Linear Integral Equations

427 Pages·2014·1.958 MB·English

by  Rainer Kress

Applied Mathematical Sciences
Rainer Kress
Linear
Integral
Equations
Third Edition
Applied Mathematical Sciences
Volume 82
FoundingEditors
FritzJohn,JosephP.LaSalleandLawrenceSirovich
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Rainer Kress
Linear Integral Equations
Third Edition
123
RainerKress
Institutfu¨rNumerischeundAngewandte
Georg-August-Universita¨tGo¨ttingen
Go¨ttingen,Germany
ISSN0066-5452 ISSN2196-968X(electronic)
ISBN978-1-4614-9592-5 ISBN978-1-4614-9593-2(eBook)
DOI10.1007/978-1-4614-9593-2
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013954393
Mathematics Subject Classification (2010): 35J05, 35P25, 35R25, 35R30, 45A05, 65M30, 65R20,
65R30,76Q05,78A45
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To the Memory of My Parents

Preface to the Third Edition
Inthefourteenyearssincethesecondeditionofthisbookappeared,linearintegral
equationshavecontinuedtobeanactiveareaofmathematicsandtheyhaverevealed
moreoftheirpowerandbeautytome.ThereforeIampleasedtohavetheopportu-
nitytomakeadjustmentsandadditionstothebook’scontentsinthisthirdedition.
Inthespiritofthetwoprecedingeditions,Ihavekeptthebalancebetweentheory,
applicationsandnumericalmethods.Topreservethecharacterofanintroductionas
opposedtoa researchmonograph,Ihavemadenoattemptsto includemostofthe
recentdevelopments.
Inadditiontomakingcorrectionsandadditionsthroughoutthetextandupdating
thereferences,thefollowingtopicshavebeenadded.Inordertomaketheintroduc-
tion to the basic functionalanalytic toolsmore completethe Hahn–Banachexten-
siontheoremandtheBanachopenmappingtheoremarenowincludedinthe text.
Thetreatmentofboundaryvalueproblemsinpotentialtheoryhasbeenextendedby
a more complete discussion of integral equations of the first kind in the classical
Ho¨lderspace setting andof bothintegralequationsof the firstand secondkindin
thecontemporarySobolevspacesetting.Inthenumericalsolutionpartofthebook,
I includeda new collocationmethodfor two-dimensionalhypersingularboundary
integralequationsand the collocationmethodfor the Lippmann–Schwingerequa-
tionbasedonfastFouriertransformtechniquesdueto Vainikko.Thefinalchapter
ofthebookoninverseboundaryvalueproblemsfortheLaplaceequationhasbeen
largelyrewrittenwithspecialattentiontothetrilogyofdecomposition,iterativeand
samplingmethods.
SomeoftheadditionstothisthirdeditionwerewrittenwhenIwasvisitingthe
InstitutMittag-Leffler,Djursholm,Sweden,inspring2013duringthescientificpro-
gramonInverseProblemsandApplications.Igratefullyacknowledgethehospital-
ityandthesupport.
vii
viii PrefacetotheThirdEdition
Overtheyearsmostofthethirty-onePhDstudentsthatIsupervisedwrotetheir
thesis on topicsrelated to integralequations.Their workand my discussionswith
themhavehadsignificantinfluenceonmyownperspectiveonintegralequationsas
presentedinthisbook.Therefore,ItakethisopportunitytothankmyPhDstudents
asagroupwithoutlistingthemindividually.Aspecialnoteofthanksisgiventomy
friendDavidColtonforreadingoverthenewpartsofthebookandhelpingmewith
theEnglishlanguage.
Ihopethatthisneweditionofmybookcontinuestoattractreaderstothefieldof
integralequationsandtheirapplications.
Go¨ttingen,Germany RainerKress
Preface to the Second Edition
Inthetenyearssincethefirsteditionofthisbookappeared,integralequationsand
integraloperatorshaverevealedmoreoftheirmathematicalbeautyandpowertome.
Therefore,Iampleasedtohavetheopportunitytosharesomeofthesenewinsights
withthereadersofthisbook.Asinthefirstedition,themainmotivationistopresent
thefundamentaltheoryofintegralequations,someoftheirmainapplications,and
thebasicconceptsoftheirnumericalsolutioninasinglevolume.Thisisdonefrom
myownperspectiveofintegralequations;Ihavemadenoattempttoincludeallof
therecentdevelopments.
Inadditiontomakingcorrectionsandadjustmentsthroughoutthetextandupdat-
ingthereferences,thefollowingtopicshavebeenadded:InSection4.3thepresen-
tationoftheFredholmalternativeindualsystemshasbeenslightlysimplifiedandin
Section5.3theshortpresentationontheindexofoperatorshasbeenextended.The
treatmentofboundaryvalueproblemsinpotentialtheorynowincludesproofsofthe
jumprelationsforsingle-anddouble-layerpotentialsinSection6.3andthesolution
oftheDirichletproblemfortheexteriorofanarcintwodimensions(Section7.8).
ThenumericalanalysisoftheboundaryintegralequationsinSobolevspacesettings
hasbeenextendedforbothintegralequationsof the firstkindin Section13.4and
integralequationsofthesecondkindinSection12.4.Furthermore,ashortoutline
onfastO(nlogn)solutionmethodshasbeenaddedinSection14.4.Becauseinverse
obstaclescatteringproblemsarenowextensivelydiscussedinthemonograph[32],
intheconcludingChapter18theapplicationtoinverseobstaclescatteringproblems
hasbeenreplacedbyaninverseboundaryvalueproblemforLaplace’sequation.
I would like to thank Peter Ha¨hner and AndreasVogt for carefullyreading the
manuscriptandforanumberofsuggestionsforimprovingit.Thanksalsogotothose
readerswhohelpedme byletting me knowthe errorsandmisprintstheyfoundin
thefirstedition.
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