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ACKNOWLEDGMENTS

TheﬁrstauthorthanksFulbrightVisitingScholarProgramfor givinghimtheopportunitytovisitU.S.A.andtheNavalPostgraduateSchoolinMonterey,CA,U.S.A.forhostinghimduring theninemonthsofhistenurein2012-2013.Thesecondauthor thanksShotaRustaveliNationalScientiﬁcFoundationofRepublicofGeorgiaforgivinghimopportunitytovisitU.S.A. andtheNavalPostgraduateSchoolinMonterey,CA,U.S.A.for hostinghimduringthefourmonthsofhistenurein2013.

Abstract

Numerical Solutions of Three Classes of Nonlinear Parabolic Integro-Differential Equation. http://dx.doi.org/10.1016/B978-0-12-804628-9.50008-9

Thisbookisconcernedwiththenumericalsolutionsofsome classesofnonlinearintegro-differentialmodels.Somepropertiesofthesolutionsofthecorrespondinginitial-boundaryvalue problemsstudiedinthemonographequationsaregiven.Three typesofnonlinearintegro-differentialmodelsareconsidered.Algorithmsoffindingapproximatesolutionsareconstructedand investigated.Resultsofnumericalexperimentswithtablesand graphicalillustrationsandtheiranalysisaregiven.Thebook consistsoffourchapters.Attheendofthebookalistofthe quotedliteratureandindexesaregiven.Eachchapterisconcludedwithadetailedsection,entitled”Commentsandbibliographicalnotes,”containingreferencestotheprincipalresults treated,aswellasinformationonimportanttopicsrelatedto, butsometimesnotincludedinthebodyofthetext. 1

Keywords: Electromagneticfieldpenetration,Maxwell’s equations,integro-differentialmodels,existenceanduniqueness, asymptoticbehavior,semi-discreteandfinitedifferenceschemes, Galerkin’smethod,finiteelementapproximation,errorestimate, stabilityandconvergence.

Chapter1 Introduction

Abstract

Thedescriptionofvariouskindsofintegro-differentialequationsandabriefhistoryoftheiroriginandapplicationsare given.Theimportanceofinvestigationsofintegro-differential modelsispointedoutaswell.Classificationofintegro-differentialequationisgiven.Themainattentionispaidonparabolic typeintegro-differentialmodels.Inparticular,threetypesof integro-differentialequationsareconsidered.Twoofthemare basedonMaxwell’sequationsdescribingelectromagneticfield penetrationintoasubstance.Thethirdoneisobtainedby simulationofheatflow.Attheendofthechapter,asatthe endofeachchapter,thecommentsandbibliographicalnotesis given,whichconsistsofdescriptionofreferencesconcerningto thetopicconsidered.

Numerical Solutions of Three Classes of Nonlinear Parabolic Integro-Differential Equation. http://dx.doi.org/10.1016/B978-0-12-804628-9.50001-6

Keywords: Electromagneticfieldpenetration,Maxwell’s system,heatflowequation,integro-differentialmodels. 3

Inmathematicalmodelingofappliedtasksdifferential,integral,andintegro-differential(I-D,forshort)equationsappear veryoften.Therearenumerousscientificworksdevotedtothe investigationofdifferentialequations.Thereisavastliterature inthefieldofintegralandintegro-differentialmodelsaswell.

Thediﬀerentialequationsareconnectingunknownfunctions, theirderivatives,andindependentvariables.Ontheotherhand, integralequationscontaintheunknownfunctionsunderanintegralaswell.

Thetermintegro-diﬀerentialequationintheliteratureis usedinthecasewhentheequationcontainsunknownfunction togetherwithitsderivativesandwheneitherunknownfunction, oritsderivatives,orbothappearunderanintegral.

Letusrecallthegeneralclassificationofintegro-differential equations.Iftheequationcontainsderivativesofunknownfunctionofonevariablethentheintegro-differentialequationiscalled ordinaryintegro-differentialequation.Theorderofanequation isthesameasthehighest-orderderivativeoftheunknownfunctionintheequation.

Theintegro-differentialequationsoftenencounteredinmathematicsandphysicscontainderivativesofvariousvariables; therefore,theseequationsarecalledintegro-differentialequationswithpartialderivativesorpartialintegro-differentialequations.

Intheapplicationsveryoftenthereareintegro-diﬀerential equationswithpartialderivativesandmultipleintegralsaswell, forexample,Boltzmannequation[66]andKolmogorov-Feller equation[288].

Volterraisoneofthefoundersofthetheoryofintegraland integro-diﬀerentialequations.Hisworks,especiallyintheintegralandintegro-diﬀerentialequations,areoftencitedtilltoday. TheclassicalbookbyVolterra[469]iswidelyquotedintheliterature.In1884Volterra[465]beganhisresearchinthetheoryof

CHAPTER1.INTRODUCTION

integralequationsdevotedtodistributionofanelectricalcharge onasphericalpatch.Thisworkledtotheequation,whichin themodernliteratureiscalledtheintegralequationoftheﬁrst kindwithsymmetrickernel.

Theworkonthetheoryofelasticitybecamethebeginning researchofVolterraleadingtothetheoryofpartialintegrodifferentialequations.In1909Volterra[466]hasstudiedaparticulartypeofsuchequationsandhasshownthatthisintegrodifferentialequationisequivalenttoasystemconsistingofthree linearintegralequationsandasecondorderpartialdifferential equations.

Theﬁrstexamplesofintegro-diﬀerentialequationswithpartialderivativesinvestigatedinthebeginningofthetwentieth centurywereinSchlesinger’sworks[417],[418],wherethefollowingequationisinvestigated:

Numerousworksinthebeginningofthetwentiethcentury weredevotedtoresearchofintegro-diﬀerentialequationsofvariouskinds.Theexcellentbibliographyinthiscaseisgiveninthe classicalbookbyVolterra[469].Inaddition,Kerimov[271],the editoroftheRussiantranslationofthisbook,hasupdated(up to1970s)thelistofreferencesonintegralandintegro-diﬀerential equations.

Letusdescribesomeclassesofmathematicalmodelsofsecondorderpromotingintensiveresearchonpartialintegro-diﬀerentialequations.

Whenwetakeintoaccounthereditaryphenomena,thequestionsofphysicsandmechanicsleadtointegro-diﬀerentialequations.Ahereditaryphenomenonoccursinasystemwhenthe phenomenondoesnotdependonlyontheactualstateofthe systembutonalltheprecedingstatesthroughwhichthesys-

Integro-differentialequationsofparabolictypeariseinthe studyofvariousproblemsinphysics,chemistry,technology, economics,etc.Oneveryimportantproblemofappliedtype isgeneratedbymathematicalmodelingofprocessesofelectromagneticfieldpenetrationintoasubstanceandisdescribedby thewell-knownMaxwell’sequations[300].Intheworks[187] and[188],complexsystemcorrespondingtononlinearpartialdifferentialequationswasreducedtointegro-differentialform.If thecoefficientofthermalheatcapacityandelectroconductivity ofthesubstancearehighlydependentontemperature,thenthe Maxwell’ssystemcanberewritteninthefollowingform(see [187]and[188]):

Insystem(1.6), ∇× W and ∇· W aretheusualvectoroperatorswithrespecttothevariables x =(x1,x2,x3). Evenonedimensionalscalarversionofthismodelisverycomplicatedand itsinvestigationhasbeenpossibleyetonlyforspecialcases.The one-dimensionalscalarcaseofthemodel(1.6),(1.7)hasthefollowingform

where a(S) ≥ a0 = const> 0isagainaknownfunctionofits argument.Investigationof(1.6),(1.7),and(1.8)typemodels

CHAPTER1.INTRODUCTION

beganintheworks[138],[187],and[188].Sincethenmany scientiﬁcpublicationsweredevotedtotheinvestigationofexistenceanduniquenessoftheirsolutionsundervarioustypesof initialandboundaryconditions.Inthisrespect,especiallysigniﬁcantaretheworks[49],[50],[141],[146],[147],[219],[220], [238],[254],[256],[302],[303],[304],[305],[306],[307],[322], [333],[334],andreferencetherein.Authorsofthisbookhave alsomadecontributioninthisdirection,forexample,see[137], [138],[139],[140],[141],[145],[146],[147],[186],[187],[188], [219],[220],[223],[238],[247],[248],[253],[254],[256],[261], andreferencetherein.

Makingcertainphysicalassumptionsinmathematicaldescriptionoftheabove-mentionedprocessofpenetrationofelectromagneticfieldintoasubstance,Laptev[306]hasconstructed anewintegro-differentialmodel,whichrepresentsageneralizationofthesystemintroducedin[187]and[188].Founded onMaxwell’ssystemthefollowingparabolicintegro-differential modelisobtained

Intheworks[303],[305],and[306]forconditionallyclosed operators,anoperatorschemeisconstructed.Thisschemeis usedfor(1.6),(1.7)typemodelstoproveexistenceanduniquenessofsolutionofinitial-boundaryvalueproblems.Intheabovementionedwork[306]Laptevpointsoutthatfortheseso-called averagedintegro-diﬀerentialmodels(1.9),itisnecessarytodevelopadiﬀerent,specialapproach.

Insystems(1.6),(1.7),and(1.9), W =(W1,W2,W3)denotes avector,whichisconnectedbyavectorofamagneticﬁeld H = (H1,H2,H3)andisthefunctionofthevariables(x1,x2,x3,t), whichwewillshortento(x,t).

Themultidimensionalscalaranaloguesofsystems(1.6),(1.7), and(1.9)havethefollowingforms: ∂U (x,t)

and

respectively.

Inequations(1.10)and(1.11)wehave x =(x1,...,xn)and thevectoroperator ∇U isgivenby ∇U = gradU =

(x,t)

1 ,...,

(x,t)

n =(D1U,...,DnU ) .

Somegeneralizationsofthemodels(1.6)-(1.11)aregivenin theworks[137],[139],[145],[219],[305],and[306].Onekindof thesemodelshastheforms: ∂U (x,t)

(x,t)

CHAPTER1.INTRODUCTION

and

Themodelsoftype(1.6)-(1.13)arecomplexandhavebeen intensivelystudiedbymanyauthors.Theexistenceanduniquenessofglobalsolutionsofinitial-boundaryvalueproblemsfor equationsandsystemsoftype(1.6)-(1.13)werestudiedin [49],[50],[137],[138],[139],[140],[141],[145],[186],[187],[188], [219],[223],[238],[247],[248],[253],[261],[302],[303],[304], [305],[306],[307],[322],[333],[334],andinanumberofother worksaswell.

Theexistencetheoremsthatareprovedin[137],[138],[139], [140],[141],[145],[187],[188],and[219]arebasedona-priori estimates,modiﬁedGalerkin’smethodandcompactnessargumentsasin[327],[328],[461],and[462]fornonlinearelliptic andparabolicequations.

Forequation(1.8)withnonhomogeneousright-handsideand a(S)=1+ S,orforequation(1.13)intheone-dimensional casewith q =2,suchtheoremforﬁrstinitial-boundaryvalue problemisprovedinsection3.6.

Theasymptoticbehavioras t →∞ ofthesolutionsofsuch modelshasbeentheobjectofintensiveresearchinrecentyears, see[28],[29],[31],[32],[33],[34],[35],[139],[141],[145],[146], [147],[217],[218],[219],[220],[223],[227],[228],[229],[231], [232],[233],[234],[235],[236],[237],[238],[239],[240],[241], [242],[243],[245],[246],[247],[248],[251],[252],[253],[254], [256],[257],[261],[276],[277],[278],[280],[282],[283],andreferencetherein.

Anothermodelconsideredinthisbookandstudiedbyone oftheauthorsofthismonograph[376]is

andalsoMSandPhDstudentsoftheappropriatespecializations.

1.1Commentsandbibliographical notes

Mathematicalmodelsofmanynaturalphenomenaandprocesses canbedescribedbytheinitial-boundaryvalueproblemsposed fornonstationarypartialdifferentialandintegro-differentialequationsandsystemsofsuchequations.Investigationandnumericalsolutionoftheseproblemsaretheactualsphereofmathematicalphysicsandnumericalanalysis.Onesuchpartialintegrodifferentialmodeldescribestheprocessofelectromagneticfield penetrationintoasubstance.Inthequasi-stationaryapproximation,thecorrespondingsystemofMaxwell’spartialdifferentialequationscanberewritteninintegro-differentialform(1.6), (1.7)(see[187]and[188]).

Mathematicalmodelsdescribingelectromagneticprocesses andmanyrelativephenomenaaregiveninmanyscientiﬁcpapersandbooks,see,forexample,[92],[136],[300],[359],[360], [399],[419],[473],andreferencestherein.

ItiswellknownthatelectromagneticfielddiffusionprocessesandmanyotherimportantpracticalprocessesaresimulatedbyMaxwell’ssystemsofpartialdifferentialequationsand Maxwell’s-typesystemsaswell(see,forexample,[195],[197], [290],[297],[298],[300],[360],[426],[485],[486],[487],[489], [491],andreferencestherein).

Manyscientiﬁcworksaredevotedtoinvestigationofvarious problemsforMaxwell’sandMaxwell’s-typesystems,see[4],[5], [56],[60],[101],[106],[120],[121],[124],[131],[139],[144],[145], [152],[158],[194],[195],[197],[208],[224],[225],[226],[247], [253],[275],[276],[290],[362],[365],[367],[397],[409],[427],

[452],[453],[454],[470],[475],[476],[477],[485],[486],[487], [489],[491],andreferencestherein.

Aswehavealreadypointedout,byusingMaxwell’ssystem [300]formathematicalsimulationoftheprocessofelectromagneticﬁeldpenetratingintoasubstance,newclassofintegrodiﬀerentialmodels(1.6),(1.7),(1.8),and(1.10)arises(see[187] and[188]).

Integro-diﬀerentialequationsariseinmanyotherpractical processesaswell,see,forexample,[11],[21],[22],[36],[37], [55],[59],[74],[84],[119],[127],[128],[164],[185],[187],[188], [295],[298],[306],[310],[311],[312],[355],[356],[357],[363], [373],[380],[384],[405],[425],[459],[463],[465],[466],[467], [468],[469],[471],[484],andinanumberofotherworksaswell.

Themotivationforstudyingintegro-differentialproblems comesfromthemanyphysicalmodelsinsuchfieldsaselectromagneticwavepropagation,heattransfer,nuclearreactor dynamics,andthermoelasticity.Besidestheintegro-differential equationsariseinmanyspheresofhumanactivityaswell.For example,thesecondorderfullynonlinearintegro-differential equationsarederivedfromthepricingproblemoffinancialderivativesandoptimalportfolioselectionprobleminamarket[59]. In[84]nonlinearintegro-differentialequationsthatarisefrom stochasticcontrolproblemswithpurelyjumpLevyprocesses areconsidered.

Manyproblemsofmodernscienceandengineeringcanbe describedbypartialintegro-diﬀerentialequations.Sincequite alotoftheseproblemsaretime-dependent,mostofthemare evolutionequationsandespeciallynonlinearevolutionparabolic equations,see[93],[177],[179],[180],[191],[216],[310],[311], [312],[322],[332],[345],[346],[358],[451],andreferencestherein).

Manyscientiﬁcworksaredevotedtoinvestigationandnumericalsolutionofparabolicintegro-diﬀerentialmodels,see[6], [18],[45],[53],[62],[63],[69],[74],[83],[86],[87],[93],[98],[103], 1.1.COMMNETSANDBIBLIOGRAPHICALNOTES

[108],[115],[118],[128],[137],[139],[145],[155],[156],[163], [175],[177],[181],[182],[191],[198],[247],[253],[263],[265], [266],[267],[276],[279],[281],[306],[308],[310],[311],[312], [313],[319],[320],[322],[323],[327],[353],[358],[370],[376], [378],[389],[393],[427],[430],[432],[433],[447],[448],[449], [450],[451],[456],[479],[483],[488],[492],[497],andreferences therein.

Studyofthemodelsoftype(1.6),(1.7),and(1.8)hasbegun intheworks[138],[187]and[188].Intheseworks,inparticular,thetheoremsofexistenceofsolutionoftheinitial-boundary valueproblem(withﬁrst(Dirichlet)boundaryconditions)for scalarequationwithone-dimensionalspacevariableareproved. Investigationsofhigherspacedimensionsformodel(1.10)carriedoutinitiallyin[137]and[140].

In[306]somegeneralizationofthesystemoftype(1.6),(1.7) wasproposed.Inparticular,assumingthatthetemperature oftheconsideredbodydependsonthetimevariable,butindependentofthespacecoordinates,thenthesameprocessof penetrationofamagneticﬁeldintothematerialissimulated bytheaveraged(astheauthorof[306]hasnamedit)integrodiﬀerentialmodels(1.9)and(1.11).

Studyofthemodelsoftype(1.9)and(1.11)hasstartedin theworks[217]and[219].

Onemustnotethatsomeworksweredevotedtothestudy ofmodelingofphysicalprocessofelectromagneticﬁeldpenetrationinthecaseofcylindricalconductors.Inthiscase,the above-mentionedintegro-diﬀerentialmodel(1.6),(1.7),written incylindricalcoordinates,wasgivenin[148].Thework[333] isdevotedtotheinvestigationofperiodicproblemforonedimensional(1.8)typemodelincylindricalcoordinates.

Interestintheabove-mentionedintegro-diﬀerential(1.6)(1.11)modelsisincreasing.Somegeneralizationsof(1.10)and (1.11)models,whichhavetheforms(1.12)and(1.13)corre-

1.1.COMMNETSANDBIBLIOGRAPHICALNOTES

spondingly,aregivenandstudiedin[219],[305],andinanumberofotherworksaswell.Equation(1.12)isinvestigated,for example,in[145],[219],[305],and[306].Equation(1.13)was investigatedin[145]and[219].Inthescientiﬁcliteraturesome moregeneralmodelshavealsoappeared,see[137],[139],[145], [219],[305],[306],and[322].

Manydiﬀerentkindsofinitial-boundaryvalueproblemswith avarietyofboundaryandinitialconditionsareconsideredfor theabove-mentionedintegro-diﬀerentialequations.Intheworks [246],[280],[281]investigationandnumericalapproximationof problemswithmixedboundaryconditions,for(1.10)and(1.11) typeone-dimensionalscalarmodels,arestudied.

Letusalsonotethatﬁrstkindinitial-boundaryvalueproblemswithnonhomogeneousboundaryconditionononesideof lateralboundaryarealsoconsideredandstudiedinmanyworks. Thistypeoftheproblemstatementisdictatedbymathematical simulationofthephysicalprocesses,see,forexample,[148]and problem(2.64),(2.65)givenin”Commentsandbibliographical notes”sectioninChapter2.

Thetheoremsprovedforinvestigatingtheasymptoticbehaviorastimetendstoinfinityinsomecasesshowthedifferencebetweenstabilizationcharacterofsolutionswithhomogeneousand nonhomogeneousboundaryconditionsofthefirstkindinitialboundaryvalueproblems.Moreprecisely,inhomogeneouscase stabilizationhasanexponentialcharacter,whereasinnonhomogeneouscaseithaspower-likeform.

Theworks[49],[50]arealsoworthmentioning,whereinvestigationofinverseproblemsformultidimensionalmodelsof (1.10)typeiscarriedout.

Anotherintegro-diﬀerentialmodelstudiedinthismonograph is(1.14).Thismodeldescribesheatﬂowinmaterialwithmemory[196],[345].Italsoarisesinthetheoryofviscoelasticity [109],[344],and[346].

Asarulewecannotfindexactsolutionsoftheconsidered nonlineardifferentialandintegro-differentialmodels.Therefore, particularattentionshouldbepaidtoconstructionofnumerical solutionsandtotheirimportanceforintegro-differentialmodels. Thefirststepsinthisdirection,forthemodelsstudiedinthis monograph,aremadeintheworks[139],[220],[274],[376],and [378].Nowtheresearchinthisdirectionhasintensified.

Letusnotethatthemodelsbeingconsideredinthismonographhavearisenfrompracticaltasks.Buttheycanbeconsideredasmodels,generalizingknownnonlinearparabolicequations,whicharestudiedinmanyknownscientiﬁcpapers,books, andmonographs,see[76],[133],[134],[169],[199],[298],[327], [332],[381],[385],[461],andreferencestherein.

Manyscientiﬁcresearchesaredevotedtonumericalsolution ofpartialdiﬀerentialandnonlinearparabolicequationsaswell, see[23],[43],[104],[167],[171],[183],[193],[210],[213],[225], [226],[301],[327],[400],[404],[407],[411],[415],[435],[439], [440],[447],andthereferenceslistedinthesepapersandbooks.

AswehavealreadymentionedthemainpartofintegrodifferentialstructuresconsideredherehasarisenfromMaxwell’s systemsofpartialdifferentialequations.TherearemanyscientificpapersandbooksonthenumericalsolutionofMaxwell’s systemsandMaxwell’s-typesystemsaswell,see[4],[5],[15], [26],[27],[54],[68],[139],[142],[145],[153],[212],[224],[247], [253],[260],[276],[314],[315],[316],[317],[336],[341],[369], [434],[442],[472],[480],[493],[498],andreferencestherein. Theresultsoftheseresearchesveryoftenareusedinconstructionandinvestigationofthenumericalalgorithmsforthecorrespondingintegro-differentialmodels.

Thedetaileddescriptionofinvestigationandnumericalsolutionoftheabove-mentionedmodelsisgiveninChapters2-4. Morecompletereferencesandcommentsaregiveninasection entitled”Commentsandbibliographicalnotes”ineachchapter.