Explicit Representation of Green s Function for Linear Fractional. Differential Operator with Variable Coefficients
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1 KSU-MH--E-R-: Verso 3 Epc Represeo of Gree s uco for er rco ffere Operor w Vrbe Coeffces Mog-H K d Hog-Co O cu of Mecs K Sug Uvers Pogg P R Kore Correspodg uor e-: oogco@ooco bsrc We provde epc represeos of Gree s fucos for geer er frco dffere operors w vrbe coeffces d Re - ouves dervves We ssue er coeffces re couous Usg e epc represeos for Gree s fuco we ob epc represeos for souo of o-oogeeous frco dffere equo w vrbe coeffces of geer pe erefore e eod of Gree s fuco wc ws deveoped prevous reserc for souo of frco dffere equo w cos coeffces s eeded o e cse of frco dffere equos w vrbe coeffces Kewords: frco dffere operor frco Gree s fuco o-oogeeous frco dffere equos vrbe coeffce Mecs Subec Cssfco: roduco sees e cocep of frco Gree s fucos for frco dffere operors ve bee roduced b SMesov 6 for e frs e 974 o represe e souos of o-oogeeous frco dffere equo w cos coeffces d sge er s cocep s oe s eeded fro e cocep of Gree s fuco for ordr dffere operor w ur uber order gve b MNr 969 o frco re uber order fer frco Gree s fuco ve bee suded b S Mesov 974 uors ve derved epc represeo for Gree s fucos of er frco dffere operors w cos coeffces W e ep of Gree fuco d soe spec fucos suc s Mg-effer fuco 993 Mer d Ross 7 obed e epc represeos of souos of soe csses of oogeeous er
2 K Mog-H d O Hog-Co frco dffere equos Es 994 Podub derved epc represeo for Gree s fuco of rbrr er frco dffere operor w cos coeffces b usg pce rsfor Hu Y 3 8 provded represeo foru of Gree s fuco for e bove eoed frco dffere operors w cos coeffces b do decoposo eod o pp o represeos of e o-oogeeous frco dffere equos Mor d So 8 gve represeo foru of Gree`s fucos for vue probe of frco dffere operors w cos coeffces b e Neu seres Bo d Jusog provde epc represeo for souo of sse frco dffere equos w cos r coeffces d sge er X Hug e 3 provded epc represeo of Gree s fuco for frco dffere operor w cos coeffces Kbs e 4 preseed eod of sovg frco dffere equos w vrbe coeffces e egborood of ordr po b power seres eod ro e surzg bove we c s sever uors provded epc represeo foru of Gree s fuco for frco dffere operors w cos coeffces bu we coud fd ou e resus o rbrr er frco dffere operors w vrbe coeffces s pper we derved epc represeo foru of Gree s fuco for rbrr er frco dffere operors w couous coeffces d Re - ouve frco dervves d pped o ge souo represeo of oogeeous frco dffere equo erefore e eod of Gree s fuco wc ws deveoped for souo of frco dffere equo w cos coeffces prevous reserc s eeded o e cse of frco dffere equos w vrbe coeffces efos d Preres efo 5 or re uber d N we defe s foows: C : { f : R : f C } ; C : C efo 5 e f C e Re-ouve ef-sde frco egr f of order w org e po s defed b f f d provded e egr ess Here s e G fuco d s ced egr
3 Represeo of Gree s uco for er rco ffere Operor w Vrbe Coeffces operor of order efo 3 5 e N d f C e R-ouve frco dervve f of order w org e po s defed b d f f f d d d s ced e frco dffere operor of order efo 4 5 or fuco N we deoe b C b e spce of cope-vued s dervve o b e e fuco f w bsoue couous f for wc ere ess os everwere fuco g b suc f f g d s cse we c g e geerzed - dervve of f d sp wre g f prcur we deoe C b C b e we c wre s foows: d C b f : b C : f C b 3 d Here C s e se of cope ubers e 5 e spce C b cosss of ose d o ose fucos f wc c be represeed e for f C 4 were b C re rbrr coss d d e 5 e N { } d R e spce C b cosss of ose d o ose fucos f wc re represeed e for f d C 5 were b d C re rbrr coss C efo 5 4 e p e spce of fucos re defed b p p p : { f : f b} : 6 3
4 K Mog-H d O Hog-Co e 3 5 e d f : f be e frco egr of order f p p d f e we ve f f 7 b f f d f C b e e foowg ods os everwere o b b f f f or ore de sees of coceps d properes of frco ccuus see c Represeo of Gree s uco e s cosder e vue probe VP for frco dffere equos E gve b 3 3 were : C d s e Re-ouve ef-sded frco dffere operor w e org ; N efo 3 e fuco G ssfes e foowg codos d s ced Gree s fuco for frco dffere operor or VP 3 3: G 33 G 34 were s e Re-ouve ef-sded frco dffere operor w org d s e preer o sud Gree s fuco ow we cosder VP of E 35 d s correspodg egr equo 36 4
5 Represeo of Gree s uco for er rco ffere Operor w Vrbe Coeffces were d efo 3 or we deoe b oc e se of fucos f wc frco dervve f s oc egrbe e erv : : { f : f } 39 oc We eed foowg prer es e 3 e ssfes e o e reos 35 d 36 f oc d o f ssfes e o e egr equo 37 Proof rs we prove e ecess e ssf e o e oc reos 35 d 36 We rewre 35 e for e 3 Sce e d e reo 3 es oc oc e reos 8 d 36 gve e foowg oc e o 3 ppg e operor o bo sde of 3 d usg 36 we ob e equo 37 d ece e ecess s proved Now we w prove e suffcec e ssfes 37 e o oc or ppg e operor o bo sdes of 37 we ve Obvous
6 K Mog-H d O Hog-Co 6 Sce oc we ve 34 Usg d 3 we ob 36 s cer 35 ppg e operor o bo sdes of 37 d usg 7 d 35 we ob e equo 35 d ece e suffcec s proved erefore we esbsed e equvece of e VP of E d e egr equo 37 Now we fd for represeo of souo of e egr equo 37 usg e eod of successve pproos e foru of successve pproos for souo of e egr equo 37 s foowg: 36 Sce s evde oc rs pproe souo s obed b e foowg: 37 ro 37 s evde oc Secod pproe souo s obed b } { 38 Here deoes es coposo of operor d u
7 Represeo of Gree s uco for er rco ffere Operor w Vrbe Coeffces 7 operor e cse Cosderg oc C d 38 we ve oc oc 39 Ccug b e duco we ob oc 3 or g s e bo sde of 3 e foowg seres s obed: oc 3 eore 3 f C e VP of E s uque souo e spce oc d s souo s represeed e for of 3: 3 Proof ppg operor o ever er of rg sde of e seres 3 we ob e seres 33 Now e us prove s seres coverge spce for rbrr fed e Usg uo-epdg d se-group properes of frco egr for 33 we derve e ese d d d
8 K Mog-H d O Hog-Co 8 d E 34 Here Z d E s vue z z of e so-ced uvre Mg-effer fuco z z E see 97 5 B e eod of upper-seres es seres 33 coverges e spce e deoe su of e seres 33 b : 35 e of 3 c be rewre s foows 36 Sce for e oc ppg e operor o bo sdes of 36 for we ve 37 Sce oc we ve 38 B d 37 e reo 36 s obed: Ne we prove of 3 s ssfed equo 35 ro 3 36 we ve 39 or we ve
9 Represeo of Gree s uco for er rco ffere Operor w Vrbe Coeffces 9 Hece 33 ro 39 d 33 we ob us of 3 ssfes e equo 35 B coror 36 of 5 we ob e uqueess resu for e VP s copees e proof of eore 3 Coror 3 e cos e e souo oc of e VP s represeed b K 33 Proof Seg cos e souo represeo 3 of VP d use e se-group properes of frco egr d u-er s epdg e e dscusso sr w e dervo of 34 gves 33 Rer 3 e represeo of 33 s cocded w uvre Mg - effer fuco E See 97 of 5 Noe uvre Mg-effer fuco ws roduced org b Y uco Rer 3 e souos 3 d 33 of VP oug re seres epresso d gve gor for ccuo of e souo drec Coror 3 e cos e e souo oc of e VP s represeed b Mg-effer fuco of wo preers s foows: E 33 Proof e for u de 33 e sce
10 K Mog-H d O Hog-Co we c rewre 33 s e foowg: K Sce we c wre s foowg: Here sce z z E we ob E us we ob 33 Rer represeed b 33 or 3 s ced Gree s fuco of VP 3-3 cse of cos coeffces eore 3 f C e ere ess uque Gree s fuco G of frco dffere operor souo of VP e spce oc d s represeed s foows: G 333 prcur f cos d e we ve G
11 Represeo of Gree s uco for er rco ffere Operor w Vrbe Coeffces Here s oegve eger d Proof e proof of eore 3 s sr o of eore 3 Usg Gree s fucos we c ob represeo of souos o o-oogeeous VP e foowg eore ods eore 33 e C e ere ess uque souo See defo 5 of VP 3-3 d s represeed s foows : G d were G s Gree s fuco of VP 3-3 oc d 334 Proof Usg e defo of frco egr d ub s eore we frs prove e equ 334 e sr w eore 3 we subsue 334 o e VP 3-3 e we c prove s e souo of Epes e cosder e frco dffere operor s cse B 333 s Gree fuco s s foows: G f we subsue 4 d 4 o 33 d 34 e we c ow e G τ s e Gree fuco of 4 Now we ccue soe ers of 4 e er 5 5 d of e seres we = usg cge of vrbe s 5 e erv of egr τ s cged o : d 5 3 Sce d / / we ve s 5 5 s ds s s ds / / 5 d d 5
12 K Mog-H d O Hog-Co usg e sr eod we c ccue us we ve e seres represeo of Gree fuco 5 5 G Now usg e foru 334 e sove e foowg VP: B eore 33 s probe s uque souo 5 s cse 3 8 / 5 5 erefore usg 334 we ve oc d f we ccue we ve e seres represeo of e VP 44 d 45: Cocusos s pper we preseed epc represeo foru for e Gree s fuco of e geer er frco dffere operor w couous vrbe coeffces e eg of Re-ouve d sowed s resu s cosse w prevous resus e cse w cos coeffces e represeo foru of e Gree s fuco for er frco dffere operor w couous vrbe coeffces w be used s powerfu oo o sove e Cpuo frco dffere equos s we s Re-ouve frco equos cowedgee: uors woud e o oous revewers ep d dvce Refereces Bo B Rvero M d ruo JJ O sses er frco dffere equos w cos coeffces pped Mecs d Copuo O: 6/c684 Hfer Ruco Y d oovs Z Opero eod for e souo of frco
13 Represeo of Gree s uco for er rco ffere Operor w Vrbe Coeffces dffere equos w geerzed Re-ouve frco dervves rc Cc pp Cross-ref 3 Hu Y uo Y u Z c Souo of e er frco dffere equo b do decoposo eod J Copu pp M Vo 5 ssue 5 M 8 9 cross-ref 4 Kbs Rvero MRodrgez-Ger ruo JJ -c souos o soe er frco dffere equos w vrbe coeffces pp M d Copu cross-ref 5 Kbs Srvsv HM d ruo J J eor d ppcos of rco ffere Equos Esever serd-oo 6 6 Mesov S Vscoesc Properes of es Meurg Moscow Mer KS d Ross B roduco o e rco Ccuus d rco ffere Equos We d Sos New Yor Mor d So K Neu-Seres Souo of rco ffere Equo erdscpr foro Sceces Vo6 No 7-37 O: 436/s7 9 Mor d So K Souo of rco ffere Equo ers of srbuo eor erdscpr foro Sceces Vo 6 No 7-83 O: 436/s67 Nr M er ffere Operors N Moscow 969 Podub rco ffere Equos cdec Press S ego 999 Podub e pce rsfor Meod for er ffere Equos of e rco order s Ep Ps Sov cd Sc No UE Kosce -3 cross-ref 3 Hug X u X e use of frco B-spes wvees Mu-ers frco ordr ffere Equos ero Jour of ffere Equos Voue rce p O:55// So SG Kbs Mrcev O rco egrs d ervves: eor d ppcos New Yor d odo Gordo d Brec Scece Pubsers 993 3
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