非线性微分方程对称和无穷级数解的符号计算研究

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1、上海交通大学博士后学位论文非线性微分方程对称和无穷级数解的符号计算研究姓名：姚若侠申请学位级别：博士后专业：物理学指导教师：楼森岳20090201PARTI：AmaplepackagetocomputeLiesymmetrygroupsandthesymmetryreductionsof(1+1)一dimensionalnonlinearsystemsAbstract．ArmedwiththecomputeralgebrasystemMaple，usingadirectalgebraicsubsti-tutionmethod，Liepointsymmet

2、ries，Liesymmetrygroupsandthecorrespondingsymmetryreductionsofonecomponentnonlinearintegrableandnonintegrableequationscouldbeoh-tainedonlybyclickingthe“Enter'’key．Abundant(1+1)一dimensionalnonlinearmathemat-icalphysicalsystemshavebeenanalyzedeffectivelybyusingtheMaplePackageLieSYMG

3、RPKeywords：Liepointsymmetry,Liesymmetrygroup，symmetryreduction，symboliccom-putationPACS：02．20．一a，02．30．Jr，02．70．一C．1IntroductionAsdescribedinthetraditionalLieapproachbyG．W．Bluman【1】，SophusLieshowedthatthesymmetrygroupsofordinarydifferentialequations(ODEs)couldbeusedtoreducetheord

4、eroftheequations，andwiththehelpofthespecialsubstitutions(invariantsolutions)therequirementofinvarianceofagivenpartialdifferentialequation(PDE)withrespecttosomeLietransformationgroupsensuresitsreducibility,andalsocouldbeusedtocheckwhetherornotaPDEcanbelinearizedandtoconstructanexp

5、licitlinearizationwhenonexists．SophusLieintroducedthenotionofacontinuousgroupoftransformationstoputordertohodgepodgeoftechniquesforsolvingODEs．Liegroupsoftransformationsarecharacterizedbyinfinitesimalgenerators，whichcouldbefoundbyusingthealgorithmgivenbyLie．ALiegroupoftransformat

6、ionsadmittedbyadifferentialequationcorrespondstoamappingofeachofitssolutionstoanothersolutionofthesamedifferentialequation．HenceitisimportantandinterestingforthestudyofPDEsaswella8ODEs．TheinterestedreadersarereferredtothebooksbyBlumanandKumei【1】’Olver【2]andOvsiannikov【3，4】formore

7、informationontheLiegroupapproachforsolvingdifferentialequations．Now，thereexistsomepackagessuchasSODEandSPDE【5】-【7】SYMMGRP．MAX18卜【11】，1ie【12】andliesymm【13]，whichearlputoutthereduceddeterminingequations．However，noneofthemcouldgivethecorrespondingsymmetryreductionsforgivenPDEs．Thisp

8、aperaimstointroduceadirectcomputeralgebr