The Asymptotic Form of Eigenvalues for a Class of Sturm-Liouville Problem with One Simple Turning Point. A. Jodayree Akbarfam * and H.

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1 Joral of Scic Ilaic Rpblic of Ira 5(: -9 ( Uirity of Thra ISSN 6- Th Ayptotic For of Eigal for a Cla of Str-Lioill Probl with O Sipl Trig Poit A. Jodayr Abarfa * ad H. Khiri Faclty of Mathatical Scic Tabriz Uirity Tabriz Ilaic Rpblic of Ira Abtract Th prpo of thi papr i to tdy th highr ordr ayptotic ditribtio of th igal aociatd with a cla of Str-Lioill probl with qatio of th for w ( λ f ( x R( x w ( o [ ab ] whr λ i a ral paratr ad f ( x i a ral ald fctio i C ( a b which ha a igl zro (o calld trig poit at poit x x ad R ( x i a cotioly diffrtiabl fctio. W pro that a a claical ca th ayptotic for of igal of ( with priodic bodary coditio w ( a w ( b w ( a w ( b a wll a with Si-priodic bodary coditio w ( a w ( b w ( a w ( b ar th a a Dirichlt bodary coditio w ( a w ( b. W alo tdy th ayptotic forla for th igal of ( with bodary coditio w ( a w ( b a wll a w ( a w ( b ad w ( a w ( b. Kyword: Ayptotic ditribtio; Trig poit; Str-Lioill probl; No-dfiit. Itrodctio May diffrtial qatio occrrig i athatical phyic ar rdcibl to th for w ( λ f ( x R( x w Archi of SID ( whr (i f R :[ a b] IR ar ti cotioly diffrtiabl fctio f ( x (ii f ( x ad hx ( i poiti ad twic x x cotioly diffrtiabl withi ( ab whr x i a itrior poit of ( ab (iii λ i a ral paratr. By ig Logr traforatio ([8]..5 x { x } { x } ( x ( f ( t dt x x x ( x ( f ( t dt x x ad w dpdt ariabl ( rdc th for whr ( ~ f w th Eqatio y y ( q( y ( * E-ail: abarfa@yahoo.co AMS: E

2 Vol. 5 No. Witr Jodayr Abarfa ad Khiri J. Sci. I. R. Ira ad ( R ~ d ~ q ~ f ( f f dx 9 ~ d ( ( ( f x λ f x. dx 9 ( x Udr thi traforatio lt cd corrpod to th d poit x ab rpctily whr c < < d. I [] it wa drid th highr ordr ayptotic forla for th poiti igal ( with Dirichlt bodary coditio y( c y( d. I [] [5] ad [8] th ditribtio of igal for th claical ca of Str-Lioill probl with priodic bodary coditio i drid. It a that i th papr th wight fctio f ( x i ( do ot chag ig i [ab]. I ([9] chaptr.. it ha b prod that th ditribtio of th i-priodic probl ha th a ayptotic xpaio a th Dirichlt probl for qatio ( whr f ( x do ot ha a trig poit. I [] Lagr ha xprd th oltio of th gi qatio i tr of th oltio of a ipl ocalld tadard qatio which gi a adqat dcriptio of th bhaior of oltio of th origial qatio. Not that firtly th ayptotic bhaior of th oltio iply dcribd thory bt rathr cbro to apply for gttig th highr ordr ditribtio of igal aociatd with priodic ad i-priodic bodary coditio o i thi papr th thod of tablihig th tiatio i bad o th foral oltio of ( cotrctd by Olr [].. Priodic Bodary Coditio Lt y y λ y y ( ( λ b two liar idpdt oltio of (. Th igal of qatio ( with priodic bodary coditio y( a y( b y ( a y ( b ar th root of Archi of SID ( W ( y y ( c y ( c ( d y ( d ( c y ( c y ( d y ( d y ( c W ( y y ( d whr W dot th wroia of y y. Sic by Lioill-Abl forla w ha W ( y y ( c W ( y y ( d W ( y y ( W ( hc ( W ( y ( c ( d y ( d ( c y ( c y ( d y ( d y ( c ( I tdyig th ayptotic bhaior of th igal it i ry coit to th ayptotic oltio gi i ([] chaptr. a follow: Thor. Th diffrtial Eqatio ( ha for ach al of ad ach ogati itgr a pair of ifiitly diffrtiabl oltio W ( ad W ( gi by thir approxiatio W ( W ( whr A ( ( ( W Bi Bi ( B ( ε ( A ( ( ( Ai ( B W Ai whr A ( ( ε ( d B ( ( R( A ( A ( < " d B ( ( R( A ( A ( > " ( ( A ( B ( R ( B ( d ε ( i i ar rror bod ad Ai Bi ( d W W (. d calld Airy fctio. ar two idpdt oltio of ( Proof. S ([] chaptr.. Not that A ( ad B ( ar ifiitly diffrtiabl. Th rror bod for larg ar ifor with rpct to. Th ayptotic for of Ai(

3 J. Sci. I. R. Ira Jodayr Abarfa ad Khiri Vol. 5 No. Witr ad Bi ( for > i ot th a a for <. Thrfor th oltio W ( or W ( ha diffrt ayptotic for for > a <. I th rt of thi papr w hall th ybol W ( W ( to igify th ayptotic for of W ( for > < rpctily a. ( Bi Bi ( 6 ( for > 6 ( La. Th ayptotic for of Ai( Ai ( Bi ( ad Bi ( ar gi by (for ( ( 6 ( Ai for > 6 ( ( Ai ( Ai ( Ai for > ( {co( ( ( 6 ( ( ( i( ( ( } ( ( for < Archi of SID 6 ( {i( ( ( ( ( co( ( ( } ( ( for < ad Bi ( for > { i( ( ( 6 ( ( ( co( ( ( } ( ( for < ( Bi ( {co( ( ( ( ( 6 i( ( ( } ( ( for < Ai ( Ai ( Γ ( whr Bi ( ( ( ( 5...(6 (6! 6 6 Γ ( Bi ( Proof. S ([] chaptr. ad. I ordr to copt th ayptotic of th 5

4 Vol. 5 No. Witr Jodayr Abarfa ad Khiri J. Sci. I. R. Ira igal w th flowig otatio: A( B( B A( ( M ( ( ( ( ( M P( ( ( ( ( R P ( R ( ( ( ( K ( ( ( K ( ( T ( ( ( T ( ( L ( ( ( L ( ( ω ( By thor ([] chaptr. ig th abo otatio w writ y ( c : W ( c Ac ( ( c 6 T ( cco ω( c} {( R ( ci ω( c ( c B( c 6 {( K ( c co ω ( c L( ci ω( c} O ( 9 6 d ( c : W ( c A c B c ( c Archi of SID ( ( 6 {( R ( c i ω ( c T ( cco ω( c} y ( c : W ( c Ac ( { R 6 ( cco ω( c ( c T ( ci ω( c} ( c B( c 6 { K ( c i ω ( c L ( cco ω( c} O (6 9 6 d ( c : W ( c A c B c ( c ( ( 6 { R ( c co ω ( c T ( ci ω( c} c A c B c L ( cco ( c} O ( ( ( ( ( 6 { K ( c i ω ( c ω 5 6 y ( d : W ( d d {( M 6 ( d A( d O( } d d P 6 d B d O d { ( ( ( } d ( d : W ( d d {( M 6 ( d( A ( d d ( ( } B d O d 6 d { P ( d( A( d B d O ( ( } 8 y ( d : W ( d (8 (9 c A c B c ( ( ( ( 6 {( K ( c co ω ( c L ( ci ( c} O (5 ω d { M 6 ( d A( d O( } d d d { P 6 ( d B( d O( } ( 6

5 J. Sci. I. R. Ira Jodayr Abarfa ad Khiri Vol. 5 No. Witr hc d ( d : W ( d d 6 { M ( d( A ( d d ( ( } B d O d 6 d B ( d O( } { P ( d( A( d d ( y ( d ( c O( ( d y ( c ( d O( a. Ad ( d ( d ( : db O ( y( d ( ( ( c cbco( w ( c O( pd ( : y ( c co( w ( c O( Th igal of th probl ( with priod bodary coditio ar th root of ( or taig ito accot of (- (aftr th ipl calclatio w gt or d d W ( O( O( y( d y ( c [ d ( pc ( ] d d d W ( O( O( O( y 6 ( c [ ( d p( c ]. ( Archi of SID Sic d ( ad pc ( ha diffrt ayptotic for ad th firt thr tr i ( do ot iflc th ladig tr of zro of ( ( [] o ayptotic ditribtio of th zro of ( i thrfor ayptotically dtrid by that zro of y ( c whr y ( c i th oltio of ( dfid by (5 i.. th zro of ( atify W ( c ; (5 a. Fro thi xprio w that th ayptotic bhaior of th igal dpd tially o th poiti part of th wight fctio o [ cd ] i.. o o [ c ]. Thi i to b xpctd by th thor of Atio-Migarlli [] or th rlt of Ebrhard ad Frillig [6]. Aalogoly to Thor i [] w ay ha: Thor. Coidr th diffrtial qatio w ( λ f ( x R( x w a x b with bodary coditio w ( a w ( b whr for o poit x [ ab ]: f ( x f ( x hx ( i a poiti ad twic cotioly x x diffrtiabl fctio withi (a b. R( x i cotioly diffrtiabl Th itgral x d H( a { a ( ( f dx f R 5( f } dx ( f 6( corg whr ( x a: x x (6 ( x { ( f ( t dt} x x x x ( ( x { f ( t dt} x x. Th th ayptotic ditribtio of th poiti igal of thi probl i gi by λ x a ( f ( t dt 5 { H( a} f t dt 6 x ( ( a 5 { H ( a} f t dt 6 x ( ( a O (. (8. Si-priodic Bodary Coditio Th igal of Eqatio ( with Si-priodic bodary coditio

6 Vol. 5 No. Witr Jodayr Abarfa ad Khiri J. Sci. I. R. Ira y( a y( b y ( a y ( b ar th zro of S ( W ( y y ( c y ( c y ( d y ( d y ( c y ( c y ( d y ( d y ( c W ( y y ( d Siilarly taig ito accot of (- w obtai: ( ( d S ( W ( ( ( ( d ( d O( O( O( y ( ( c [ d ( pc ( ]. 6 Coparig ( with S ( w that th oly diffrc i th ig of o xprio which do ot iflc of th ditribtio of th igal thrfor th zro of S ( atify W ( c. W ha th prod th followig a claical ca ( [9] chaptr... Thor. If R( x i a ooth fctio th th igal of th i-priodic probl ad th priodic probl ha th a ayptotic xpaio a th Dirichlt bodary probl for qatio ( (i.. of th for (8.. Othr Ca W ow coidr qatio ( with th bodary coditio y ( c y ( d. q( t I thi ca w a that (9 Archi of SID c ( t th igal of th probl ( (9 ar th root of y ( c y ( d y ( c y ( d. or taig ito accot (5 ( (9 ( (aftr th ipl calclatiow obtai d W ( a c ( Fro ( ad ( w ha fod that for larg whr F ih i{ w ( c } F ih F ( A ( c db( c R ( c ( c ( A( c B ( c L ( c H ( A ( c db( c T ( c. ( c ( A( c B ( c K ( c By taig th logarith i ( w obtai whr F ih log c c i F ih F ih ( ( ( c ( c ( i ( A ( c( i ( ( i( c B ( c( i ( c ( c ( ( i ( ( i ( F ih i( c B ( c ( i A ( i ( c A ( c ( c ( c ( c ( c B ( c ( ( c ( i ( B ( c ( c A ( c. ( ( ( ( ( d ( d ( 6 O( O( ( c. ( 6 ( d Diidig throghot by ad lttig w that th igal of thi ca atify ( Now w ha ia ib ( F ih B ( c c ( 8

7 J. Sci. I. R. Ira Jodayr Abarfa ad Khiri Vol. 5 No. Witr ( c A B( c i ( A B ( c O ( ( c ( i( c A ib ( F ih B c ( ( ( c A c B c i( c ( A B ( c O (. Coqtly whr F ih F ih Ti T ( i L T O( ( c 5B T L. B 8( c ( Sbtittig xprio ( ito forla ( ad ig th xpaio of th logarith w fid: or ( c [ it ( i ( L T O ( ] ( c i T ( c ( c ( L T O(. ( c Firt ad cod approxiatio i O( ( c ( Archi of SID (5 btittig thi approxiatio (5 ito ( w fid th followig approxiatio: ( ( q( t ( ( c ( t ( O(. c dt (6 Lt coidr th diffrtial Eqatio ( whit bodary coditio y( c y ( d. By applyig th abo iilar thod w that th igal of thi probl ar zro of y ( c : W ( c thrfor th highr-ordr ayptotic ditribtio of igal i of th for (8. Th igal of Str-Lioill Eqatio ( with bodary coditio y ( c y( d coicid with zro of ( d ( c : W c thrfor th ditribtio of igal of thi ca i of th for (6. Acowldgt Thi wor ha b pportd by th Rarch Ititt for Fdatal Scic Tabriz Ira. Rfrc. Jodayr Abarfa A. Highr-ordr ayptotic approxiatio to th igal of th Str-Lioill probl i o trig poit ca. Bll. Iraia Math. Soc. No. -5 (99.. Olr F.W. Ayptotic ad Spcial Fctio. Acadic Pr Nw Yor (9.. Boir A. Eigal of priodic Str-Lioill probl by th Shao-Whittar aplig thor. Uphi Mat. Na 6: -96 (95.. Lagr R.E. O th zro of xpotial ad itgral. Bll. A. Math. Soc. : -9 (9. 5. Dorodity A.A. Ayptotic law of ditribtio of th charactritic al for crtai pcial for of diffrtial qatio of th cod ordr. Uphi Mat. Na AMS tralatio 6 (Sr. 6: -96 ( Ebrhard W. ad Frilig G. Th Ditribtio of th Eigal for Scod Ordr Eigal Probl i th Prc of a Arbitrary Nbr of Trig Poit. Rlt Math. : - (99.. Atio F.V. ad Migralli A.B. Ayptotic of th br of zro ad of th igal of gral wightd Str-Lioill probl. J. Ri Ag. Math. 5: 8-9 ( Abraowitz M. ad Stg A. Hadboo of Mathatical Fctio. Dor Pblicatio Ic. Nw Yor (9. 9. Lita B.M. ad Sargja I.S. Str-Lioill ad Dirac Oprator. Klwr Acadic Pblihr (99. 9

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

Australian Journal of Basic and Applied Sciences, 4(9): , 2010 ISSN Australia Joural of Basic ad Applid Scics, 4(9): 4-43, ISSN 99-878 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