Oscillations of Hyperbolic Systems with Functional Arguments *

Transcription

1 Avll ://vmd/gs/9/s Vol Iss Dcmr 6 95 Prvosly Vol No Alcons nd Ald mcs AA: An Inrnonl Jornl Asrc Oscllons of Hyrolc Sysms w Fnconl Argmns * Y So Fcly of Engnrng nzw Unvrsy Isw 9-9 Jn E-ml: so@nzw-c Noro Yosd Drmn of mcs Unvrsy of oym oym Jn E-ml: nor@sc-oymc Rcvd y 5 6; ccd Ocor 6 Hyrolc sysms w fnconl rgmns r sdd nd sffcn condons r ond for vry solon of ondry vl rolms o wly oscllory s ls on of s comonns s oscllory n cylndrcl domn Ron-y ondry condon s consdrd roc sd s o rdc ml-dmnsonl oscllon rolms o on-dmnsonl oscllon rolms y sng som ngrl mns of solons ywords: Oscllon; yrolc sysms; fnconl rgmns AS SC No: 5B5; 5R Inrodcon W r concrnd w oscllon of sysm of yrolc qons w fnconl rgmns U H U A U F B U τ Ω m P U wr s ondd domn n n Llcn n R nd n R w cws smoo ondry s

2 Y So nd N Yosd f f F U U U U P B A H τ τ τ srscr dnong rnsos I s sy o s cn wrn n followng sysm: f m τ for Ω ondry condon o consdrd s followng: on BC wr ; R C ;[ C nd dnos n ror norml vcor o In cs on n on nd nc ondry condon BC rdcs o on If on n BC cn wrn n form on morovr f on n BC cn wrn s on

3 AA: Inrn J Vol Iss 6 [Prvosly Vol No ] 5 W ssm : H H H H H 5 H 6 H 7 H C C[ ;[ C[ ;[ C C τ C[ ; R C[ ; R ξ mn ξ m f C [ ;[ [ ;[ [ ; R [ ; R ξ C R; R ξ ξ r conv n m ; ; lm lm lm ; ; ; ; ; m ; ; τ ; ; ; m ; for ξ ξ ξ m ; ; s nondcrsng nd conv n for ξ > nd Dfnon : By solon of sysm w mn vcor fncon sc C [ ; R C [ ;R nd ssfy n Ω wr mn mn { nf τ } mn{ nf } mn{ mn{ nf }} m Dfnon : A solon of sysm s sd o wly oscllory n Ω f ls on of s comonns s oscllory n Ω cf Ldd Lsmnm nd Zng 97 Dfnon 6

4 6 Y So nd N Yosd In 9 oscllons of dly yrolc qons v n frs nvsgd y sv nd Bnov 9 cf sv 99 sv nd Bnov 96 Pr nd rn 99 nvsgd oscllory rors of solons of cold yrolc qons Oscllon of yrolc sysms w dvng rgmns ws sdd y L 997 nd n oscllon rsls v n slsd y svrl ors s g L Agrwl ng nd L nd rfrncs cd rn Howvr ll of m rn o cs wr mrcs H r dgonl mrcs or H ros of s r s o drv sffcn condons for vry solon of ondry vl rolm BC o wly oscllory n cylndrcl domn W no mrcs H r no ncssrly dgonl mrcs Oscllon rsls In s scon w sls lmm nd wo oscllon orms for ondry vl rolm BC wo mls r lso gvn n s scon o llsr oscllon rsls w w λ w n w on s nonngv nd corrsondng gnfncon cn cosn so > n s Y nd L 99 orm In cs on n w cn coos λ nd If / on n r s λ > nd gnfncon > n W s noon : { ; } ; { } Lmm: If ssfy ondry condon BC nd s gnfncon corrsondng o smlls gnvl λ n w on wr Ψ d Ψ

5 AA: Inrn J Vol Iss 6 [Prvosly Vol No ] 7 Proof: I s vdn Snc on nd on w on 5 From ondry condon BC w s on on Hnc w osrv 6 Comnng -6 ylds dsrd dny W no f on n / nd If nd n / nd wr d dnos volm of W s noon :

6 Y So nd N Yosd F f d [ Θ ] m{ ± Θ } ± { ; } W no # nd for ± ; for som w # For ml w l n w osrv nd nc { } nd wr { } { ± ± } { ± ; } { } orm : Assm yoss H -H old If followng condons r ssfd : H H H 9 ; on [ r s fncons Θ cllry nd for C Θ som [ > ; dnos sclr rodc ; ; R sc Θ s os - f for som { m} nd for ny c > Θ c d 7 n vry solon of ondry vl rolm BC s wly oscllory n Ω wr m F mn mn Θ Θ Ψ Θ Proof: Sos r ss solon Ψ τ of rolm BC wc s no wly oscllory n Ω n c comonn s nonoscllory n Ω nd

7 AA: Inrn J Vol Iss 6 [Prvosly Vol No ] 9 W sly s r s nmr sc > for Lng w δ wr δ sgn w s w > n [ r ss nmr sc w > w > w τ > w > n [ Procdng s n roof of om of So nd Yosd 5 w osrv followng dny olds : wr r s d d V δ δ δ W W V sc δ Sng m V w s Y V δ δ m W Θ V Y 9 rfor Y > or Y on [ for som If Y on [ n nd rfor V δ δ W Θ V W Θ lf nd sd of s osv n vw of yoss H wrs rg nd sd of s oscllory s s conrdcon Hnc w concld Y > on [ Snc Y Y > on [ w on Y on [ for som Hnc Y Y for In vw of fc V Y Θ nd Y s nondcrsng w on

8 9 Y So nd N Yosd V Y Y Y V Θ δ δ Y Θ Y Θ W Θ Θ Snc Y Y nd V > for from w v nd rfor V V Y Θ Y Θ for som Snc s nondcrsng from 9 nd w on Y Θ Y Ingrng ovr [ ] ylds s s Y Θ Y Y Y s s conrdcs yoss 7 nd comls roof orm : Assm yoss H -H old Evry solon of ondry vl rolm BC s wly oscllory n Ω f for ny ds for ll lrg s s lmnf ds Proof: Sos r ss solon of rolm BC wc s no wly oscllory n Ω Argng s n roof of orm w osrv

9 AA: Inrn J Vol Iss 6 [Prvosly Vol No ] 9 olds nd nc d d V W δ δ δ for som > Lng V V δ δ W w no V > > for som cf roof of orm Ingrng wc ovr [ ] w on V c c s s ds δ for som consns c nd c lf nd sd of ov nqly s osv wrs rg nd sd s no ondd from low y yoss s s conrdcon Rmr : W fnd { ± ; } for som w # Hnc orm olds r f yoss H nd condon 7 r rlcd y r s fncons Θ C [ ; R sc Θ s H rscvly oscllory rodc nd Θ ± Θ c dnos d sclr Rmr : Undr sm yoss of orm conclson of orm olds r f for ny for ll lrg lmnf lms s s Eml : W consdr yrolc sysm s s ds ds nd

10 9 Y So nd N Yosd sn sn sn sn 5 w ondry condon > 6 Hr n m / / / / / / τ τ / / / / ξ ξ ξ f sn f sn sn sn nd / W osrv I s sy o s λ sn Ψ / F / nd F / / sn / Snc w cn coos sn sn Θ Θ w on sn sn Θ Θ nd rfor

11 AA: Inrn J Vol Iss 6 [Prvosly Vol No ] 9 ± ± sn sn d d c nd ± ± sn sn d d c for ny > c Hnc follows from orm nd Rmr vry solon of rolm 5 6 s wly oscllory n On sc solon s U sn sn sn Eml : W consdr yrolc sysm cos sn cos cos / / 7 w ondry condon > sn Hr / n m / / / /

12 9 Y So nd N Yosd / / / τ / τ / / ξ ξ ξ f cos f cos sn cos nd W fnd / I s sy o cc λ nd sn Coosng Θ sn Θ sn w s Θ 6sn Θ 6sn Snc [ ± Θ ] d 6sn d ± nd [ ± Θ ] d 6sn d ± orm nd Rmr mly vry solon oscllory n In fc cos U cos sn s sc solon of 7 s wly

13 AA: Inrn J Vol Iss 6 [Prvosly Vol No ] 95 Rfrncs Agrwl R P F W ng nd W N L Oscllon of solons of sysms of nrl y rl fnconl dffrnl qons Com Al Ldd S V Lsmnm nd B Zng 97 Oscllon ory of Dffrnl Eqons w Dvng Argmns rcl Dr Inc Nw Yor L Y 997 Oscllon of sysms of yrolc dffrnl qons w dvng rgmns Ac Snc 5 n Cns L W N Oscllon rors for sysms of yrolc dffrnl qons of nrl y J Anl Al 69 L W N Forcd oscllon rors for crn sysms of rl fnconl dffrnl qons Al Com sv D P 99 Oscllon of solons of yrolc dffrnl qons of nrl y w mm odsn Vss Ucn Zvd Prlozn 5 9 sv D P nd D D Bnov 9 Oscllon rors of solons of yrolc qons of nrl y Procdngs of Colloqm on Qlv ory of Dffrnl Eqons Szgd 77 7 sv D P nd D D Bnov 96 Oscllon rors of solons of clss of yrolc qons of nrl y Fncl Evc 9 Pr N nd rn 99 Oscllory vor of solons of cold yrolc dffrnl qons Anlyss 56 So Y nd N Yosd 5 Oscllons of rolc sysms w fnconl rgmns oym J 5 Y Q X nd Z Y L 99 Inrodcon o Rcon-Dffson Eqons Bng Scnc Prss