Journal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi

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1 Joural o Egirig ad aural Scic Mühdilik v F ilimlri Drgii Sigma Rarch ricl / raşırma Makali GREE S FUCTIO OF DISCOTIUOS OUDRY VLUE PROLEM WITH EIGEPRMETER I OUDRY CODITIOS Fama YDI KGÜ* Yildiz Tchical Uivri Facul o Chmical-Mallurg Dparm o Mahmaical Egirig Davupaa Campu Elr-ISTUL Rcivd/Gliş: 6..9 Rvid/Düzlm: 6.. ccpd/kabul: 5.3. STRCT W corucd Gr ucio oluio o dicoiuou Surm-Liouvill problm wih igparamr i boudar codiio. Kword: Dicoiuou Surm Liouvill problm igparamr Gr ucio igvalu igucio. MSC umbr/umaraları: 3L 35R. SIIR KOŞULUD ÖZ PRMETRE ULU SÜREKSİZ SIIR DEĞER PROLEMİİ GREE FOKSİYOU ÖZET u makald ıır koşullarıda özparamr bulua ürkiz Surm-Liouvill problmlri içi Gr okiou işa dilmişir. ahar Sözcüklr: Sürkiz Surm-Liouvill problmlri özparamr Gr okiou özdğr özokio.. ITRODUCTIO I hi papr w ablih h Gr ucio or Surm-Liouvill uaio -. o h irval wih h igparamr-dpd boudar codiio..3 ad h ramiio codiio * akgu@ildiz.du.r l:

2 6..5 whr h ral valud ucio i coiuou o havig h ii limi lim ad lim i a ral igparamr ad i a arbirar ral umbr. Ma opic i mahmaical phic ruir ivigaio o igvalu ad igucio o boudar valu problm. I pi o h ac ha h gral hor ad mhod o boudar valu problm wih coiuou coici ar highl dvlopd vr lil i kow abou a gral characr o imilar problm wih dicoiui. Som problm wih ramiio codiio which ari i mchaic uch a hrmal coducio problm or a hi lamiad pla wr udid i []. I rc ar coiuou rul hav b obaid or h boudar valu problm wih igparamr dpd boudar codiio. Som o h rul ca b i [ ]. I paricular [ 6 9 ] coai ma rrc o problm i phic ad mchaic. Som pcial ca o h problm.-.5 ari rom applicaio o h mhod o paraio o variabl o h varid aorm o phical problm [ 5 ]. I mu b od ha ampoic ormula o igvalu ad igucio o hi problm ar ivigad i []. I hi papr w will coidr igparamr dpd boudar codiio ad will d om rul o h adard Surm-Liouvill problm o dicoiuou ca. I paricular w will coruc Gr ucio or h problm.-.5 uig a mhod dcribd i [].. SOME SIC SOLUTIOS CCORDIG TO TRSMISSIO CODITIOS W di wo udamal oluio or or. ad or or. o h dirial uaio. which ai o o h boudar codiio i..3 ad boh o h ramiio codiio..5 a ollow. F. dı kgü Sigma 8 5-5

3 7 L b a oluio o h uaio. o aiig h iiial codiio..3 I ha b how i [] ha h oluio o. wih.3 i uiu o. ow coidr h dirial uaio. o oghr wih h pcial p iiial codiio...5 W will prov ha h iiial codiio di a uiu oluio which i alo a ir ucio o h paramr C or ach id. Coidr h uc... did b h rcurrc ormula. d.6 L ma Q. I i obviou ha or ach poiiv ral umbr > hr i > K uch ha K or all ad. Thu K Q d K Q..7 iducio i ollow rom.7 ha d Q or Gr Fucio o Dicoiuo oudar Valu Sigma 8 5-5

4 8! Q K or Hc h ri. covrg uiorml or valu o aiig i h irval or. Morovr w ca obai h ollowig uaio b diriaig h uaio.6 d... viru o.9 ach o h ri d ad covrg uiorml or o h irval. Hc i ollow rom. ad. ha h diriad ri ad alo covrg uiorml i o h irval. ow akig. ad. io accou w hav F. dı kgü Sigma 8 5-5

5 Gr Fucio o Dicoiuo oudar Valu Sigma Thu aii h dirial uaio. o h irval ad h did b uaio. aii h boudar codiio. ad h iiial codiio. ad.5. Hc h ucio h dirial uaio. o ramiio codiio. ad.5. W ow ur o h uiu oluio aiig h iiial codiio ad o h uaio. o.3 which i a ir ucio o or id. Th ucio will b did i rm o ad b h codiio...5 pplig h am chiu a i h diiio o uaio. wih codiio.-.5 ha a uiu oluio ir ucio o or id w ca prov ha h which i alo a. Thu aii h dirial uaio. h. boudar codiio.3 ad h ramiio codiio. ad.5 o Th Wrokia i i i i i i.6 W i ar idpd o or i ad i ca ail b how ha..7 9

6 F. dı kgü Sigma Thorm : Th igvalu o h problm.-.5 coi o zro o h ucio. Thorm ca b provd ail b h am chiu a or Thorm i []. Lmma : L.Th h ollowig igral uaio hold..8 Si Co τ Si τ τ dτ Si Co τ Si τ τ dτ.9 Proo: Coidr h oluio o h dirial uaio... or Muliplig boh id b Si τ ad h igraig w g τ Si τ dτ τ Si τ dτ τ Si τ τ dτ.3 w obai. r igraig b par wic h ir igral ad uig h codiio i. Si Co τ Si τ τ dτ Similarl prormig h am calculaio or ad uig h codiio. ad.5 w g Si Co τ Si τ τ dτ or. Lmma : L ampoic uaio hold or or big a compl umbr. L Im.. Th h ollowig Co O.

7 O Co Co Si Si. O Si.3 O Si Co Co Si. Thorm : L ad Im. Th ampoic rpraio o characriic ucio i O Si Si Co Co..5 Proo: Thi rul i a dirc couc o uaio.6 ad Lmma. 3. SYMPTOTIC REPRESETTIO OF GREE FUCTIO L u coidr Surm-Liouvill uaio L 3. oghr wih igparamr-dpd boudar codiio.-.3 ad ramiio codiio.-.5. Hr i coiuou ucio o h irval. W ca rpr h gral oluio o 3. i h ollowig orm or or Y. 3. W applid h adard mhod o variaio o h coa o 3. hu h ucio ad aii h liar m o uaio 3.3 or ad Gr Fucio o Dicoiuo oudar Valu Sigma 8 5-5

8 3. or. Sic i o a igvalu ad W i i or i ach o h liar m i 3.3 ad 3. ha a uiu oluio which lad. d d d d Whr ad ar arbirar coa. Subiuig h prio o 3. w obai h oluio o 3. d d Y 3.5 or ad d d Y 3.6 or. Subiuig h oluio io h igparamr-dpd boudar codiio.-.3 ad h ramiio codiio.3-. w obai h rolv o h boudar valu problm.-.5 a d d Y. 3.7 F. dı kgü Sigma 8 5-5

9 3 W ca ow ail id h Gr ucio o h problm.-.5 rom h rolv 3.. Thorm 3: I h uaio u L ha ol rivial oluio h or a ucio which i coiuou o h irval[ ] b a hr i a oluio o h uaio Lu giv b b a d G u ξ ξ ξ. Proo o hi horm ca b oud i [3] whr ξ G do h Gr ucio or h opraor L. From Thorm 3 w hav d G Y. ad rom 3.7 h Gr ucio o h problm.-.5 ca b rprd a ollow G. 3.8 Fiall i viw o h codiio. ad uig h ampoic ormula i.-.5 w obai h ampoic rpraio o Gr ucio G or a Gr Fucio o Dicoiuo oudar Valu Sigma 8 5-5

10 O Si Co Co Si Si O Si Si Co Co Co Si O Co Si Co O Si Co Co Si Si O Si Si Co Co Co Si O Co Si Co G REFERECES / KYKLR [] Tikhoov.. Samarkii.. Euaio o Mahmaical Phic Prgamo: Oord ad w York 963. [] kdoğa Z. Dmirci M. Mukharov OSh. Dicoiuou Surm Liouvill Problm wih Eigparamr-Dpd oudar ad Tramiio Codiio ca pplicada Mahmaica 86: [3] idig P.. row P.J. Wao.. Srum Liouvill Problm wih oudar Codiio Raioall Dpd o h Eigparamr II. Joural o Compuaioal ad pplid Mahmaic 8:7 69. [] Fulo C.T. Two-Poi oudar Valu Problm wih Eigparamr Coaid i h oudar Codiio Proc. R. Soc. Ediburg [5] Mukharov O.Sh. Kadakal M. Muharov F.S. O Dicoiuou Surm Liouvill Problm wih Tramiio Codiio Joural o Mahmaic o Koo Uivri : [6] Shkalikov.. oudar Valu Problm or Ordiar Dirial Euaio wih a Paramr i oudar Codiio Trud. Sm. Provk [7] Tuc E. Muharov O.Sh. Fudamal Soluio ad Eigvalu o O oudar Valu-Problm wih Tramiio Codiio pplid Mahmaic ad Compuaio 57: [8] Yakubov S. Compl o Roo Fucio o Rgular Dirial Opraor Logma Sciiic ad Tchical w York 99. F. dı kgü Sigma 8 5-5

11 Gr Fucio o Dicoiuo oudar Valu Sigma [9] Tiu I. Yakubov Ya. Compl o Rooh Fucio or Thrmal Coducio i a Srip wih Picwi Coiuou Coici Mahmaical Modl ad Mhod i pplid Scic 77: [] Voiovich.. Kaalbaum.Z. Sivov.. Gralizd mhod o Eig-vibraio i h Thor o Diracio auka: Mokow 997. [] dı kgu F. "O oudar Valu Problm which hav Spcral Paramr i oudar Codiio PhD Thi Mahmaical Egirig Dparm Y.T.Ü 8. [] kdoğa Z. Dmirci M. Muharov Oh. Gr Fucio o Dicoiuou oudar Valu Problm wih Tramiio Codiio Mah. Mh. ppl. Sci. 3: [3] aimark M.. Liar Dirial Opraor Par I Gorg G. Harrap & Co.Ld. Lodo