UTILIZAREA METODEI NUCLEELOR DEGENERATE MODIFICATĂ LA REZOLVAREA APROXIMATIVĂ A ECUAŢIILOR INTEGRALE LINIARE DE TIP FREDHOLM
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1 UTILIZRE METODEI NULEELOR DEGENERTE MODIFITĂ L REZOLVRE PROXIMTIVĂ EUŢIILOR INTEGRLE LINIRE DE TIP FREDHOLM Mr S II dr Vse ăruţşu strct I ths rtce we propose ppromto method or Fredhom er ter equto souto whch s the deeerte ere method wth the cudrture method The coverece order o ths method s coosey coected wth the coverece order o deeerte ere method d wth the order o ppromto o the cudrture ormu whch s used Itroducere Utzre uceeor deeerte re oc îcă d 97 de către Gourst [] ş Schmdt [8] î 98 de Leesue [] ş m recet de Mute [6] [7] So [9] Keo ş Xu [] e utzeză rezovre promtvă ecuţor tere de tp Hmmerste r utoru î [] e utzeză rezovre promtvă ecuţor tere de tp Uryso Utzre ue ormue de cudrtură vez DV Ioescu [] petru ecuţ teră tştă e coduce rezovre uu sstem m smpu ş costrure ue souţ promtve m rpde chr dcă ordu de covereţă ceste este m mc decât ordu de covereţă d metod uceeor deeerte Vom prezet î coture metod uceeor deeerte câtev ormue de cudrtură mportte m es cee de tp Guss după cre vom costru ou ecuţe tştă căre souţe promeză souţ ecuţe tere dte ş vom vede pe u eempu cum ucţoeză cestă metodă utzâd petru rezovre sstemeor re cre se ue prormu MTHD Metod ecuţor propte Petru îceput vom euţ cee două teoreme e u vo Newm ecesre î demostrre teoreme u Ktorovc Teorem Fe X u spţu Bch ş :X X u opertor r ş cotuu cu < tuc opertoru perturt I- este ectv ş I- XX ude XX este spţu r pcţor re ş cotue de X X I- - - I N Teorem Fe X ş Y două spţ Bch ST:X Y două pcţ re ş cotue ste îcât S ectvă ş S - ot < tuc pcţ perturtă ST este ectvă ş ST - YX * Prezetăm î coture metod ecuţor propte Fe X u spţu ormt :X X u opertor r X ş ecuţ: I-y cre se presupue că re o ucă souţe Metod ecuţor propte costă î eere uu t opertor Ã:X X propt de ş uu t vector X propt de ste îcât ecuţ: I-Ã y să ă o ucă souţe r erore y- y să u depăşescă u umăr dt
2 Teorem Ktorovc Fe X u spţu Bch Ã:X X do opertor r ş cotu βγ cu β< ste îcât I- - β ş - γ tuc ecuţe ş dmt câte o sură souţe y respectv y ş re oc evure: β y- y γ β Rezovre ecuţor tere re de tp Fredhom cu metod uceeor deeerte Fe K[][] [] două ucţ dte ş se cosderă ecuţ teră ră: y λ K s y s ds [] ş λr u prmetru Fe :[] [] y vem sup K s y s ds y [ ] [ ] [ ] K s ds Ecuţ se m pote scre y λy su I-λy Teorem Dcă λ < tuc ecuţ re o ucă souţe y[] de orm yλ R λ s s ds [] ude R:{λR λ <}[][] R Rλs K s λ cu KsKs ş K s K u K u s du Vom pc metod ecuţor propte descrsă î teorem u Ktorovc rezovre umercă ecuţe tere Fe ecuţ teră tştă: y K s y s ds c [] Teorem Fe p q r ş γ R ste îcât pq< q r - γ [] ş p sup [ ] K s ds r q K s K s ds tuc ecuţe tere ş u câte o sură souţe y respectv sup [ ] y y- d [] ş re oc evure: qr y γ p p q Teorem 6 E Gourst ş E Schmdt Presupuem că K s B s s[] B [] este u uceu deeert cu {B }-r depedete Dcă y este o souţe ecuţe tuc estă R ste îcât y B ude s s ds B s s ds Oservţe Teorem u Stoe-Weerstrss e sură că orce uceu cotuu pote promt cu u uceu deeert orcât de e Formue de cudrtură de tp Guss
3 Fe î eer o ucţe :[] R teră ş e puctee [] pe cre e vom um odur O ormuă de tpu: R d î cre costtee ş odure sut ese ste îcât restu R să e u câd ucţ este îocută cu u poom orecre de u umt rd p se umeşte ormuă de cudrtură Formuee de cudrtură e u Guss sut ormuee de tpu î cre se pot determ tât odure cât ş costtee ste îcât restu să e u câd ucţ este îocută cu u poom orecre de rdu - ş re orm: d d ϕ 6 restu su orm R d stt de DV Ioescu d ϕ oeceţ ormue u Guss sut: ] [!] [! u 7 Dcă î ormu de cudrtură u Guss se îocueşte cu se oţ următoree ecuţ: Puâd coeceţ sut depedeţ de ş ş se determă dor d prmee ecuţ de m sus: 8 ş dec ormu de cudrtură u Guss se pote scre su orm: ] [ R d ude sut dţ de ecuţe 8 Restu ormue de cudrtură u Guss este: d R ϕ 9
4 Restu 9 se pote răt că este de orm:! R ude [!] âd ormu u Guss corespuzătore este: 6 ϕ d 8 d [ 8 ] î cre Restu se scre su orm: ude 7 6 R ude 7 Prezetre metode u metod uceeor deeerte ecuţe tere se tşeză ecuţ y K s y s ds [] K s depedete Souţ ceste ecuţ este: B s s[] B [] este u uceu deeert cu {B }- r y B ude s s ds B s s ds este souţ sstemuu Metod propusă costă î promre tereor cre pr î rezovre sstemuu cu o ormuă de cudrtură ste promre souţe ecuţe tere v de orm: ude y B m este souţ sstemuu m s s B s m Ordu de covereţă ceste ormue este sum dtre ordu de covereţă d metod uceeor deeerte ş ordu de covereţă ormue de cudrtură 6 Eempu umerc Fe : R cos Ne propuem să ăsm o souţe promtvă petru ecuţ teră y s s y s ds
5 Să oservăm îceput că ucţ d euţ este cotuă î deorece s m m Vom ee K K ude K ss- ;!!! s s s s N ş ude N!!!!!!! eâd ş utzâd ormu de cudrtură u Guss cu m vem: ude r De semee 8 8 r B B s s s - 6 s Se u următoree vor ţe petru ecuoscute: ; ; Gve B B Merr vem: 6 ste: y!! -6
6 um souţ ectă ecuţe tere dte este chr y îsemă că promre souţe ecuţe tere dte este orte uă este de pt souţ ectă dtortă pro rezovăr sstemuu cre m us Note orce [] ăruţşu V Deeerte Kere Method or Noer Iter Equtos Buet Mtemtque de Soc Mth Roume Vo9 Nr 999 pp7- [] Gourst M Sur u cs eemetre de equto de Fredhom Bu Soc Mth Frce 97 pp6-7 [] Ioescu DV udrtur umerce Bucureşt Edtur Tehcă 97 [] Keo H d Xu Y Deeerte ere method or Hmmerste equtos Mth Komp9 99 pp-8 [] Leesue H Sur methode de M Gourst pour resouto de equto de Fredhom Bu Soc Mth Frce 98 pp -6 [6] Muteu I ză ucţoă u-npoc Edtur Uverstăţ Beş-Boy 99 [7] Muteu I ză ucţoă ptoe spece u-npoc Edtur Uverstăţ Beş-Boy 99 [8] Schmdt E Zur Theore der ere ud cht ere Iterechue Mth 6 97 pp6-7 [9] So IH Error yss or css o deeerte-ere methods Numer Mth 976 pp -8
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