Partial differential equations and solitary waves theory

761 Pages·2009·3.02 MB·English

by  Abdul-Majid Wazwaz

NONLINEAR PHYSICAL SCIENCE
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SERIES EDITORS
AlbertC.J.Luo NailH.Ibragimov
Department ofMechanical and Industrial Engi- DepartmentofMathematicsandScience
neering BlekingeInstituteofTechnology
SouthernIllinoisUniversityEdwardsville S-37179Karlskrona,Sweden
Edwardsville,IL62026-1805,USA Email:[email protected]
Email:[email protected]
INTERNATIONAL ADVISORY BOARD
PingAo,UniversityofWashington,USA;Email:[email protected]
JanAwrejcewicz,TheTechnicalUniversityofLodz,Poland;Email:[email protected]
EugeneBenilov,UniversityofLimerick,Ireland;Email:[email protected]
EshelBen-Jacob,TelAvivUniversity,Israel;Email:[email protected]
MauriceCourbage,Universite´Paris7,France;Email:[email protected]
MarianGidea,NortheasternIllinoisUniversity,USA;Email:[email protected]
JamesA.Glazier,IndianaUniversity,USA;Email:[email protected]
ShijunLiao,ShanghaiJiaotongUniversity,China;Email:[email protected]
JoseAntonioTenreiroMachado,ISEP-InstituteofEngineeringofPorto,Portugal;Email:[email protected]
NikolaiA.Magnitskii,RussianAcademyofSciences,Russia;Email:[email protected]
JosepJ.Masdemont,UniversitatPolitecnicadeCatalunya(UPC),Spain;Email:[email protected]
DmitryE.Pelinovsky,McMasterUniversity,Canada;Email:[email protected]
SergeyPrants,V.I.Il’ichevPacificOceanologicalInstituteoftheRussianAcademyofSciences,Russia;
Email:[email protected]
VictorI.Shrira,KeeleUniversity,UK;Email:[email protected]
JianQiaoSun,UniversityofCalifornia,USA;Email:[email protected]
Abdul-MajidWazwaz,SaintXavierUniversity,USA;Email:[email protected]
PeiYu,TheUniversityofWesternOntario,Canada;Email:[email protected]
Abdul-Majid Wazwaz
Partial Differential
Equations and
Solitary Waves Theory
With 14figures
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•
.. HIGHEREDUCATIONPRESS
Author
Abdul-MajidWazwaz
DepartmentofMathematics
SaintXavierUniversity
Chicago,IL60655,USA
E-mail:[email protected]
ISSN1867-8440 e-ISSN1867-8459
NonlinearPhysicalScience
ISBN978-7-04-025480-8
HigherEducationPress,Beijing
ISBN978-3-642-00250-2 e-ISBN978-3-642-00251-9
SpringerDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2009920462
(cid:2)c HigherEducationPress,BeijingandSpringer-VerlagBerlinHeidelberg2009
Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyright LawofSeptember9,
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areliabletoprosecutionundertheGermanCopyrightLaw.
Theuseofgeneral descriptive names, registered names, trademarks, etc. inthis publication does not
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Coverdesign:FridoSteinen-Broo,EStudioCalamar,Spain
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Thisbookisdedicated tomywife, ourson,
andour threedaughtersforsupportingmein
allmyendeavors

Preface
PartialDifferentialEquationsandSolitaryWavesTheoryisdesignedtoserveasa
textandareference.Thebookisdesignedtobeaccessibletoadvancedundergrad-
uateandbeginninggraduatestudentsaswellasresearchmonographtoresearchers
in applied mathematics, science and engineering.This text is differentfrom other
texts in that it explains classical methods in a non abstract way and it introduces
and explainshow the newly developedmethodsprovidemore concise methodsto
provideefficientresults.
Partial Differential Equations and Solitary Waves Theory is designed to focus
readers’attentionsontheserecentlydevelopedvaluabletechniquesthathaveproven
theireffectivenessandreliabilityoverexistingclassicalmethods.Moreover,thistext
alsoexplainsthenecessaryclassicalmethodsbecausetheaimisthatnewmethods
wouldcomplementthetraditionalmethodsinordertoimprovetheunderstandingof
thematerial.
The book avoids approaching the subject through the compact and classical
methods that make the material impossible to be grasped, especially by students
whodonothavethebackgroundintheseabstractconcepts.Compacttheoremsand
abstracthandlingofthematerialarenotpresentedinthistext.
The book was developed as a result of many years of experience in teaching
partialdifferentialequationsandconductingresearchworkinthisfield.Theauthor
has taken account on his teaching experience, research work as well as valuable
suggestions received from students and scholars from a wide variety of audience.
Numerousexamplesandexercises,ranginginlevelfromeasytodifficult,butcon-
sistentwiththematerial,aregivenineachsectiontogivethereadertheknowledge,
practiceandskillinpartialdifferentialequationsandsolitarywavestheory.Thereis
plentyofmaterialinthistexttobecoveredintwosemestersforseniorundergradu-
atesandbeginninggraduatesofMathematics,Science,andEngineering.
Thecontentofthebookisdividedintotwodistinctparts,eachisaself-contained
andpracticalpart.PartIcontainselevenchaptersthathandlethepartialdifferential
equations by using the newly developed methods, namely, Adomian decomposi-
tionmethodandVariationalIterationMethod.Someofthetraditionalmethodsare
usedinthispart.Withadiversereadershipandinterdisciplinaryaudienceofapplied
viii Preface
mathematics, science, and engineering, attempts are made so that part I presents
bothanalyticalandnumericalapproachesinaclearandsystematicfashiontomake
thisbookaccessibletomanywhoworkinthisfield.
PartIIcontainssevenchaptersdevotedtothoroughlyexaminesolitarywavesthe-
ory.Sincethediscoveryofsolitonsin1965,mathematicians,engineers,andphysi-
cists have been intrigued by the rich mathematical structure of solitons. Solitons
playaprevalentroleinpropagationoflightinfibers,surfacewavesinnonlineardi-
electrics,opticalbistability,opticalswitchinginslabwaveguides,andmanyother
phenomenainplasmaandfluiddynamics.
Chapter1providesthebasicdefinitionsandintroductoryconcepts.Initialvalue
problemsandboundaryvalueproblemsarediscussed.InChapter2,thefirstorder
partialdifferentialequationsarehandledbythenewlydevelopedmethods,namely,
the Adomian decomposition method (ADM) and the variational iteration method
(VIM).Themethodofcharacteristicsisintroducedandexplainedindetail.Chapter
3 deals with the one-dimensional heat flow where homogeneous and inhomoge-
neousinitial-boundaryvalueproblemsareapproachedbyusingthedecomposition
method,thevariationaliterationmethodandthemethodofseparationofvariables.
Chapter 4 is entirely devoted to the two-dimensional and three-dimensional heat
flow.Chapter5providesthereaderwithacomprehensivediscussionoftheliterature
relatedto the one-dimensionalwave equation.Thedecompositionmethodandthe
variationaliterationmethodareusedin handlingthe waveequationsina bounded
and an unbounded domain. Moreover, the method of separation of variables and
the D’Alembert method are also used. Chapter 6 presents a comprehensive study
onwave equationsintwo-dimensionalandthree-dimensionalspaces. Chapter7is
devotedtotheLaplace’sequationintwo-andthree-dimensionalrectangularcoordi-
natesandinpolarcoordinates.Moreover,theLaplace’sequationinannulusformis
alsoinvestigatedbyusingthedecompositionmethodandtheseparationofvariables
method.Chapter8introducesacomprehensivestudyonnonlinearpartialdifferen-
tialequations.Eventhoughthesubjectisconsidereddifficultandmostlyaddressed
indistinctbooksindependentoflinearPDEs,butitwillbehandledsuccessfullyand
elegantlybyusingthenewlydevelopeddecompositionmethodandthevariational
iteration method. Chapter 9 provides the reader with a variety of linear and non-
linearapplicationsselected frommathematicalphysics,populationgrowthmodels
andevolutionconcepts.Theusefulconceptof solitonsandthe recentlydeveloped
conceptofCompactons arethoroughlyexaminedbyusingbothtraditionalandnew
methods.Chapter10isconcernedwiththenumericaltechniques.Emphasisinthis
chapterwillbe oncombiningthe decompositionseriessolution,the variationalit-
erationmethod,andthePade´approximantstoprovideapromisingtoolthatcanbe
appliedforfurtherapplications.Chapter11isconcernedwiththeconceptsofsoli-
tonsand compactons.In this chapter,the solitonsand compactonsare determined
by using prescribed conditions, a necessary condition for the applicability of the
decompositionmethod.
PartIIofthisbookgivesaself-contained,practicalandrealisticapproachtosoli-
tarywavetheory.Thedissipationandthedispersioneffectsarethoroughlyinvesti-
gated.Solitonsplayaprevalentroleinmanyscientificandengineeringphenomena.
Preface ix
Thenewlydiscoveredcompactons:solitonswithacompactsupportarealsostudied.
PartIIofthisbookisdevotedtousemainlytheHirota’sbilinearmethod,combined
withsimplifiedversiondevelopedbyHeremanandthetanh-cothmethod.Chapter
12 presents discussions about the dissipation and dispersion effects, analytic and
nonanalytic solutions, conservation laws and multiple-soliton solutions, tanh-coth
method,andHirota’sbilinearmethodcombinedwiththeHereman’ssimplifiedform
oftheHirota’smethod.InChapter13,thefamilyoftheKdVequationsisstudied.
Multi-solitonsolutionsareobtainedforonlycompletelyintegrableequationsofthis
family.Compactonssolutionsarealsoexamined.Chapter14isconcernedwithKdV
andmKdVequationsofhigherorders.Thesinglesolitonsandthemultiple-soliton
solutionsforcompletelyintegrableequationsareaddressedbyusingtheHirota’sbi-
linearmethod.Inaddition,theHirota-Satsumaequationsandthegeneralizedshort
waveequationswereinvestigatedformultiple-solitonsolutions.
Chapter15investigatesmanyKdV-typeofequationswheresolitonsolutionsand
multi-solitonsolutionsareobtainedbyusingtanh-cothmethodandHirota’smethod
respectively.Chapter16isentirelydevotedtostudyafamilyofwell-knownphysi-
calmodelsforsolitonsandmulti-solitonsolutionsaswell.Someoftheseequations
are Boussinesq equation,Klein-Gordonequation,Liouvilleequation,sine-Gordon
equation, DBM equation,and others. Chapter 17 providesthe reader with a com-
prehensivediscussionoftheliteraturerelatedtoBurgers,Fisher,Huxley,FitzHugh-
Nagumoequationsandrelatedequations.Mostoftheseequationsarecharacterized
bythedissipationphenomenathatgivekinkssolutions.Chapter18presentsacom-
prehensive study on two distinct types of equations that appear in solitary wave
theory.ThefamilyofCamassa-Holmequationsis examinedto obtainthenonana-
lyticsolutionofpeakons.Ontheotherhand,theSchrodingerandGinzburg-Landau
equationsofdifferentordersarestudiedinthischapter.
Thebookconcludeswithsixusefulappendices.Moreover,thebookintroduces
the traditionalmethods in the same amountof concernto providethe reader with
theknowledgeneededtomakeacomparison.
IdeeplyacknowledgeProfessorLouisPennisiwhomadeveryvaluablesugges-
tions that helped a great deal in directing this book towards its main goal. I also
deeplyacknowledgeProfessorMasaakiItoandProfessorWillyHeremanformany
helpfuldiscussionsandusefulremarks.Iowethemmydeepestthanks.
I am deeply indebted to my wife, my son and my daughterswho providedme
with theircontinuedencouragement,patienceandsupportduringthe longdaysof
preparingthisbook.
The author would highly appreciate any note concerningany constructivesug-
gestion.
SaintXavierUniversity Abdul-MajidWazwaz
Chicago,IL60655 E-mail:[email protected]
2009