International Journal of Scientific & Engineering Research, Volume 6, Issue 12, December ISSN
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1 Itertiol Jourl o Scietiic & Egieerig Reserc, Volume 6, Issue, December-0 0 ISSN -8 A Clss o New Bloc Geerlized Adms Implicit Ruge-Kutt Colloctio Metods Kumleg G.M., Collom J. P, Omgwu S. Abstrct I tis pper, te reormultio o te bloc geerlized Adms metods ito bloc geerlized Adms implicit Rug-Kutt metods or step umbers,,, is cosidered. Tis is becuse o te useuless o bloc implicit Ruge-Kutt metods or te solutio o sti ordir dieretil equtios. Te ew metods proposed i tis pper tur out to be A-stble d possess te stbilit properties o te Ruge-Kutt metods d ve implicit structure or ccurte d eiciet implemettio. Numericl exmples obtied demostrte te ccurc d eiciec o te ew bloc metods. Idex Terms A-stble, Geerlized Adms metods, Rug-Kutt metods d Sti Ordir Dieretil Equtios. Itroductio 0 ttp://
2 Itertiol Jourl o Scietiic & Egieerig Reserc, Volume 6, Issue, December-0 ISSN -8 I erlier wor, Collom d Kumleg 0 costructed d implemeted clss o bloc Adms Moulto implicit Ruge KuttBmir metods or sti dieretil equtios. Te Bmir metods were sow to be A-stble d results obtied coverged to te exct solutio. Yubu 00, costructed oe-step bloc implicit Ruge Kutt colloctio metods or solvig sti ODEs, te scemes were obtied bsed o colloctio t Gussi poits d were sow to possess lrge regio o bsolute stbilit. Oter utors suc s Y d Adegboe 007, 0 ve costructed implicit bloc Ruge -Kutt metods bsed o te Qude s tpe or te solutio o sti ODEs. Te implicit Ruge-Kutt metods ve bee sow to be ver eective i te tretmet o sti iitil vlue problems s c be see i te wor o Axelso 6, Cipm 7. Te implicit Ruge-Kutt metods were ull developed bsed o te Gussi, Lobtto d Rduqudritures b Butcer d Wer 6. Gozlez-Pito et l 7 developed sigle Newto scemes or te solutio o te stge equtios o some implicit Ruge-utt metods suc s Guss, RduIIA d Lobtto IIIA wit our implicit stges. I tis pper, te reormultio o te bloc geerlized Adms metods s Bloc geerlized implicit Ruge- Kutt BGAIRK metods is cosidered or te solutio o iitil vlue problems o te orm ' x,, x 0 0, x [ b, ], R, be pproximtio to x, x,,,... d let 0 s te orm bii were i s i t ci, ij j, i,..., s j Te Butcer tbleu or te metod is give s c s c s c s s s ss b b x A implicit Ruge-Kutt metod s For te implicit Ruge Kutt metods, t ever step sstem o lgebric equtios wit s compoets s to be solved. Tis icreses te computtiol cost cosiderbl. Te New Metod b s Te costructio o bloc geerlized Adms metods or step umbers,, 0 ve bee cosidered i Kumleg 0usig te cotiuous iite dierece pproximtio d te iterpoltio d colloctio criteri described b Lie d Norsett 8 d Oumi et.l. Tese bloc metods were sow to ve good stbilit properties mig tem useul i te solutio o sti ODEs. To urter improve o te good stbilit properties o tese bloc metods, te reormultio o te metods ito bloc implicit Ruge- Kutt metods or step umbers,, is cosidered i tis pper. Te reormultio o te bloc geerlized Adms metodis cosidered ere becuse o te dvtges tt te implicit Ruge-Kutt metods ve over te covetiol lier multistep metods. Te bloc geerlized Adms metods cosidered b Kumleg 0 or step umbers,, re s ollows; Bloc geerlized Adms metod or step umber wit uiorm order 0 ttp://
3 Itertiol Jourl o Scietiic & Egieerig Reserc, Volume 6, Issue, December-0 ISSN -8 0 ttp:// Bloc geerlized Adms metod or step umber wit uiorm order Bloc geerlized Adms metod or step umber wit uiorm order respectivel. Derivtio o BGAIRK Metod Te bloc GAMs re reormulted ito te BGAIRK Metod o order our b solvig tem simulteousl to rrive t 8 6 Its GAIRK metod wic is our stge implicit Ruge Kutt colloctio mil o uiorm order our is give s,,,, x x x x 7
4 Itertiol Jourl o Scietiic & Egieerig Reserc, Volume 6, Issue, December-0 ISSN -8 Te crcteristic coeiciets o te BGAIRK metod re displed i orm o te Butcer tbleu s ollows: Derivtio o BGAIRK Metod Te bloc GAMs re lso reormulted ito te BGAIRK Metod o order ive b solvig tem simulteousl to rrive t Its GAIRK metod wic is ive stge implicit Ruge Kutt colloctio mil o uiorm order ive is give s x, x, x, x, x, Te crcteristic coeiciets o te BGAIRK metod re displed i orm o te Butcer tbleu s ollows: ttp://
5 Itertiol Jourl o Scietiic & Egieerig Reserc, Volume 6, Issue, December-0 ISSN -8 Derivtio o BGAIRK Metod Te bloc GAMs re reormulted ito te BGAIRK Metod o order six b solvig tem simulteousl to rrive t Te GAIRK metod or 0 wic is six stge implicit Ruge Kutt colloctio mil o uiorm order six is give s x, x, x, x, x, 6 x, Te crcteristic coeiciets o te BGAIRK metod re displed i orm o te Butcer tbleu s ollows: ttp://
6 Itertiol Jourl o Scietiic & Egieerig Reserc, Volume 6, Issue, December-0 ISSN -8 Absolute Stbilit Regios o te GAIRK Metods I tis sectio, te plots o te regio o bsolute stbilit o te ewl costructed GAIRK metods or s,, re cosidered. Tis is doe b reormultig te GAIRK metods s geerl lier metods itroduced b Burrge d Butcer, Butcer d Wer6 d urter cosidered b Collom 007. Te regios o bsolute stbilit o te GAIRK metods re s sow i ig.. Imz Imz Rez Rez Fig. Absolute Stbilit Regios o te Fig. Absolute Stbilit Regios o te GAIRK metod GAIRKmetod Imz Rez Fig. Absolute Stbilit Regios o te GAIRK metod Te regios i Fig d sows tt te GAIRK metods or d re A-stble sice teir regios o bsolute stbilit cotis te wole o te let-d l ple wile Fig revels tt te GAIRK metod or step umber is A-Stble. 0 ttp://
7 Itertiol Jourl o Scietiic & Egieerig Reserc, Volume 6, Issue, December-0 6 ISSN -8 Numericl Experimets I order to scerti te ccurc d eiciec o te GAIRK metods, umericl exmples re cosidered. Results obtied re compred wit LobttoIIIA metods o orders d 6 d te RduIIAmetod o order becuse o teir possessio o excellet stbilit d covergece properties or sti problems see Deer d Verwer 8, Butcer 87, Hirer d Wer 6.I ll te problems solved, 0. d0 x. Exmple. See Collom d Kumleg 0 ' 00si x, x si x0,0cos x 0.0e x Exmple. See Yubuet.l 0 ' 00 x x, 0, x x Exmple. e 00x - x - 000x - x ' , 0 0, x 000 e, 0, x e x e e Exmple ' x e Exmple gmr order LobttoIIIA order gmr order rdiia gmr order 6 LobttoIIIA order6 0.00E E E E E E00.0E-0.0E-0.8E-0.7E-0.E-0.E-0.7E-0.E-0.6E-0.68E-0.0E-0.E-0.7E-0.76E-0.E-0.0E-06.0E-0 8.8E-06.E-0 8.6E-0.07E-0.08E-08.E E E-06.E-0.E-06.8E E E-08.E E-06.8E-07.6E-08.E-08 8.E-0.E-07.E-06 6.E-08.E-08.E-08.E-0.8E E-07.0E-08.8E-08.E-0.E-0.7E-08.E-07.E-0.E-08.06E-0.6E-0.6E-08.E-07.07E-0.60E-08.8E-.0E-0 0 ttp://
8 Itertiol Jourl o Scietiic & Egieerig Reserc, Volume 6, Issue, December-0 7 ISSN -8 Exmple gmr order LobttoIIIA order gmr order rdiia order gmr order 6 LobttoIIIA order6 0.00E E E E E E00.0E-0.0E-0.8E-0.7E-0.E-0.E-0.7E-0.0E-0.6E-0.68E-0.0E-0.E-0.7E-0.E-0.E-0.8E-0.0E-0 8.8E-0.E-0.07E-0.07E-0 7.6E-06.E-0 8.E E-0.7E-0.E-0.70E E-0 8.E-06.8E-0.07E-0.8E-0.E-08.E E-07.E-0.0E-0 6.E-06.E-0.E-06 7.E-08.08E-0.7E-0.E-06.E-.E-07 7.E-0.E-06 7.E-0.0E-07.6E-.E E-0 6.E-07 8.E-0.7E-08.7E-.0E-0 6.7E- Exmple Gmr LobttoIIIA gmr Gmr LobttoIIIA order order rdiia order order 6 order6 0.00E E E E E E00.E E-0.E-0.E-0.E-0 7.8E-0.0E-0 7.7E-0.E-0 6.0E-0.E-0 6.8E-0.E-0 6.7E-0.E-0.76E-0 7.E-0.86E-0.0E-0.7E-0.E-0.77E-06.E-0.8E-0.8E-0.E-0 8.6E-0.E-06.8E-0.0E-0.6E-0.6E-0.0E-0.E-06.6E-0.7E-0.76E-0.06E-0.E-0.0E-06.7E-06.86E-0.E-0.6E-0.E-0.E-07.00E-06.6E-0.8E-0.E-0.E-06 8.E-07.0E-07.E-0.6E-0.7E-0.7E E E-07.06E-0 0 ttp://
9 Itertiol Jourl o Scietiic & Egieerig Reserc, Volume 6, Issue, December-0 8 ISSN -8 Exmple gmr order LobttoIIIA order gmr order rdiia gmr order 6 LobttoIIIA order6 0.00E E E E E E00.E-0.E-0 7.E-.08E-08.E-.E-0 8.7E-0 6.6E-0.7E-.00E-08.8E-.66E-.E-0 8.E-0.76E-.0E-0.7E-.0E-0 8.0E-0.00E-0.E- 8.E-0.E-.E- 7.E-0.E-0.8E- 7.E-0.0E- 7.E- 6.67E-0.E-0.6E- 6.7E-0.E-.E- 6.06E-0.7E-0.E- 6.0E-0.E-.68E-.8E-0.0E-0.8E-.E-0.6E-.E-.6E-0.E-0.6E-.E-0.7E-.E-.8E-0.0E-0.E-.E-0.E-.E- Coclusio Te umericl results rom Tbles to revel te ccurc o te ewl costructed clss o bloc Geerlized Adms implicit Ruge Kutt metods or step umbers, d. It c be see clerl rom Tble, d tt our ew metods perorm better t te Lobtto IIIA metod o order, Lobtto IIIA o order 6 d te Rdu IIA metod o order or te problems solved i exmples, d. However, or exmple, Tble revels tt our ew metods compete well wit te Rdu IIA metod o order. It ws lso observed tt te ew metods ve better stbilit regios t te covetiol bloc geerlized Adms metods. Reereces [] O.Axelso, A Clss o A-Stble Metods. BIT,,8-6. [] J.C.Buctcer, Te Numericl Alsis o Ordir Dieretil Equtios, Jo Wile d Sos, Cicester 87.. [] J. C.Buctcer, G.Wer, Ruge Kutt metods: Some Historicl otes, Jourl o Applied Numericl Mtemtics, - 6. [] K. Burrge,J.C. Butcer, No Lier Stbilit or Geerl Clss o Dieretil Equtios Metod. BIT 0, 8-0. [] F.H.Cipm, A-stble Ruge-Kutt Processes. BIT,, [6] J.P. Collom, G.M. Kumleg, A Clss o Bloc Hbrid Adms Moulto Implicit Ruge-Kutt Bmir Metods or Sti Ordir Dieretil Equtios. Itertiol Jourl o Mtemtics d Computtios [7] K. Deer d J. G. Verwer, Stbilit o Ruge Kutt metods or Sti Nolier Dieretil Equtios. Nort-Holld, Amsterdm.8 [8] J.P.Collom, J.N. Ndm d G.M.Kumleg, O Some Properties o te Bloc Lier Multistep Metods. Sciece World Jourl,, [] S.Gozlez-Pito, S. perez-rodriquez, J.I. Motijo, O te umericl solutio o sti IVP s b Lobtto IIIA ruge-utt metod. Jourl o Computtiol d Applied Mtemtics, 8,-8 7. [0] E. Hirer, d G. Wer, Solvig Ordir Dieretil Equtios II, Spriger-Verlg, Berli 6. [] G.M. Kumleg, Cotiuous Geerlized Adms Metods i Bloc Form or Te Solutio o Nolier Ordir Dieretil Equtios. PD Tesis Upublised Uiversit o Jos ttp://
10 Itertiol Jourl o Scietiic & Egieerig Reserc, Volume 6, Issue, December-0 ISSN -8 [] I Lie d S. P. Norsett, Super covergece or Multistep Colloctio. Mtemtics o Computtio. 8, [] P. Oumi, D.O. Awoemi, S.N. Jtor d U.W. Sirise, New Lier Multistep Metods wit Cotiuous Coeiciets or First Order Iitil Vlue Problems. Jourl o te Nigeri Mtemticl Societ., 7. [] D.G. Yubu, New Bloc Implicit Ruge-Kutt Colloctio Metods or Solvig Sti Dieretil Equtios. Abcus A [] D.G. Yubu, N.H. Mj, S.S. Bub, A.I. Ms, A Fmil o Uiorml Accurte Order Lobtto-Ruge-Kutt Colloctio Metods. Comput. Appl. Mt Geore Mic Kumleg s obtied PD degree i Numericl lsis. Presetl He lectures mtemtics course i te Uiversit o Jos, Jos Nigeri, PH E-mil: umleg_g@oo.com Josu Pm Collom s obtied PD degree i Numericl lsis. Presetl He is ssocite Proessor o mtemtics i umericl lsis i te Uiversit o Jos, Jos Nigeri, PH-070. E-mil: collomjp@oo.com Smso Omgwu is curretl pursuig PD degree progrm i mtemtics umericl lsis i Uiversit o Jos, Jos Nigeri, PH E-mil: smsoomgwu@oo.com 0 ttp://
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