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1、Asymptoticapproachesinmechanics:NewparametersandproceduresIgorVAndrianovInstitutfuÈrAllegmeineMechanik,RWTHAachen,Templergraben64,D-52056Aachen,Germany;igor±andrianov@hotmail.comJanAwrejcewiczTechnicalUniversityofèoÂdzÂ,DepartmentofAutomaticsandBiomechanics,1/15StefanowskiegoS
2、t.,90-924èoÂdzÂ,Poland;awrejcew@ck-sg.p.lodz.plRemGBarantsevSt.PetersburgStateUniversity,Box159,195256,St.Petersburg,Russia;brem@mail.ruThissurveyisdevotedtorecentachievementsinthe®eldofasymptoticapproaches.Hereweconsidertheasymptoticsinrelationtocompletelynewandsometimesunexp
3、ectedparameters.Someproceduresleadingtoimprovementandisolationoftheessentialanalyticalstructureoftheperturbationseriesarepresented.Itisalsoshownthatmanyproblemsofperturbationtheory,whichseemtoberelativelysimpleata®rstglance,arestillfarfromcompletelysolved.Differentasymptoticte
4、chniquestosolvethesameproblemandtheirin¯uenceontheresultsarebrie¯yillustratedanddiscussed.Thisreviewpapercontains310references.@DOI:10.1115/1.1521436#Asymptoticsshouldbeseenastheframeworkformodel-questionsstillappear:Howtoformulatetheboundarycon-ing,andtheframeworkinwhichthewh
5、oleproblem,itsfor-ditionsinordertoachievethesmallesterrorsofthemulation,solution,interpretation,andre®nement,shouldbeasymptoticapproximations?Howtoproceediftheconsid-embedded.Crigton(1994)eredproblemdoesnothavesmallorlargeasymptoticparam-eters?Theseandrelatedproblemsaredealtwi
6、thinthissurvey.1INTRODUCTIONAlthoughthegivenexamplespresentmainlytheauthors'researchandtheyareaddressedtothe®eldofNonlinearInastate-of-the-artreview@1,2#wehaveillustratedanddis-MechanicsandMechanicsofSolids,theareaofapplicationscussedaroleofPadeÂapproximants,continuedfractions
7、,asisfarwider.Inordertobringmorelighttofurtherconsider-wellasothermethodsdevotedtosummationsandintegra-ations,wegiveourgeneralpointofviewonasymptotictionsinmodernasymptoticanalysis.Amongothers,wehavemethods.Eachoftheappliedasymptoticmethodspossessesemphasizedthatalloftheasympt
8、oticmethodsarefarfromitsownimportantmeaning,butsometimesisnot
