Avlble t htt://vuedu/ Al Al Mth SSN: 9-966 Vol ue Decebe 5 878-89 Alcto d Aled Mthetc: A tetol oul AAM he hted cob olyol tegl oetol t o olvg Rcct deetl euto o ctol ode A Nety B Aghel d R D Detet o Mthetc Uvety o Md Bbol ty@uc Detet o Mthetc ehh Bch lc Ad Uvety ehh bghel@ehhuc Detet o Mthetc Ne Bch lc Ad Uvety Ne d@uec Receved: Octobe ; Acceted: ue 6 5 Abtct th tcle we hve led cob olyol to olve Rcct deetl euto o ctol ode o do o we hve eeted geel oul o the cob oetol t o ctol tegl oeto Ug the u ethod the oluto o th oble educe to the oluto o yte o lgebc euto he uecl eult o the ele eeted th e deotte the ececy o the eet ethod eywod: Fctol deetl euto; Oetol t; cob olyol; u ethod Rcct euto MSC No: A8; 6A; 7B6 toducto Fctol clculu h bee oe o the ot ctg ue tht hve ttcted the tteto o lge gou o chol tcully the eld o thetc d egeeg h due to the ct tht boudy vlue oble o ctol deetl euto c be eloyed to 878
AAM: te Vol ue Decebe 5 879 el vou tul heoe My chol d utho deet eld uch hyc lud low electcl etwo d vcoeltcty hve tteted to toduce odel o thee heoe though ug ctol deetl euto [Oldh d Se 97 Ro 975 lb et l 6 odluby 999 Lhth et l 9] teeted ede c chec othe boo d e the elted ltetue to get uthe oto bout ctol clculu [lb et l 6 odluby 999] We ow tht ot ctol deetl euto do ot led theelve to ccute lytcl oluto Coeuetly we hould ue ote d uecl techue to d oluto o ctol deetl euto Vou ethod hve bee eloyed the lt ew decde to d uch oluto hee ethod clude ctol tl deetl euto d ctol tego-deetl euto cotg ctol devtve Ado decooto ethod [Mo d Noo 6 Ry et l 6 Wg 6] Vtol teto ethod [c 8 Odbt d Mo 6 Abbbdy7] Hootoy ly ethod [Hh 9 Zugt ] d othe ethod [e Swel et l Eee et l ] Attet to d ccute d ecet ethod to olve ctol Rcct euto hve vted lot o ctve eech oect Schol d utho hve eeted vou lytcl d uecl ethod o olvg th euto Alytcl ethod clude the ADM d VM Abbbdy 7 Aothe och though whch we c olve ctol Rcct euto to ue HM Abbbdy 7 the eet eech we hve eloyed cob othogol olyol to d oluto to the Rcct deetl euto o ctol ode whch D y y b y g y d D ge cuto ctol devtve oeto o ode d b d g td o el ucto o R he uoe o th tudy to geele cob tegl oetol t to ctol clculu hu thee tce hve bee ued log wth the u ethod to educe the oluto o th oble to the oluto o yte o lgebc euto ele th ecto evel deto o ctol clculu e eeted he deto clude the cob olyol the hted cob olyol d oe o the oete Fctol Clculu Deto A el ucto codeed to be the ce C R thee et el ube o tht d t d to be the ce d oly C N whee C C
88 A Nety et l Deto he Re-Louvlle ctol tegl oeto o ode o ucto C gve by d Deto he Cuto' ctol devtve o deed d D D whee C d N oety Fo C N d the ollowg oete ty D! D cob olyol he cob olyol whch e eeeted by e othogol wth egd to the weght ucto w o the tevl e d whee! d the oece ucto Oe c ely otce tht the weght ucto w belog to L d oly he ollowg thee te-ecuece to elto eult the cob olyol
AAM: te Vol ue Decebe 5 88 c b whee c he cob olyol o degee e geeted by he hted cob olyol A eult o chgg vble we obt ew othogol olyol wth weght ucto w o the tevl [] whch clled hted cob olyol hee olyol hve the ollowg othogolty oete d w whee Fo we c wte ollow: 5 6 Fo elto 5 d 6 we c ely otce tht the ollowg oete e ted oety oety d d
88 A Nety et l oety he hted cob olyol c be cheved the ollowg o: whch oety 5 Fo w d l B l 7 l whee B t Bet ucto he oto o ucto the Sobolov ce the o y N Suoe N the et o ll o-egtve tege the weghted Sobolev ce H c be deed the uul wy whch dcte t e oduct eo d o by w v u w d eectvely w w tcully L w H w d w w H c be eued w w w w w w Now we c uoe the ucto H w eeted the ollowg oul: 7 whch the coecet e geeted by:
AAM: te Vol ue Decebe 5 88 w d 8 ctce oly the t - te hted cob olyol e te to ccout he we hve: wth 9 uch o y tht : te deol vecto ce h uue bet oto y y w w Guo d Wg ce to the cocluo tht o y H geec otve cott C deedet o y ucto d et o tht: C w w w N d he oetol t o ctol tegl We c ee Re-Louvlle ctol tegl oeto o ode o the vecto by: whee the oetol t o Re-Louvlle ctol tegl o ode heoe the oetol t o Re-Louvlle ctol tegl o ode the the eleet o th t e te : B l l Now we dee the eo vecto E l E he u o o vecto E deed ollow Guo d Wg
88 A Nety et l!! B L B L E whee d L M Reult Le Let Now we uoe tht the e heoe he H whee h H wth d w h l l t lt t oo: We deote We hve we code wth
AAM: te Vol ue Decebe 5 885 d lyg 8-9 the ollowg obted: H h h h whee d w h l l t lt t he oo colete Now we code the Rcct euto wth ctol ode o the o D y y b y g d y whee d b e el cott coecet d D td o the Cuto ctol devtve o ode Ug Deto we c ewte Euto : Dy y b y g o olve oble - we ote y D d g by the hted cob olyol : D y 5 G g g 6 Fo 5 we get d D y 7 Coeuetly y y 8
886 A Nety et l d y y y 9 Moeove we hve D y D y Ug 6 d 8- oble - c be ewtte : y b y y G Alyg le th elto educe to the ollowg elto C b H C C G whee y C y C d C y c be clculted the e wy By lyg the tycl u ethod ee Cuto et l 988 yte o lgebc euto L F L b H C obted F G C C 5 Nuecl eult th ecto we led the ethod eeted th e d olved oe ele he ele eoted th ecto wee elected o lge collecto o oble to whch th ethod could be led Ele We code the ollowg ctol Rcct deetl euto wth tl codto D y y y g 5 y 5 he ect oluto o th oble o y th w oud to be o the o log By ug the ethod tht w elboted evou ecto we hve the oto 5-
AAM: te Vol ue Decebe 5 887 Ug d u ethod the oble 5-5 e toed to the ollowg elto H G 5 Lettg d we obt 7778 9 57 5 5 G H 7778 7778 68778 68778 96785 85975 9678 85975 Now o 5 we coclude tht 6957 879 So y 69 7 Fgue he ote oluto the ce 5 d Ele o Ele We code the ollowg ctol Rcct deetl euto wth tl codto D y y y he ect oluto o th oble 5 y 56 e y e By ug the ethod tht w elboted evou ecto we hve the oto 5-
888 A Nety et l Ug d u ethod the oble 5-55 e toed to the ollowg elto Lettg H 57 d 8 we obt 95 9 6 5 6857 7778 778 79 6667 79 7558 698 6899 557 t ey to ee tht H h h h 6 8 8 6 6958 67 58788 So Now o 56 we coclude tht y 5 58 7578 689 8856 758 h 9 58 86 88 9977 567 h 689 8856 758 676 98 7 85857 5579 98 85 586 79 h 8595 5695 598 955 796 999
AAM: te Vol ue Decebe 5 889 Fgue he ote oluto the ce 5 d Ele o Ele A l ele we code the ollowg ctol Rcct deetl euto wth tl codto he ect oluto o th oble whee t D y y 57 y 58 y the Beel ucto o the t d We uoe tht d t ey to ee tht 5 5 5 5 5 the e wy the evou ele by ug the oble 57-58 e toed to the ollowg elto: H 5 5
89 A Nety et l Now ug u ethod we educe the oble to olve the ollowg yte o lgebc euto Now o 59 we coclude H 5 5 59 So 657 578 887 y 55778 79 ble the ote oluto o tet oble d obted by deet vlue o d ug the eeted ethod ble he ote oluto o ele d EX y EX EX EX 5 8 9 5 5 8 9 5 5 8 9 5 969 68 669 69 65 7 8 69 765 7 9996 77 969 8 76 96 88 78 59 89 79 565 58 6 785 8 66 9999 7 7 6 5 65959 5565 9769 6686 65959 556 9989 675 579 658798 869 5 997
AAM: te Vol ue Decebe 5 89 5 Cocluo th e we hve ooed uecl ethod o olvg Rcct deetl euto o ctol ode he hted cob olyol tegl oetol t w develoed to olve th euto he uecl eult howed th ethod oweul ew d teetg All o the uecl coutto th tudy hve bee doe o C lyg oe og wtte MALE REFERENCES Abbbdy S 6 Hootoy etubto ethod o udtc Rcct deetl euto d coo wth Ado decooto ethod Al Mth Cout 7:85-9 Abbbdy S 7 A oto oluto o ole euto wth Re- Louvlle' ctol devtve by He' vtol teto ethod Cout Al Mth 7:5-58 Abbbdy S 7 A ew lcto o He vtol teto ethod o udtc Rcct deetl euto by ug Ado olyol Cout Al Mth 7 59-6 Cuto C Hu MY uteo A Zg A 988 Sectl Method Flud Dyc etce-hll Eglewood Cl N Eee GH ghvd H d Al M Nuecl oluto o the hgh thel lo oble eeted by ctol deetl euto Cou Nole Sc Nue Sul 6 :56-6 Guo BY Wg LL cob oto o-uoly cob-weghted obolev ce Ao heoy 8: Hh Abdul O Mo S 9 Hootoy ly ethod o ctol V Cou Nole Sc Nue Sul :67-68 c M 8 he ote d ect oluto o the ce- d te-ctol Buge euto wth tl codto by vtol teto ethod Mth Al Al 5: 76-8 e M d Eee G H Alytcl d uecl oluto o deet bolc het euto eeted the o o ult-te ctol deetl euto oul o Scece d echology cto A Scece vol 5 o 85-9 lb AA Svtv HM ullo 6 heoy d lcto o ctol deetl euto Eleve BV Netheld Lhth V Leel S Vudh 9 heoy o ctol dyc yte Cbdge Acdec ublhe Cbdge Mo S Noo MA 6 Nuecl ethod o outh-ode ctol tegodeetl euto Al Mth Cout 8:75-76 Odbt Z Mo S 6 Alcto o vtol teto ethod to ole deetl euto o ctol ode t Nole Sc Nue Sul 7:7-79 Oldh B Se 97 he ctol clculu Acdec e New Yo d Lodo
89 A Nety et l odluby 999 Fctol deetl euto Acdec e S Dego CA Ry SS Chudhu S Be R 6 Alytcl ote oluto o ole dyc yte cotg ctol devtve by oded decooto ethod Al Mth Cout 8:5-55 Ro Ed B 975 he ctol clculu d t lcto : Lectue ote thetc vol75 Sge-Velg Bel Swel NH hde MM d Mhdy AM Nuecl Stude o Solvg Fctol Rcct Deetl Euto Al Al Mth 7: 595-68 Wg 6 Nuecl oluto o ctol dv-buge euto by Ado decooto ethod Al Mth Cout 8: 8-55 Zugt M Mo S Alweh A Alytcl ote oluto o yte o ctol lgebc-deetl euto by hootoy ly ethod Cout Mth Al 59: 7-5