Table Of ContentLoewy 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
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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
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