 2 2 ?t ?x wherea>0isaconstant,describessmalltransversaloscillationsofastretched stringorlongitudinaloscillationsofanelasticrod.Letusgiveabriefderivationofthisequation. Foramorerigorousargument,see [Vla84,SD64,TS90]. Transversaloscillationsofastring Weassumethatastringoflengthlis u stretchedwiththeforceT. Wechoosethe directionoftheaxisOxalongthestring initsequilibriumcon?guration. Letx=0 u(x,t) correspondto the left end of the string. Thentherightendofthestringisgiven 0 x l x byx=l. SeeFig. 1. 1. |
| Illustrations: |
85 black & white illustrations, biography |
| Publication: |
US |
| Imprint: |
Springer-Verlag New York Inc. |
| Returns: |
Returnable |
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