Regular order reductions of ordinary and delay-dif

arXiv:physics/9810015v1[physics.comp-ph]9Oct1998Regularorderreductionsofordinaryanddelay-differentialequationsJ.M.Aguirregabiriaa,1,Ll.Belb,A.Hern´andezaandM.RivasaaF´ısicaTe´orica,FacultaddeCiencias,UniversidaddelPa´ısVasco,Apdo.644,48080Bilbao(Spain)bLaboratoiredeGravitationetCosmologieRelativistes,CNRS/URA769,Universit´ePierreetMarieCurie,4,placeJussieu.Tour22-12,Boˆıtecourrier142,75252ParisCedex05(France)AbstractWepresentaCprogramtocomputebysuccessiveapproximationstheregularorderreductionofalargeclassofordinarydifferentialequations,whichincludesevolutionequationsinelectrodynamicsandgravitation.Thecodemayalsofindtheregularorderreductionofdelay-differentialequations.Keywords:Ordinarydifferentialequations;Delay-differentialequations;Orderreduction;Lorentz-Diracequation;Abraham-Lorentzequation;ChaoticscatteringPACS:02.30.Hq,02.30.Hq,04.50.+h,04.90.+e,41.60.-m,47.52.+jProgramLibraryIndexsection:4.3DifferentialEquations1Correspondingauthor.e-mail:[e-mail].:[phone](ext.2585)FAX:[phone]PreprintsubmittedtoElsevierPreprint2February2008PROGRAMSUMMARYTitleofprogram:ODEredCatalogidentifier:Programobtainablefrom:CPCProgramLibrary,Queen’sUniversityofBelfast,N.IrelandLicensingprovisions:noneComputers:ThecodeshouldworkonanycomputerwithanANSICcompiler.IthasbeentestedonaPC,aDECAlphaServer1000andaDECAlphaAXP3000-800S.Operatingsystem:ThecodehasbeentestedunderWindows95,DigitalUnixv.4.08(Rev.564)andOpenVMSV6.1.Programminglanguageused:ANSICMemoryrequiredtoexecutewithtypicaldata:Itdependsonthenumberofequationsandretardedpointsstoredinmemory.Manyinterstingproblemscanbesolvedin1Mbyte.No.ofbytesindistributedprogram,includingtestdata,etc.:84.695Distributionformat:ASCIIKeywords:Ordinarydifferentialequations;Delay-differentialequations;Orderre-duction;Lorentz-Diracequation;Abraham-Lorentzequation;ChaoticscatteringNatureofthephysicalproblemIndifferentphysicalproblems,includingelectrodynamicsandtheoriesofgravitation,thereappearsingulardifferentialequationswhoseorderdecreaseswhenaphysicalparametertakesaparticularbutveryimportantvalue.Typicallymostsolutionsoftheseequationsareunphysical.Theregularorderreductionisanequationoflowerorderwhichcontainspreciselythephysicalsolutions,whicharethoseregularinthatparameter.Theprogramcomputesthesolutionoftheregularorderreductionforalargesetofordinaryanddelay-differentialequations.MethodofsolutionThebasicintegrationroutineisbasedonthecontinuousPrince-Dormandmethodofeighthorder.Ateachintegrationstep,successiveapproximationsareperformedbyusingthepolynomialinterpolatingthesolutionthathasbeencomputedinthepreviousapproximation.TypicalrunningtimeItdependsheavilyonthenumberandcomplexityoftheequationsandonthedesiredsolutionrange.Itwasatmostacoupleofsecondsinthetestproblems.2LONGWRITE-UP1IntroductionIndifferentphysicaltheoriesthereappearsingularevolutionequationsthatsharesomecommonproperties:mostoftheirsolutionsareunphysicalbecausetheirorderishigherthanexpectedexceptforaparticular,butimportant,valueofaparameterforwhichtheorderreducestowhatonewouldexpectonphysicalgrounds.Forinstance,theLorentz-Diracequation[1]thatdescribesthemotionofachargedparticlewithradiationreactionisofthirdorder,sothatinitialpositionandvelocitywouldnotbeenoughtodetermineitsevolutionandmostofitsmathematicalsolutionsare‘runaway’,i.e.theaccel-erationincreaseswithoutlimit.Nevertheless,inthelimitwhenthechargegoestozerotheLorentz-Diracequationbecomesasecond-orderequationwhilethenonphysicalsolutionsdivergeforthatvalueoftheparameter.Equationsofthiskindshowstrongnumericalinstabilities,whichpreventfromintegratingthemforwardintime:evenifthephysicalinitialconditionsarechosen,theintegrationerrorintroducesaninitiallysmallcontributionfromthenonphysicalsolutionswhichthenblowout[2,3].Thestandardrecipetoavoidthisproblemistointegratebackwards[2,3],butthisisimpossibleinmanycasesbecausethefinalstatefromwhichtointegrateisunknown[4].Anaturalapproachtothiskindofproblemsisprovidedbytheconceptof“regularorderreduction”,whichisanevolutionequationwiththerightorderthatcontainspreciselythephysicalsolutionsandbehavessmoothlyfortheparticularvalueoftheparameterforwhichtheorderofthesingularequationdecreases[5].InthecontextoftheLorentz-DiracequationthisconceptwasdiscussedbyKerner[6]andSanz[7].Orderreductionshavebeenalsousedtoreplacethedelay-differentialequationsthatappearintheelectrodynam-ics[8,9]andinnon-linearoptics[10],aswellastoanalysefourth-orderequa-tionsthatappearintheoriesofgravitationwithaquadraticLagrangian[11]andinthestudyofquantumcorrectionstoEinsteinequations[12].Exceptinrathertrivialcases,theorderreductioncannotbecomputedexactlyandsomeapproximationschemeisnecessary.Onemayuseapowerexpan-sion[7],buttheexplicitexpressionsbecomequicklytoocomplex.Severalyearsago,oneofus(Ll.B.)wrotearoutinetofindtheregularorderreductionofdelay-differentialequations.Tocomputenumericallytheorderreductionofsingularordinarydifferentialequationswehaveproposedandanalysedinsomecasesamethodofsuccessiveapproximations[4,13].Thegoalofthispaperistwofold:wewanttomakewidelyavailablethecodefor3t