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Loewy Decomposition of Linear Differential Equations Texts and Monographs in Symbolic Computation A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria SeriesEditor:PeterPaule,RISCLinz,Austria FoundingEditor:B.Buchberger,RISCLinz,Austria EditorialBoard RobertCorless,UniversityofWesternOntario,Canada HoonHong,NorthCarolinaStateUniversity,USA TetsuoIda,UniversityofTsukuba,Japan MartinKreuzer,Universita¨tPassau,Germany BrunoSalvy,INRIARocquencourt,France DongmingWang,Universite´PierreetMarieCurie–CNRS,France Forfurthervolumes: http://www.springer.com/series/3073 Fritz Schwarz Loewy Decomposition of Linear Differential Equations 123 FritzSchwarz InstituteSCAI FraunhoferGesellschaft SanktAugustin Germany ISSN0943-853X ISBN978-3-7091-1285-4 ISBN978-3-7091-1286-1(eBook) DOI10.1007/978-3-7091-1286-1 SpringerWienHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012945905 (cid:2)c Springer-VerlagWien2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Birgit • Preface Theaimofthisbookistocommunicatesomeresultsonsolvinglineardifferential equationsthat have been achievedin the last two decades.The key conceptis the factorization of a differential equation or the corresponding differential operator, andtheresultingdecompositionintouniqueobjectsoflowerorder.Althoughmore than100yearsold,theseresultshadbeenforgottenforalmostacenturybeforethey werereawakened. Several new developments have entailed novel interest in this subject. On the onehand,methodsofdifferentialalgebraleadtoabetterunderstandingofthebasic problemsinvolved.Insteadofdealingwithindividualequations,thecorresponding differential operators are considered as elements of a suitable ring where they generate an ideal. This proceeding is absolutely necessary if partial differential equationsandoperatorsareinvestigated.InparticulartheconceptofaJanetbasisfor thegeneratorsofanidealandtheLoewydecompositionoftheidealcorresponding tothegivenequationsareoffundamentalimportance.Inordertoapplytheseresults for solving concrete problems, the availability of computer algebra software is indispensableduetheenormoussizeofthecalculationsusuallyinvolved. Proceedingalongthese lines, forlargeclasses oflinear differentialequations– ordinary as well as partial – a fairly complete theory for obtaining its solutions in closed form has been achieved. Whenever feasible, constructive methods for algorithm design are given, and the possible limits of decidability are indicated. Thisproceedingmayserveasa modelfordealingwithotherproblemsinthearea ofdifferentialequations. IamgratefultoDimaGrigoriev,ZimingLi,MichaelSingerandSergeyTsarev for numerous discussions and suggestions. Thanks are due to Hans-Heinrich Aumu¨llerforcarefullyreadingthefinalversionofthemanuscript,andtoWinfried Neun at the Zuse Institut in Berlin for keeping the ALLTYPES website running. The ideal working environment at the Fraunhofer Institut SCAI is gratefully acknowledged. SanktAugustin,Germany FritzSchwarz July2012 vii • Contents 1 Loewy’sResultsforOrdinaryDifferentialEquations................... 1 1.1 BasicFactsforLinearODE’s.......................................... 1 1.2 FactorizationandLoewyDecomposition ............................. 3 1.3 SolvingLinearHomogeneousOde’s.................................. 11 1.4 SolvingSecond-OrderInhomogeneousOde’s........................ 16 1.5 Exercises................................................................ 20 2 RingsofPartialDifferentialOperators.................................... 21 2.1 BasicDifferentialAlgebra............................................. 21 2.2 JanetBasesofIdealsandModules .................................... 23 2.3 GeneralPropertiesofIdealsandModules ............................ 25 2.4 DifferentialTypeZeroIdealsinQ.x;y/Œ@x;@y(cid:2)...................... 30 2.5 DifferentialTypeZeroModulesoverQ.x;y/Œ@x;@y(cid:2)................ 32 2.6 LaplaceDivisorsLxm.L/andLyn.L/................................. 34 2.7 TheIdealsJxxx andJxxy............................................... 45 2.8 LatticeStructureofIdealsinQ.x;y/Œ@x;@y(cid:2)......................... 47 2.9 Exercises................................................................ 59 3 EquationswithFinite-DimensionalSolutionSpace...................... 61 3.1 EquationsofDifferentialTypeZero................................... 61 3.2 LoewyDecompositionofModulesM.0;2/............................. 63 3.3 LoewyDecompositionofIdealsJ.0;2/andJ.0;3/...................... 66 3.4 SolvingHomogeneousEquations ..................................... 73 3.5 SolvingInhomogeneousEquations.................................... 75 3.6 Exercises................................................................ 78 4 DecompositionofSecond-OrderOperators............................... 81 4.1 OperatorswithLeadingDerivative@xx................................ 81 4.2 OperatorswithLeadingDerivative@xy................................ 86 4.3 Exercises................................................................ 90 ix