Neumann Asymptotic Eigenvalues of Sturm-liouville Problem with Three Turning Points

Transcription

1 Australan Journal of Basc and Appled Scences 5(5): ISSN Neumann Asymptotc Egenalues of Sturm-loulle Problem wth Three Turnng Ponts E.A.Sazgar Department of Mathematcs Yerean state Unersty Armena. Abstract: e consder the dfferental equaton y q y y I " ( ) ( ) [] (*). In ths paper we study the asymptotc egenalues of Sturm-Loulle problem wth Neumann conons n three turnng ponts case that are here the zeros of φ() n (*). e fnd second term of asymptotc Neumann egenalues wth three turnng ponts case. Key words: Turnng pont Egen alues Neumann conons. INTRODUCTION e consder the boundary alues problems whch are the followng form: y q y y I " ( ) ( ) [] where λ = ρ s the spectral parameter; φ and q are real functons. e suppose that ( ) ( ) ( ) () () where < < < < L ε N φ () > for I = [ ] and φ s twce contnuously dfferentable functon on I. On the other words φ has n I there zeros = of order R the zeros of φ are called turnng ponts. Dfferental equatons wth turnng ponts play an mportant role n mathematcs and other branches of natural scences for eample n elastcty optcs geophyscs. In fact dfferental equatons wth turnng ponts hae arous applcatons. The mportance of asymptotc analyss n obtanng the soluton of Sturm-Loulle equaton wth turnng ponts was realzed by Leuny Oler Eberhard Frelng n (997). Neamaty and Sazgar (9) authors obtaned asymptotc Neumann egenalues of Sturm-Loulle problem n two turnng ponts. In ths paper we obtaned the asymptotc egenalues of equaton () n tree turnng ponts case wth Neumann conon.. Notatons: In the real second-order dfferental equatons y" qy ( ) ( y ) () φ has n I there zeros of order l = where R s een R s odd and R s een. Let ε > be fed suffcently small and let D [ ] D [ ] I [ ] U[ ] U[ ]. () Correspondng Author: E.A.Sazgar Department of Mathematcs Yerean state Unersty Armena. E-mal: EA. Sazgar@Gmal.com 89

2 Aust. J. Basc & Appl. Sc. 5(5): In (Eberhard 99) dstngushed four dfferent type of turnng ponts: for < ν < m. I f l s eenand n I II f l s een and n I T III f l sodd and n I IV flsoddand ni l ( )( ) l ( )( ) l ( )( ) l ( )( ). (5) e know from [] that s of type I s type IV and s of type II. mn{ } l we use for conenence the abbreatons θ = mn{µ µ µ } [] O ( ). (6) e know from [] that the sectors S - s n form of S arg ]. The Fundamental Systems: Now let (λ) be the soluton of equaton (). The fundamental system of solutons (FSS) for equatons () when Tν = IV can be represented n the form (see[] page 9) IV () ( ) { ( ) [] ( ) e p D s (7) () ( ) e [] D IV ( ) p () ( ) sn e [] D (8) Snce s of type IV we hae the followng FSS for ρ S - II () ( ) e [] D ( ) ( ) csc e [] cos e [] D () () (9) II ( ) csc e [] cos e [] D ( ) () ( ) (sn e []) D () () () 9

3 Aust. J. Basc & Appl. Sc. 5(5): If ν be a turnng pont of type I then the estmates for I ν ( ρ) I ν ( ρ) are obtaned from the correspondng estmates for I ν ( ρ) I ν ( ρ) by substtutng there n ρ by ρ. In ths paper the FSS of () for the sector S arg ] are the followng form () ( ) e [] ( ) ( ) csc e [] () () () ( ) e [] ( ) () ( ) sn e [] () Snce s type of IV V () ( ) e [] ( ) ( ) csc e [] e () () () V () ( ) e [] ( ) () ( ) sn e [] Snce s of type of II we also hae the followng FSS () U () ( ) e [] ( ) () () ( ) csc e [] cos e [] (5) U ( ) csc e [] cos e [] ( ) () ( ) sn []. () () (6) The ronskan of FSS satsfes n followng form ( ) ( ( ) ( )) [] ( V ( ) V ( )) [] 9

4 Aust. J. Basc & Appl. Sc. 5(5): ( U ( ) U ( ) [] as. (7). Asymptotc Form of the Soluton: Let us consder the dfferental equaton () wth followng conons C( ) C'( ). By applyng the FSS { ( ρ) ( ρ)} for I ε we hae C( ) c ( ) c ( ) by deraton from C ( ρ) we can wrte C'( ) c ' ( ) c ' ( ) for I e nfer by usng Cramer s rule leads to the followng equaton C( ) ( ' ( ) ( ) ' ( ) ( )) ( ) (8) where (ρ) = -ρ[]. By substtutng (-) n to account we dere () ( ) () e Ek ( ) ( ) C ( ) () csc e E ( ) () k (9) where E k ( ρ) = [] + ( ) n ( ) kn k e [ b ( )] kn α =β = - α = -α - = I β k () () < δ < β k () < β k () <... < β kν() () < ma{r+() R - ()} where nteger-alued functons ν and b kn are constant n eery nteral D jε j = and R ( ) ma{ ( t)} R ( ) ma{ ( t)}. Smlarly n order to fnd the soluton n ( ) ( ) ( ) by usng Cramer s rule we hae p () () k C( ) ( ) ( ) csc csc e E ( ). e obtaned the leadng term of C( ρ) n ( ) as follows too C e Ek () ( ) ( ) ( ) csc csc csc ( ) 5. Derate of Solutons and Asymptotc Egenalues: Let us consder boundary alue problem L = L (φ () q () b) for equaton () wth boundary conons y ( λ) = y` ( ) = y` ( b) =. 9

5 Aust. J. Basc & Appl. Sc. 5(5): The boundary alue problem L for b ( ) has a countable set of poste egenalue. e consder () R( ) csccsc e T( ) R( ) e () R( ) snsn e T( ) R( ) e R( ) e. Now for fed ( ) and usng (5-6) we determne the connecton coeffcents B (ρ) B (ρ) C( ) B ( ) B ( ) C'( ) B ( ) ' B ( ) '. () The deraton of u ( ρ) and u ( ρ) are n the followng form () () ' ( ) ( ) csc { e cos e }[] () ' ( ) sn ( ) e []. () By substtutng () n () we obtan C ( ρ) n the followng form () () () C'( ) ( ) B( ) csc e cos e [] B( ) sn ( ) e [] () Such that B (ρ) and B (ρ) are n the followng form () () () B( ) ( ) R( ) e cos R( ) e R( ) e [] () () () B( ) ( ) R( ) e R( ) e []. () e suppose L R e R e R e () () () ( ) ( )cos cos ( ) (5) () () L R( ) e R( ) e [] H e e () () csc cos [] (6) (7) 9

6 Aust. J. Basc & Appl. Sc. 5(5): () sn []. H e (8) Thus by usng (5-8) and substtutng them n () we obtan the leadng term of C ( p) as follows C'( ) () ( ) ( HLHL). (9) Now we account H L and H L n followng form () () () () () csc { ( ) ( )cos ( )cos HL R e R e R e () () () () ( )cos ( )cos R e R e R e ( )cos }[] () () () HL R( )sne R( )sn e []. () () () hen then e so R ( ). By mnus () from () we hae () () () csc ( ){ cos cos HL HL R e e () () () () () e cos e sn e }[] () By substtutng () n (9) leads to the equaton () () () C'( ) () () csc R ( ){ e cos e cos e () () () () () () t () e e } cos sn () () t () () () t t C '( ) () () csc csc e {[] cos e cos e () () e e cos sn } () Consequently we hae C ( ρ) n the followng form () () C'( ) () () csc csc csc e E( ) (5) 9

7 Aust. J. Basc & Appl. Sc. 5(5): So that E( ρ) are defned as followng form ( t) ( t) ( t) ( t) E( ) [] cos e cose cos e sn e. (6) By applyng C ( ρ) = consequently E ( ρ) = therefore ( ) ( ) ( ) ( ) [] cose cose cos e sn e. (7) By usng Moaer s rule we hae e so we can rewrte [] cos cos cos ( t) ( t) ( t) e e e ( t) ( ) ( ) cos e cos e sn e. (8) e know [] = O ( ) hence we hae ( t) ( t) O( ) cos e cos e cos ( t) ( t) e sn e KO e ( t) ( ) (cos sn ) (9) () Such that K s a fed number and are gen n the followng form ( t) ( t) K cos e cos e K O( ) e cos sn ( t) () ( t) k LO( ) e. () 95

8 Aust. J. Basc & Appl. Sc. 5(5): K Here we called L = cos sn O( ) ( t) k () by factorng - () can be wrtten as follows O( ) ( t) k where k O ( ) () k () (5) snce ρ = O(k) then the egenalues of equaton () are obtaned n the followng form k k O ( ) () k (6) REFERENCES Abramowtz M. J.A. Stegun 96. Handbook of Mathematcal functons Appl. Math. Ser.55 U. S. Got. Prntng Offce D.C. ashngton Eberhard. G. Frelng A. Schneder 99. Connecton formula for second order dfferental equatons wth a comple parameter and hang an arbtrary number of turnng ponts Math. Nachr. 65: 5-9. Dyachenko A.X.. Asymptotcs of the egenalues of an ndefnte Sturm-Loulle problem Math. Notes 68(I): -. Jodayree Akbarfam A Hgher-order asymptotce appromatos to the egenalues of the Sturm - loulle problems n one turnng pont case Bulletn Iranan Math. Socety (): 7-5. Leung A Dstrbuton of egenalues n the presence of hgher order turnng ponts Trans. Amer. Math. Soc. 9: -5. Neamaty A. and E.A. Sazgar 9. The Neumann conons for Sturm-Loulle problems wth turnng ponts Int. J. Contemp. Math. Scences. (): Neamaty A. and E. A. Sazgar 9. The Negate Neumann Egenalues of Second Order Dfferental Equaton wth Two Turnng Ponts Appled Mathematcal Scences (): Oler F..J Connecton formula for second order dfferental equatons hang an arbtrary number of turnng ponts of arbtrary multplctes SIAM J. Math Anal. 8: Oler F..J Connecton formula for second order dfferental equatons hwth multple turnng ponts SIAM J. Math Anal. 8: