On One Generalization of Boundary Value Problem for Ordinary Differential Equations on Graphs in the Three-dimensional Space
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1 WSEAS TRASACTIOS o MATEMATICS D G Gordez V Medze T D Dts O Oe Geerzto o Bodr Ve Proem or Ordr Deret Eqtos o Grps te Tree-dmeso Spe DGGORDEZIAI Deprtmet o Mtemts Ft o Et d tr Sees Ts Stte Uerst Uerst str43 Ts GEORGIA dgord37@otmom V MELADZE Deprtmet o Compter Sees St Adre Frst Georg Uerst 53- Cdze e Ts GEORGIA _medze@otmom TD DAVITASVILI Deprtmet o Compter Sees Ft o Et d tr Sees Ts Stte Uerst Uerst str43 Ts GEORGIA t_dts@otmom Astrt: - Te preset work s te geerzto o odr e proem or ordr deret eqtos o grps Ts proem s estgted d orretess o te stted proem s proed Te spe tteto s ge to ostrto d reser o deree oges Estmto o preso s ge Te orms o doe-sweep metod tpe re sggested or dg te soto o oted deree seme I ts work te odr-e proems or Posso s eqtos te tree-dmeso spe o some twodmeso strtres wt oe-dmeso ommo prt s ge d estgted Ts teqe o estgto e es pped to te more ompe t dt d eqtos Te deree seme or mer soto o ts proem s ostrted d estmto o preso s ge S proems e prt sese d te e sed or mtemt modeg o spe proems o pss egeerg eoog d so o e-words: - Deret eqtos o grps Deree seme Itrodto I te work te odr e proem or ordr deret eqtos o grps s estgted; orretess o te stted proem s proed Te spe tteto s ge to ostrto d reser o deree oges w s tte oer ppers o oter tors Estmto o preso s ge; doe-sweep metod tpe orms re sggested or dg te soto o deree seme -3 ISS: ISB: ISS: Isse 8 Vome 8 Agst 9
2 WSEAS TRASACTIOS o MATEMATICS D G Gordez V Medze T D Dts Proeedgs o te d WSEAS It Co o FIITE DIFFERECES FIITE ELEMETS FIITE VOLUMES BOUDARY ELEMETS It s posse to ote some works deoted to te teoret estgto o odr e proems osdered o grps see or empe 4 d te tertre metoed tere Cert ts st s ompete I te preset work tere re ge some geerztos o te oe metoed proem: te tree-dmeso spe o some twodmeso strtres wt oe-dmeso ommo prt te odr-e proem or Posso s eqto s stted d estgted Ts teqe o estgto e es pped to te more ompe t dt d eqtos Oos s proems e prt sese d te e sed or mtemt modeg o spe proems o pss egeerg eoog d so o 5- Cert ts st s ompete Ordr deret eqtos o te seod order o grps Let s osder grp G V E V were s set o tops o ts grp s ode o te grp d E s set o rs o te grp { } Deote te r Γ O e r trode o oordte sstem wt te org te ode d te oordte were s egt o te re Γ Let s stte te oowg proem: d te tos w stses te deret eqtos d d d d q odr odtos d odtos o ojtos β β 3 were q d 4 d C > C C q C ost > re te ge tos d re te ge mers Teorem Tere ests qe regr soto o proem -4 e ests qe tos C C w stses eqtos odr odtos d odtos o ojtos 3 4 Te proo o ts teorem see 3 Deree seme or mer soto o proem -4 O Γ we trode orm mes wt step : ω { ; ; } I o te mes ω we ssttte deret opertor te deree opertor we ot te oowg deree seme: q 5 6 β 7 β ISS: Isse 8 Vome 8 Agst 9
3 WSEAS TRASACTIOS o MATEMATICS D G Gordez V Medze T D Dts were 8 Teorem Tere est o more te oe soto o te deree seme Teorem 3 Let C Te te soto o te deree seme 5-8 orm oerges to te soto o te proem -4 t te rte o O we were m Te proo o te teorems 3 see 3 Remrk Let C Te were d Ψ O Θ O Ψ Θ q 4 Let C Te sted o odto 8 we osder te oowg ppromto o te ojto odtos: q Te te error o ppromto Θ 5 5 q w e te order O Ideed d Θ d 5 5 q } O O s d d ojto odto d q d d q Tereore Ψ O Θ O 4 C So te se o deree seme te oowg teorem s tre 4 Teorem 4 Let C Te te soto o te deree seme orm oerges to te soto o te proem -4 t te rte o O we were m ISS: Isse 8 Vome 8 Agst 9
4 4 Vrt o doe-sweep metod or deree eqtos 5-8 Let s wrte te deree seme 5-8 s sstem o te er ger eqtos: q β β Itrode te oowg deottos: m Te te sstem o eqtos e rewrtte te oowg orm: β β 3 m 4 Sppose tt or te soto o deree eqto te reto ods: 5 te 6 Sstttg epresso 5 te eqto we ot From ts eqto we dee : Comprg ts eqt wt te eqt 6 we ot: 7 WSEAS TRASACTIOS o MATEMATICS D G Gordez V Medze T D Dts ISS: Isse 8 Vome 8 Agst 9
5 WSEAS TRASACTIOS o MATEMATICS D G Gordez V Medze T D Dts Usg te odr odtos to dee d we ot: 8 Rerret retos 7 8 ow to dee oeets d As tereore repetg te resog rom t e proed tt < d Ts we e proed tt mes o rerret orms 7 8 qe e deed es o te oeet Wrte ot orms 5 se o : Isert tese eqtes 4 d tke to ot retos 3 te we ot: m As < rom te st eqt we ot: m m m Coet orms o doe-sweep metod d wrte tem dow order o ppto: m m m 9 5 O oe geerzto o te proem -4 Let s osder oe geerzto o proem - 4 Isted o grp et's osder --pes R 3 w re oded ommo odr e γ O tese pes we osder te o oordte sstems: Cosder te oowg proem: d tos deret eqtos w sts te ISS: Isse 8 Vome 8 Agst 9
6 WSEAS TRASACTIOS o MATEMATICS D G Gordez V Medze T D Dts < < odr odtos < < β β 3 ϕ 3 d ojto odtos were β β 4 C C re ge tos ϕ C 5 Teorem 5 Tere ests o more te oe regr soto o proem -5 Proo It s set to proe tt te omogeeos proem wt te omogeeos odr odtos orrespodg to te proem -5 s o tr soto Cosder ts proem: < < 6 < < 3 Mtp te eqto 3 o to d tegrte t t rst o te ter wt respet to re d te o te ter wt respet to re Frter sm p tese eqtes rom to : 3 To ge te et d-sde o ts eqt rst we eed to e sre te ress o te oowg eqtes teret tke to ot odtos 7-3: d d d d d d d ISS: Isse 8 Vome 8 Agst 9
7 d d Aogos: d d d d d d d Tkg tese eqtes to ot rom te 3 we ot: d d 33 From te st eqto we ot tt e ost Tkg to ot odr odtos 7-9 we ot tt Te teorem s proed Itrode te deotto Let s osder te proem -5 t spposg tt ϕ d L Itrode te oowg set o tos: { } : L L e s Sooe spe o te rst order o d eqze to zero o te odr s met sese o tre see or empe 3 Let s dee tos o te set 3 R te oowg w: Deote { } : β β Itrode ts spe te sr prodt d te orm ded ts prodt: Te geerzed soto o te proem -5 wt te omogeeos odr odtos we te to or w te eqt tkes pe: WSEAS TRASACTIOS o MATEMATICS D G Gordez V Medze T D Dts ISS: Isse 8 Vome 8 Agst 9
8 WSEAS TRASACTIOS o MATEMATICS D G Gordez V Medze T D Dts F or to were d F see Teorem 6 Tere ests qe geerzed o te proem -5 soto Proo It e es sow tt te s otos d oere o 3 From ts o te ss o L- Mgrm Teorem te ress o sttemet o te teorem mmedte oows 6 Deree seme or mer soto o te proem -5 Let s dee te meses: ω γ { ω ω γ γ γ ; { ; } { ; } { ; } { ; } 3 { ; } 4 } ω ω ω ω Itrode so te oowg deottos: ± ± ± ± Let's pt oormt to te proem -5 te deree seme ω 36 γ 37 γ γ 38 ϕ 3 39 β β 4 γ 4 4 were It s es to sow tt t proem -5 s set smoot soto te 36 ppromtes te eqto wt error O d te odto 4 ppromtes te odto 5 wt error O ISS: Isse 8 Vome 8 Agst 9
9 WSEAS TRASACTIOS o MATEMATICS D G Gordez V Medze T D Dts So te seme 36-4 ppromtes te proem -5 wt error O For te mer soto o te proem - 5 t s posse to ostrt te seme wt error O For ts prpose te odto 5 e ppromted s oows: 4 t ; Aordg to 4 β tereore 4 e rewrtte s oows: F were ; F Frter or smpt o te sttemet we w osder tt so 43 Te proem -5 t te epted ssmptos 43 e soed orm te orm o row kπ π k e d e s kπ were k d k k k k π e e π ; kπ ϕ k ϕ k kπ kπ kπ kπ e e e e kπ ϕ k ϕ k kπ kπ kπ kπ e e e e 44 ϕk k -t oeet o Fre row or to ϕ It s es to ote tt te row 44 represets te soto o te proem -5 3 ϕ k C Smr t s posse to ostrt te soto o te seme 36-4 t ssmptos 43 It s posse to ser te soto o 36-4 te orm o te sm o mes re ω Te teoog o ostrto o te soto o deree seme repets te resogs pped we orm 44 s oted Smpe geersto o metods or estmto o r o deree semes or retgr re se o med odr odtos te osderto o oe o te prtes o te retge em s odto ows to proe te oergee o soto o te deree seme 36-4 to set smoot soto o t proem wt speed O Te qestos o oergee or speed deree seme s re ot estgted Reerees DGorgez TDts Mpres Medze O te Soto o Bodr Ve Proem or Deret Eqtos Ge Grps // Apped Mtemts Iormts d Mes Ts 8 3 pp-4 Pokor YV Pek OM Boroskk AV Prde VL Lzre P Sro SA Deret eqtos o geometr grps Rss M: Fzmtt 4 p7 3 Smrsk AA Teor o deree semes Te trd edto M: k 989 p6 4 Pokor YV Prde VL O trsmsso odtos te Strm-Loe proem o etwork Rss Sorem Mt Proz o Der Ur Cst Prozod ; trsto J Mt S Y 3 5 o ISS: Isse 8 Vome 8 Agst 9
10 WSEAS TRASACTIOS o MATEMATICS D G Gordez V Medze T D Dts 5 rse AB dr o rers d wter reserors M: P Rer Trsport 955 p9 6 Lose A Teor o er eetr s M: g Soo 987 p67 7 met P Grps modes or wes t strktres // Wes Rdom Med pp-4 8 Akkerms E Comtet A Desos J Motm G Teer C Spetr determt o qtm grps // A Ps 84 pp-5 9 kos E Mstorks Estee Uqeess d Fte Deree Soto or te Dret proem o te Srodger-Mwe eqtos WSEAS Trstos o Mtemts Isse 4 Vome 8 Apr 9 pp77-83 kos E Mstorks ss Ftd O te Soto o p-lp or o-ewto d ow WSEAS Trstos o Mtemts Isse 6 Vome 8 Je 9 pp39-45 Aedre Greeko Fst Agortms d MATLAB Sotwre or Soto o te Dret Bodr Ve Proems or Ept Prt Deret Eqtos Doms wt Compted Geometr WSEAS Trstos o Mtemts Isse 4 Vome 7 Apr 8 pp73-8 Smrsk AA Adree BG Deree metods or ept eqtos M: k 976 p35 3 Los J-L Mgees E Proèmes mtes o omogèes et pptos Dod Prs 968 p37 ISS: Isse 8 Vome 8 Agst 9
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