Differential Equation of Eigenvalues for Sturm Liouville Boundary Value Problem with Neumann Boundary Conditions
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1 Ierol Reserc Jorl o Aled d Bsc Sceces 3 Avlle ole ISSN 5-838X / Vol 4 : Scece Exlorer Plcos Derel Eqo o Eevles or Sr Lovlle Bodry Vle Prole w Ne Bodry Codos Al Kll Gold Dere o Mecs Azr Brc Islc Azd Uversy Azr Ir Corresod Aor: Al KllGold ABSTRACT: Eevles derel eqo o Sr Lovlle odry vle role s rerded s co o odry os o rs order derel eqo wc s re or ec relr sel-djo odry codo. We sdy Sr Lovlle derel eqo y '' qy ω y LyWere q w: I A R d A B R w > q w L I J [ ] o I d we dce Ne's eevles re re odry o ollow derel eqo: ' q w Keywor: Eevles Ne odry codos INTRODUCTION We Sr Lovlle derel eqo w odry codo os w ce - s cosdered odry vle role s creed. Bodry codos c e sed s eerl codos sere codos erreled codos Drcle codos d Ne codos. We js rodce sere odry codos d s rerc se d Ne codos e ecessry roosos d deos re reseed. Relr sel-djo odry vle role s rodced d orl col-ler Lre or wc s or rov roosos s rodced. I wll e sow Eevles re coos d ere s orl co ccord o d ' or every d re or o ec coressed ervl o. Te Eevles derel eqo o Sr Lovlle role w Ne odry codos s oed. Bodry Codos Sered odry codos s deed s ollows Ko 994: loc A y A y ' A A R A A B y B y ' B B R B B Prerc se o sered odry codo s deed s ollows:
2 Il. Res. J. Al. Bsc. Sc. Vol cos y s y ' cos β y s β y ' β I sered odry codos A B e odry codo s clled Ne odry codos. Deos d Proosos Sr Lovlle role w eerl odry codos s kow s sel-djo : v D Lv Lv Ass ervl d coeces {qw} re ve o s clled relr : R s. P q ω L c] c Sr Lovlle odry vle role o s clled relr re relr. I s clled ovosly coos o I or ec ε > ere Colex co deed o closed ervl [ ] s δ > or every oe ervls: β... β I were β δ e: β ε Ass s rel co so L e: Teore: sse A l loc Alos everywere o s xed d F s. I s roved F ACloc A ovosly coos loclly o A. A c colleco ro oe ervls β... β s cosdered: F ACloc A B s w F L F β F s s s s β β β F β F s β δ ε Te s s d F ow: s F F s F s l s l [ F F ] F' Ad e rooso s roved. Norl secc co Ko 994: Secl co s eqvle o Eevle ervl [ ] wc s clled orl co or Sr- Lovlle eqo : 998
3 Il. Res. J. Al. Bsc. Sc. Vol w l-ler Lre or Ko 994: We v s swers o Sr- Lovlle eqo l-ler Lre or or v s deed s ollows: [ v] v ' ' v Asse v re swers o Sr- Lovlle eqo Ly y' ' qy w µ d η Ko 994 e: Teore: sce v s swer o e eqo: v ' ' q µ w v ' ' qv µ vw v' ' qv ηvw v' ' qv ηvw v ' ' q v v ' ' q v µ v w ηv w v ' ' v ' ' µ η v w B: [ v ' v ]' ' v ' v ' ' v ' ' ' ' v ' ' v v ' ' [ v ' v ']' ' ' v v ' ' µ η v w [ v ' v ' ] µ η v w Coy o Eevles We odry codos re sere d cos w d d q re o deed o [ ] deed s ollows: ~ Ω W W C D q~ w~ Ω s Were ~ [] q o q d w~ re deed slrly. Oerwse For sce X R R M C M C L L L were A B or W s deed s ollows: 999
4 Il. Res. J. Al. Bsc. Sc. Vol W W A B q w A : or s xed ec rx W w dw Ω. Ass W Asse C D q Eevle o Sr- Lovlle eqo s w eerl odry codo W e s coos W s or ec ve ε > ere s oe δ > so W Ω d W W C C D D q q w w δ W ε W Uor coverece eore Ko 994 Norl co ccord o s or ec were d ' covere d or o o ec servl ro s: l l ' ' 5 re Y y y' Teore: sse or ec y swer ro eqo d secl cos ere s: U Asse sere odry codos s re or ec scely sll secl co s re o coressed servl s seleced e or coverece ro coy owrd d swer o y d y ' re re o reer sce : Fro coy: ε > ε l Now: 3
5 Il. Res. J. Al. Bsc. Sc. Vol ε ε > : N N T s ws roved s orly covere. Rerd ' c e wre: w q w q Cosder coy o d : Eevles Derel eqo o Sr Lovlle w Ne odry codos Ao 997 Sr Lovlle odry vle role w odry codos 3 were d β s cosdered d s ssed d d: ' w q Eseclly q w s coes ove relos s re. Teore: or sll vles o H: µ η v e [v] Becse: I co co ' ' ] [ co ' ' s cos co ' ' s cos v I e s [v] I ddo w β d codo ' s cos β β : B A B Alos everywere
6 Il. Res. J. Al. Bsc. Sc. Vol Cosder orle[ v] [ v] µ η vw s : v ' µ η vw s ' s s w s Sce s s orl d s s ove eqo y vle: ' ' l y l s eqly we d y dvso o O e oer d: ' ' ' ' ' s d s [ q s s s w s ] d s [ q s s s w s q s s q s s s w s s w s ] d s q s s q s [ s s ] d s [ s s ] w s d s s w s By dvso od s eqly y l o sdes we d ve revos cs ere s: l q s w s s q w cosder s s d Ts: ' l q w Ad e rooso s roved y ove reloss: ' q w 3
7 Il. Res. J. Al. Bsc. Sc. Vol REFERENCES Ao Z. 5. Sr-Lovlle Teory Mecl Srveys d Moors.Aerc Mecl Socey. Bley PB. 966.Sr-Lovlle Eevles v Pse-Fco. J. SIAM Al. M. 6: Bley PB Ever WN Wed J Zel A Relr roxos o slr Sr-Lovlle roles.resls Mecs. 3: 3-. Bley PB Ever WN Zel A. 99. Co eevles o slr Sr-Lovlle roles. Resls Mecs.: De M Heler B Ievles Vro I Ne role or sr-lovlleoerers.j. D. Eqos. 4: Es MSP Ko Q W H Zel A Ieqles Ao Eevles o Sr-Lovlle Proles. J. Ieqles d Alcos. 3: Ever WN Rce D.967. O ecessry d s_cecodos or e exsece o creodory solos o ordry d_erel eqos.qesoes M. 3: 7-5. Ko QZel A.994. Ler ordry d_erel eqos. WSSIAA v.3 Secl vole dedced o W.Wler Ieqles d Alcos eded y R. P. Arwl Re SY C YC.. Srce ses/odes oe-desol se-e crysls.als o Pyscs Wed J Secrl Teory o Ordry D_erel Oerors vole 58 o Lecre Noes Mecs 58. Srer Verl Berl. 33
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