2005 Boundary Value Problems for Third-Order Nonli

BoundaryValueProblemsforThird-OrderNonlinearOrdinaryDifferentialEquationsByP.L.Sachdev,N.M.Bujurke,andV.B.AwatiInthispaper,wedescribehowtoanalyzeboundaryvalueproblemsforthird-ordernonlinearordinarydifferentialequationsoveraninfiniteinterval.Severalphysicalproblemsofinterestaregovernedbysuchsystems.Theseminumericalschemesdescribedhereoffersomeadvantagesoversolutionsobtainedbyusingtraditionalmethodssuchasfinitedifferences,shootingmethod,etc.Thesetechniquesalsorevealtheanalyticstructureofthesolutionfunction.Forillustrativepurposes,severalphysicalproblems,mainlydrawnfromfluidmechanics,areconsidered;theyclearlydemonstratetheefficiencyofthetechniquespresentedhere.1.IntroductionThird-ordernonlinearordinarydifferentialequationsholdingoveraninfiniteintervalappearinvariousbranchesofphysicsandengineeringandhavebeenatopicofextensiveanalysisinavarietyoffluidflowproblems.Bothnonlinearityandinfiniteintervaldemandnovelmathematicaltoolsfortheiranalysis.ForspecifictypeofboundaryconditionssomeoftheapproximatesolutionssuchasDirichletseriessolutions[1]aremoreuseful,especiallyinobtainingderivedquantities,thanpurenumericalschemes.TheaccuracyAddressforcorrespondence:P.L.Sachdev,DepartmentofMathematics,IndianInstituteofScience,Bangalore560012,India;e-mail:[e-mail]ATICS115:303–318303C
2005bytheMassachusettsInstituteofTechnologyPublishedbyBlackwellPublishing,350MainStreet,Malden,MA02148,USA,and9600GarsingtonRoad,Oxford,OX42DQ,UK.304P.L.Sachdevetal.aswellasuniquenessofthesolutionsoobtainedcanbeconfirmedusingotherequallypowerfulseminumericalschemes.Oneofthemethodsinthiscategoryrequiresintroductionofnewvariables,thusconvertingthird-orderequationsintosecond-orderequationswhosesolutionmaybeobtainedbyusingpowerseries.Sometimesathird-ordernonlinearODEmaybereducedtoasecond-orderEmden–Fowlerequationorfirst-orderAbelequation,whichitselfcanbesolvedinaparametricformbutthesolutionssoobtainedaretooimplicittobeofmuchuseinpractice.AbroadclassofphysicalproblemsofinterestundertheabovementionedcategoryhasbeenpresentedbySachdev[2].Severalapproachesforfindingsolutionsweregiventhere.TheseproceduresincludeDirichletseriessolution,reductionofequationstoEmden–Fowler/Abeltype,phaseplaneanalysis,perturbationmethods,etc.WeseeksolutionofequationsofthetypeF−(p1−DF)F+q1F+KF2−MFF=0,(1)satisfyingF(∞)=0,where,p1,q1,D,K,andMareconstants[3]andaprimedenotesderivativewithrespecttotheindependentvariableη.TheseequationsadmitDirichletseriessolutions;necessaryconditionsfortheexistenceanduniquenessofthesesolutionsmayalsobeobtained.Ouraimhereistopresentqualitativefeaturesofsolutionsofphysicalproblemsofinterest.Weobservethatthemajorityoftheseproblemshaveariseninfluiddynamics—stretchingboundaryinextrusionprocess,themanufacturingofpolymerandmetalsheets,coolingofaninfinitemetallicplate,boundarylayeralongaliquidfilmincondensationprocess,andthermalcapillaryflowinaviscouslayer,etc.EquationsofBlasius[4]andFalknerandSkan[5]areclassicexamplesoftheclasspresentedhere.Hartree[6]presentednumericalsolutionsforthisfamilyofself-similarproblems.OtherexamplesinwhichsolutionofFalkner–Skanequationoccursareaxisymmetricflowduetostretchingflatsurface[7],magnetohydrodynamicsfreeconvection[8,9],andfreeconvectiveflowsinsaturatedporousmedia,thermallydrivencavityflowsinporousmedia[10],andthermalcapillaryflowsinviscouslayers[11].Boundaryvalueproblemsoveraninfiniteinterval,whicharenoteasilyamenabletoavailablenumericalschemes,areofmuchinterest.AlargemajorityoftheproblemsquotedabovearegovernedbydifferentialequationsofthetypeF+AFF+BF2=0,(2)withboundaryconditionsF(0)=α1,F(0)=β1,F(∞)=0,(3)BoundaryValueProblems305orF(0)=α2,F(0)=β2,F(∞)=0,(4)whereAandBareconstants.Equation(2)isaspecialcaseof(1).Theboundaryconditions(3)includestudiesduetonaturalconvectionandflowsduetostretchingofsurfaceswhile(4)arerelevanttotheanalysisoffreeconvectionandthinlayerflows.SpecialcasesofEquation(1)withtheboundaryconditions(3)or(4)arelistedbelow:(i)F+2FF−F2=0,F(0)=0,F(0)=1,F(∞)=0,[7].(5)(ii)F+35FF−15F2=0,F(0)=0,F(0)=1,F(∞)=0,[12].(6)(iii)F+12FF=0,F(0)=0,F(0)=1,F(∞)=0,[9].(7)(iv)F+FF−βF2=0,F(0)=0,F(0)=1,F(∞)=0,(8)whereβ=2m1+misaparameter[13].(v)F+3FF−2F2=0,F(0)=c,F(0)=1,F(∞)=0,(9)wherecisthesuctionparameter[12].(vi)F+α+12FF−αF2=0,F(0)=0,F(0)=1,F(∞)=0,(10)whereαpertainstothetemperaturedistributionprescribedonthewall[14].(vii)F+AFF+BF2=0,F(0)=0,F(∞)=0,F(0)=−1,(11)whereAandBareconstants[10].(viii)F+2FF−F2=0,F(0)=0,F(∞)=0,F(0)=−1,(12)seeRef.[11].(ix)5F+6FF+3F2=0,F(0)=0,F(∞)=0,F(0)=−1,(13)seeRef.[11].Weobservefromexamples(5)to(13)thatthesignofAisalwayspositivewhilethatofBmaybepositiveornegative.Boundaryconditionsfortheexamples(5)–(10)areoftype(3)andwhilethosefor(11)–(13)areoftype(4).306P.L.Sachdevetal.Itisinterestingtoobservethatthelargeclassofequationsofthetype(2)maybereducedtoageneralizedformofEmden–Fowlerequation.SubstitutingF=Z(F)in(2),wehaveZ2Z+ZZ2−AFZZ−BZ2=0,whereZ=dZdF,F=dFdη.(14)Now,usingthetransformationZ(F)=√2ξ,ξ=ξ(F),(1