Australian Journal of Basic and Applied Sciences, 4(9): , 2010 ISSN

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1 Australia Joural of Basic ad Applid Scics, 4(9): 4-43, ISSN Th Caoical Product of th Diffrtial Equatio with O Turig Poit ad Sigular Poit A Dabbaghia, R Darzi, 3 ANaty ad 4 A Jodayr Akbarfa, Islaic Azad Uivrsity Nka Brach, Nka, Ira 3 Dpartt of Mathatics, Uivrsity of Mazadara, Babolsar, Ira 4 Dpartt of Mathatics, Uivrsity of Tabriz, Tabriz, Ira Abstract: W cosidr th diffrtial quatio y qy y for I ( ) ( ) : [,], (*) whr th wight fuctio f is ral ad has o zro i th op itrval (,), so calld turig poit, ad th assuptio that is odd ordr This turig poit is adittd to b pol of th fuctio q Usig th asyptotic solutio as wll as th distributio of positiv ad gativ igvalu, w driv th caoical product of a particular solutio of th Stur-Liouvill i o turig poit cas Ky words: Turig poit; Sigular poit; Eigvalus; Stur-Liouvill; Ifiit products INTRODUCTION W cosidr th diffrtial quatio y qy y ( ) ( ), () Hr = is th igvalu paratr W assu that th wight fuctio f is ral ad has o zro i th op itrval (,), so calld turig poit, ad th assuptio that is odd ordr This turig poit is adittd to b pol of th fuctio q Diffrtial quatios with turig poits hav various applicatios i athatics, lasticity, optics, gophysics ad othr brachs of atural scics (s (Ebrhard t al, 994; Goldvizr t al, 997) ad th rfrcs thri) Th iportac of asyptotic aalysis i obtaiig iforatio o th solutio of a Stur-Liouvill quatio with ultipl turig poits was ralizd by Lug (Lug, 977), Olvr (977a; 977b), Hadig (Hadig, 977), ad Ebrhard, Frilig ad Schidr i (Ebrhard t al, 994) Th rsults of Dorodicy (Dorodicy, 96), Kazarioff (Kazarioff, 958), Lagr, (93), Dyachko, (), Marasi ad Jodayr i (6) ad Naaty (999), Naaty ad Dabbaghia (7) brig iportat iovatios to th asyptotic approiatio of solutios of Stur-Liouvill quatios It is cssary to poit out that applyig asyptotic solutios for studyig ivrs probl i turig poits cass, is or coplicatd ad practically is ot covit to us Espcially i drivig th asyptotic forulas, o should apply Bssl fuctio typ I additio a or difficult ad challgig task is to shap th asyptotic bhavior of th solutios ad corrspodig igvalus So th ivrs probl of rcostructig th pottial fuctio fro th giv spctral iforatio ad corrspodig dual quatio caot b studid by usig th asyptotic fors I fact, i asyptotic thods o caot grally prss th act solutio i closd for Idd i thods coctd with dual quatios, th closd for of th solutio is dd Th rprstig solutio of th ifiit product for plays a iportat rol for ivstigatig th corrspodig dual quatios I th prvious articl (Khiri ad Jodayr, 3), authors cosidrd th followig Stur-Liouvill quatio () It is assud that q() is a ral fuctio that is Lbsgu itgrabl o th itrval [,], = is th spctral paratr ad ( ) ( ) ( ), 4 whr < <, N, > for [,], is a twic cotiuously diffrtiabl o [,], ad () Corrspodig Author: A Dabbaghia, Islaic Azad Uivrsity Nka Brach, Nka, Ira E-ail: adabbaghia@iaukaacir 4

2 Aust J Basic & Appl Sci, 4(9): 4-43, has o zro i [,], so calld turig poit Th solutio y(,) of such ad quatio with iitial coditios y(, ) =, y(, ) = was foud to hav th ifiit product for ( ) y (, ) ( ) () p ( ), k k zk ( ) y (, ) ( ) () p( ) f ( )cos ( rk( )) p ( ) ( uk( ) ) f ( ),, k k k k () (3) whr th squc {u k ()} rprsts th squc of positiv igvalus ad {r k ()} th squc of gativ igvalus of th Dirichlt probl associatd with () o [,], for ach i (, ] Th squc { k ()}, for ach fid, <, rprsts th squc of gativ igvalus of th Dirichlt probl for Eq() o th closd itrval [,], whr p ( ) ( t) dt for, f( ) ( t) dt for, k zk ( ) for k,,, p ( ),, k,,, Ad ad th positiv zros of J( ), J, rspctivly I this papr w obtai th k k ( ) 3 ifiit product rprstatio of solutio of () i a cas whr th wight fuctio has o zro that it is adittd to b pol of th fuctio q otatios ad Prliiary Rsults: Lt C(,) b th solutio of () corrspodig to th iitial coditios C(,)=, C(,)= I ordr to rprst th solutio C(,) as a ifiit product w us a suitabl fudatal syst of solutios (FSS) for Eq() as costructd i [4] Itroducig so triology at this poit w writ I [, ] [, ] [,], ), (,], : ( 4 A) l with arg (, ], ad []: = + O( - ) ) () is ral ad has i (,) o zro l, of ordr l whr l is odd I th triology of [4], l is of typ IV Th fuctios : I, R, ( ): ( ) ( ), ar o-vaishig ad ral-aalytic; dot k : ( ) 3) q has th for q ( ) A( ) B( ) C( ), ( I, ),, 43

3 Aust J Basic & Appl Sci, 4(9): 4-43, With costats A l, B ad a boudd ral-aalytic fuctio C 4) For [,] lt ( ): () (): a{, ()} R t dt with t t Accordig to th typ of l w kow fro [4, Thor,6] that i th sctor S { arg [,]}, 4 thr ist a fudatal syst of solutios of () { IV IV Z (, ), Z (, )}, ad such that t dt [],, Z (, ) si ( l 3) si ( l ) [] [], i i t dt i 4 t dt i si si, t dt [],, IV N i ( t) dti Z (, ) si 4 [],, si ( l 3) (5) W ( ) [] I th squl w also d { Z (, ), Z (, )} Fro [4] w hav i si ( l 3) si ( l ) IV N 4 4 Z (, ) k V V, si si whr ( ) i O() 4 ( ) V ( ) icsc ( ), ad O() V ( ) icsc ( ) ( ) i 4 ( ) Cosqutly Z ( ) IV N (, ) ( ) csc i si ( l 3) si ( l ) i [] si si i i (6) 44

4 Aust J Basic & Appl Sci, 4(9): 4-43, Siilarly i IV N Z (, ) ( ) [] si ( l ( ) 3) 3asyptotic For of th Solutio: W cosidr th diffrtial quatio () with th followig coditios C(,) =, C(,) = (7) Applyig th { (, ), (, )} Z Z for I,, whr l is of typ IV W hav C (, ) CZ(, ) CZ(, ) That usig of Crar's rul lads to th quatio C (, ) ( (, ) (, ) (, ) (, )) Z Z Z Z Takig (4)-(5) ito accout w driv C (, ) whr si ( l 3) ( t) dti 4 M( ) si ( t) dt ( t) dt () [] [], i () t dt i ( ) t dt [] ( ) M M [],, si ( l ) ( t) dti si t dti 4 4 M( ) i si si ( l 3) By virtu of (9) ad (), th followig stiats ar also valid: C (, ) si ( l 3) ( t) dti ( t) dti 4 () [[]],, si Siilarly, usig of (6), (7) ad (8) for = w fid that ( t) dt () [[]],, (8) (9) () () l ( t) dt i( ) C (, ) i () ( ) csc ( ) ( t) dt si () ( ) csc ( ) si ( l 3) 45

5 Aust J Basic & Appl Sci, 4(9): 4-43, l i( ) ( t) dt () ( ) csc i 4 ( ) 4distributio of th Eigvalu: W cosidr th boudary valu probl [[]] L L q b ( ( ), ( ), ) () for Eq() with boudary coditios y(, ), y(, ), y( b, ) Th boudary valu probl L for Fro (Taarki, 97) w hav ( b) O( ), b ( t) dt ad for = l siilarly fro () w hav ( ) 4 ( ) O( ) ( t) dt b(, ) has a coutabl st of gativ igvalus { ( b)} Th spctru {l } of boudary valu probl L for l < b, cosist of two squcs of gativ ad positiv igvalus: { ( b)} { ( b)} { ( b)}, N, 4 ( b) O( ), b ( tdt ) 4 ( b) O( ) ( tdt ) such that 5Mai rsults: Sic th solutio C(,r) of th Stur-Liouvill quatio dfid by a fid st of iitial coditios is a (3) (4) (5) (6) tir fuctio of r for ach fid [,], thus it follows fro th classical Hadaard's factorizatio thor that such solutio is prssibl as a ifiit product For fid b(, ) by Halvors's rsult (Halvors, 987), C(b, ) is a tir fuctio of ordr Thrfor w ca us Hadaard's thor to rprst th solutio i th for whr h(b) is a fuctio idpdt of l but ay dpd o b ad th ifiit ubr of gativ igvalus, { ( b)} zro st of C(b, l) for ach b Sic Cb (, ()) b for th, ths l (b) corrspod to igvalus of th boudary valu probl L o th closd itrval [,b], < b < W rwrit th ifiit product as ( b) Cb (, ) hb () ( ) h() b( ) ( b) with z (8) 46

6 Aust J Basic & Appl Sci, 4(9): 4-43, ( ): ( ) z h, b h b ( b) whr z R ( b) Now (3) iplis that z z O( ) ( ) b It follows fro th rsults of [6] that th ifiit product z is absolutly covrgt o ay copact subitrval of (, ) Th fuctio is cotiuous ( b) ( b) ad so th O-tr is uiforly boudd i b Thor Lt C(, ) b th solutio of () satisfyig th iitial coditios C(, ) =, C(, ) = Th for < <, (, ) () ( ) ( C R ) (9) z whr th squc ( ),, rprsts th squc of gativ igvalus of th boudary valu probl L o [,]proof Lt { ( b )} fid b =, < < th accordig to [] w hav b th igvalus of th boudary valu probl L o [,b], for ( ) sih R ( ) log ( ) O( ) () z R ( ) C (, ) Thus fro (), (8), w obtai h ( ) () R ( ) ( ) ( ) z Siilarly for b = agai by Hadaard's thor w that C (, ) A( ( ) ) () whr A is costat Lt, =,, b th squc of positiv zros of th Bssl fuctio of ordr h, th (s []) O( ), R ( ) ( ) So th ifiit product R ( ) ( ) ar absolutly covrgt Cosqutly w ay writ as bfor, ( ( )) R ( ) C (, ), A () 47

7 Aust J Basic & Appl Sci, 4(9): 4-43, whr A A R ( ) ( ) Thor For b =, l i( ) () V( ) R ( ) R c (, ) whr V t dt ( ) li ( ) ( ( )) ( ) proof Accordig to (Jodayr ad Migarlli, ) th ifiit product ( ( )) R ( ) is a tir fuctio of l, whos roots ar prcisly ( ), Morovr ( ( )) R ( ) log ( )[ i R( )] J ( i R ( ))( O( )) Uiforly o th Circls R ( ) Thus it follows fro (), l i( ) C (, ) () V ( ) R ( ) ( ( )) R ( ) A Siilarly for b =, < <, th boudary valu probl L o [,] has a ifiit ubr of positiv ad gativ igvalus which ar dotd by { ( )} { ( )} { ( )}, rspctivly By Hadaard's thor, th solutio o [,], < < is of th for C (, ) g ( ) ( )( ) ( ) ( ) (4) Lt,,,, b th positiv zros of J( z), drivativ of th Bssl fuctio of ordr o Th distributio of is of th for O(), (s []) Cosqutly, w hav O( ), R ( ) ( ) O( ) R ( ) ( ) (5) (6) 48

8 Aust J Basic & Appl Sci, 4(9): 4-43, Cosqutly, th ifiit products R ( ) ( ) ad R ( ) ( ) ar absolutly covrgt for ach (,) Thrfor w ay writ ( ( )) R ( ) ( ( ) ) R ( ) C (, ) g( ) with g ( ) g( ) R ( ) ( ) R ( ) ( ) Thor 3 For < <, C (, ) () ( R( R ) ( )) 8 si ( l 3) ( ( )) R( ) ( ( ) ) R( ) si (7) (8) (9) proof: Fro Las ad 3 of [] th ifiit products ( ( )) R( ) ( ( ) ) R( ) ar tir fuctios of l for fid, thos roots ar prcisly ( ) ad ( ), ³, rspctivly Morovr ( ( )) R( ) ( ( ) ) R( ) R ( ) 4 R O 4 ( ) ( ) cos( ( ) ) ( ), R R as Thus by usig of th asyptotic pasio of C(,l) i [] w gt g ( ) C (, ) ( ( )) R ( ) ( ( ) ) R ( ) si ( l 3) () ( ( ) ( )) R R 8 si REFERENCES Abraowitz, M, JA Stgu, 964 Hadbook of Mathatical Fuctios, Appl Math Sr 55, U S Govt Pritig Offic, Washigto, DC 49

9 Aust J Basic & Appl Sci, 4(9): 4-43, Dorodicy, AA96 Asyptotic laws of distributio of th charactristic valus for crtai spcial fors of diffrtial quatios of th scod ordr, Ar Math Soc Trasl Sr, (6): - Dyachko, AX Asyptotics of th igvalus of aidfiit Stur-Liouvill probl, Math Nots, 68(I): -4 Ebrhard, W, G Frilig, K Wilck, Idfiit igvalu probls with svral sigular poits ad turig poits, Math Nachr, 9: 5-7 Ebrhard, W, G Frilig, A Schidr, 994 Coctio forula for scod-ordr diffrtial quatios with a copl paratr ad havig a arbitrary ubr of turig poits, Math Nachr, 65: 5-9 Evs, HW, 979 Fuctios of a copl variabl, Pridl, Wbr ad Schidt, pp: Goldvizr, AL, VB Lidsky, PE Tovstik, 979 Fr vibratio of thi lastic shlls, Nauka, Moscow Halvors, SG, 987 A fuctio thortic proprty of solutios of th quatio ( q), Quart J Math Oford, (38): Hadig, J, 977 Global phas-itgral thods, Quart J Mch Appl Math, 3: 8-3 Jodayr, A, Akbarfa, AB Migarlli, Th caoical product of th solutio of th Stur- Liouvill quatio i o turig poit cas, Caad Appl Math Quart, 8(4): 35-3 Kazarioff, ND, 985 Asyptotic thory of scod ordr diffrtial quatios with two sipl turig poits, Arch Ratio Mch Aal, : 9-5 Khiri, H, A Jodayr Akbarfa, 3 O th ifiit product rprstatio of solutio ad dual quatios of Stur-Liouvill quatio with turig poit of ordr 4+, Bull Iraia Math Soc, 9(): 35-5 Lagr, RE, 93 O th asyptotic solutio of ordiary diffrtial quatios with a applicatio to Bssl fuctios of larg ordr, Tras Ar Math Soc, 33: 3-64 Lug, A, 977 Distributio of igvalus i th prsc of highr ordr turig poits, Tras Ar Math Soc, 9: -35 Marasi, HR, A Jodayr Akbarfa, 7 O th caoical solutio of idfiit probl with turig poits of v ordr, J Math aal Appl, doi: 6/aa 6 49 Naaty, A, A Dabbaghia, 7 Asyptotic for of th solutio of Stur-Liouvill probl with turig poits of odd-v ordr, Far East J Appl Math, 9(): 6-7 Naaty, A, 999 Th caoical product of th solutio of th Stur-Liouvill probls, Ira J Sci Tchol Tras A () Olvr, FWJ, 977 Coctio forulas for scod-ordr diffrtial quatios havig a arbitrary ubr of turig poits of arbitrary ultiplicitis, SIAM J Math Aal, 8: Olvr, FWJ, 977 Coctio forulas for scod-ordr diffrtial quatios with ultipl turig poits, SIAM J Math Aal, 8: 7-54 Taarki, JD, 97 So gral probls of th thory of ordiary liar diffrtial quatios ad pasios of a arbitrary fuctio i sris of fudatal fuctios, Math Z 7: