A 4-Step Implicit Collocation Method for Solution of First and Second Order Odes

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1 Itertiol Jourl o Sciece d Tecolog Volume No. November A -Step Implicit Colloctio Metod or Solutio o First d Secod Order Odes Z A. Adegboe I. Aliu Deprtmet o Mts/Stts/Computer Sciece Kdu Poltecic Kdu.Nigeri. ABSTRACT Te pproc o colloctio metod pproimtio will be dopted i te derivtio o discrete scemes or direct itegrtio secod order ordir dieretil equtio wic re combied togeter to orm block metod. Te metod is eteded to te cse i wic te pproimte solutio to secod order (specil or geerl) s well s irst order Iitil Vlue Problems(IVPs) c be clculted rom te sme cotiuous iterpolt d is o order ive wic is A-stble d s implicit structure or eiciet implemettio. Te metod produces simulteousl pproimtio o te solutio o iitil vlue problems t block o our poits (i=). Numericl results re give to illustrte te perormce metod. i Kewords: Colloctio metod Block metod Cotiuous iterpolt First d Secod Order ODEs. INTRODUCTION I erlier work (Y d Adegboe (7))te utors costructed d implemeted ew Qude s tpe our-step block brid multistep metod or ccurte d eiciet prllel solutio o irst order Ordir Dieretil Equtios(ODEs) Te result coverged better to te ect solutio wit A-stble regio o bsolute stbilit.tese sme utors i (Y d Adegboe (8)) derived mil o -step block metod or specil secod order ODEste eiciec o te metod were tested o bot lier d o-lier specil secod order d te stbilit plot o te block metod ws mde. I Bdmus d Adegboe () te utors obtied two dieret brid block scemes o Qude s tpe rom sigle cotiuous ormultio d umericl eperimets were pplied or te purpose o compriso.also i Y d Adegboe () Reormulted te Qude s Tpe Four-Step Block Hbrid Multistep Metod Ito Ruge-Kutt Metod For Solutio O First Ad Secod Order Odes. Tis pper is prt o reserc eort to reormulte or eiciet d ccurte use o lier multi-step metods. Eorts re directed towrds geertig -step implicit colloctio metod or te solutio o iitil vlue problems o te orm ( ) ( o ) (.) ( ) ( o ) ) (.). ( o ( ) ( o ) ) (.) ( o A umber o umericl metods or tese clsses o problems ve bee etesivel developed. O te cotrr sigle cotiuous iterpolt metod or (..d.) is ot commol discussed. I literture dieret cotiuous iterpolt metods re used or (.) d (..) see Sirise etl () Awoemi () d Adee etl ()Ades etl (8) Kode () to metioed but ew. We cosider te umericl solutio o te IVP or wic te clcultio o te secod derivtive cost little more t irst derivtive loe. Tere re severl iterrelted ims i te serc or suc metod suc s ig order low error costts stisctor stbilit propert suc s A-stbilit low implemettio costs d sel strtig. We prticulrl wis to empsize te combitio o multi-step structure wit te use o o-grid poits we seek metod tt re multistge d multi-vlue becuse it will be coveiet to eted te cotiuous ormultio to te lower order cse b cosiderig polomil. ( ) i- () i ( ) ( ( )) (.) IJST IJST Publictios UK. All rigts reserved.

2 Itertiol Jourl o Sciece d Tecolog (IJST) Volume No. November were t deotes te umber o iterpoltio poit + =--- t-; d m deotes te distict colloctio poits [ +k ] = ---m- cose rom te give step [ +k ]. Here d re smoot rel N-dimesiol vector uctios. Te umericl costt coeiciets ( = ---t-) d ( = ---m-) o (.) re to be determied sice te re selected so tt ccurte pproimtios o well beved problems step size c be costt or cge i te umericl itegrtio process. Deiitio.: For irst order odes o te orm multistep metod is tpicll epressed s d Lier multistep metod (LMM) te lier (.) is sid to be o order P i but d is clled error costt. Deiitio. A itervl o te rel lie is sid to be itervl o bsolute stbilit i te metod is bsolutel stble or Deiitio. A umericl metod (.) is sid to be A stble i its regio o bsolute stbilit cotis te wole o te let d l ple Re Te pper is orgized s ollows i sectio we sow ow te ew metod ws costructed tis leds to sectio were te Implemettio Strtegies is discussed I sectio some umericl test or te ew metods rom wic te coclusio (sectio ) re drw re preseted. re te irst d secod crcteristic polomils respectivel. Tese metods owever ve direct pplictios to secod order dieretil equtio o te orm I literture see Lmbert (99) Ketgle d S (99). compctl Deiitio. A lier multistep metod o te orm (.7) d more (.) (. CONSTRUCTION OF THE PRESENT METHOD I tis sectio we cosider te costructio o multistep Colloctio Metod or costt step size d give recursive epressio or te coeiciets. Te vlues o t d m re rbitrr ecept or Colloctio t te mes poits. Let + be pproimtios to ( + ) were + =( + ) = k-. See Awoemi (99).Speciicll or tis metod we llow t = m = i.e. iterpoltio poits re [ + +]. Wile Colloctio poits re s [ + + +] d we use power series pproimtio s bsis uctio. Te pproimte solutio to equtio () is o te orm (.8) We dieretite equtio () twice to ve: (9) To obti our solutio mtri equtio our iterpoltio d colloctio coditios log wit equtios (.8) d (.9) ields te ollowig sstem o equtios wic we writig i mtri equtio orm we ve IJST IJST Publictios UK. All rigts reserved.

3 Itertiol Jourl o Sciece d Tecolog (IJST) Volume No. November IJST IJST Publictios UK. All rigts reserved. (.) Solvig (.) or =() usig mtri iversio teciques d substitutig teir vlues ito equtio(.8)some lgebric mipultios ields te proposed cotiuous sceme o te orm Hece our cotiuous sceme is Evlutig te cotiuous sceme t d its secod derivtive t ields (.)

4 Itertiol Jourl o Sciece d Tecolog (IJST) Volume No. November 7 7 (.). IMPLEMENTATION STRATEGIES To obti strter we dieretite te Cotiuous Sceme o equtio (.) oce d evlute t Z we = d ( ) (.) () () () () 9 To solve (.) o te sub-itervl. We combie equtios (.) d (.) Solvig te block implicit discerete scemes simulteousl we obtied te ollowig block scemes we d Tlor series epsio re dopted to evlute (.) Were te orm (.) s order T wit error costts simulteousl provides vlue or d. Y 7 T d 9 Y 9 9 Y Y 7 7 (.) Were te orm (.) s order T wit error costts ( ) T We To solve (.) o te sub-itervl. dieretite te Cotiuous Sceme o equtio (.) oce d evlute t d we obtied te ollowig block scemes we wit respect to (.) d simulteousl provides vlue or d.were Y we solvig problem o te orm (.) d Y we solvig problem o te orm (.) ece to solve (.) o te sub-itervl we combie (.) d (.).. NUMERICAL EXPERIMENTS To demostrte te eiciec o te preset metod tree test emples were solved. Emple () [see Y d Bdmus (8)] IJST IJST Publictios UK. All rigts reserved. 7

5 Itertiol Jourl o Sciece d Tecolog (IJST) Volume No. November To stud te eiciec o te metod we preset some umericl emples widel used b severl utors suc s Y d Bdmus () Yusup Y. d Oumi P () Ykub etl(7) d Y d Adegboe (8) teir pproimte solutios were compred wit te teoreticl solutio. Te metod is pplied to solve irst order specil d geerl secod order iitil vlue problems i ordir dieretil equtios t block o our poits directl witout reductio to sstem o irst order. Problem. ( ) () =.. Teoreticl Solutio: ( ) e Problem. ( ) () =.. Teoreticl Solutio: ( ) cos si Problem. ( ) =. Teoreticl Solutio: ( ) e Problem. ( ) () =. Teoreticl Solutio: ( ) Tble : Absolute Errors o Problem. LMM(YAHAYA ()) Preset Metod. 9.E- 8.E+..E-.E-9..E-.E-..E-.E- Tble : Absolute Errors o Problem. NUMEROV (ONUMANYI() Preset Metod..E-7.E-9..E-7.E-9..E-7.E-. 7.E-.E-9 Tble : Absolute Errors o Problem. Ykub(7) Preset Metod. 7.8E-.97E-9..E-.E-8..9E-.E E-.9E-9 Tble : Absolute Errors o Problem. Lmm(Y(8)) Preset Metod..E-.E-7..E-.E-7..E-.E E-.E-7. CONCLUSION Troug te pproc preseted i tis pper we c give te error costts d te cotiuous orm is lso vilble or dese pproimtio to te solutio o irst order specil d geerl secod order ordir dieretil equtios t block o our poits. Te metod requires less work wit ver little cost (we compred wit clssicl d improved RK) d possesses gi i eiciec (we compred wit LMM); te metod is sel strtig wit o overlppig o solutio models. REFERENCES []. Y Y.A. d Adegboe Z.A. (7). A New Qude s Tpe -step block Hbrid Multstep Metod or Accurte d Eiciet Prllel Solutio o Ordir Dieretil Equtios. Abcus No B: []. Y Y. A d Adegboe Z.A (8) A Fmil o -step block Metods or Specil Secod Order IVPs.MAN proc []. Bdmus.A.M d Adegboe Z.A. (). Compriso o Two New Qude s Tpe blocks Hbrid Metod or Solutio o Ordir Dieretil Equtios J. o Reserc i Psicl Sciece No 7-7. []. Y Y.A. d Adegboe Z.A. (). Reormultio o Qude s Tpe Four-step block Hbrid Multstep Metod ito Ruge-Kutt Metod or Solutio o First d Secod Order Ordir Dieretil Equtios. Abcus 8No : -. []. Sirise.U.W.Kumleg G.M.d Y Y. A. (). A New Butcer Tpe Two-Step Block Hbrid Multstep Metod or Accurte d Eiciet Prllel Solutio o Ordir Dieretil Equtios Abcus No A -7. []. Lmbert J D (99). Numericl metod or ordir dieretil sstems. New York. Jo Wile d Sos:9p. IJST IJST Publictios UK. All rigts reserved. 8

6 Itertiol Jourl o Sciece d Tecolog (IJST) Volume No. November [7]. Ketgle R d Edwrd B S (999) Fudmetls o dieretil equtios. Secod editio New York. Te Bemi/Cummigs publisig comp Ic. [8]. Ades O. A Ake A.T d Udo M. O (8) Improved cotiuous metod or direct solutio o geerl secod order ordir dieretil equtios Jourl o te Nigeri ssocitio o Mtemticl Psics. Vol :9- [9]. Awoemi D.O (999) A clss o cotiuous metods or secod order iitil vlue problems i ordir dieretil equtios Itertiol ourl o Computer Mtemtics. Vol 7: 9-7 []. Yusup Y. d Oumi P () New Multiple Fiite Dierece Metod troug Multi Step Colloctio or = () Abcus 9; 9-9. []. Ykub D.G.Mrkus S.Admu M.Y.d Bub S.S (7). A New Fomultio or Smmetric Ruge Kutt Metod Iitil Vlue Problems o Ordir Dieretil Equtios Abcus Vol No A 9-. []. Kode S.J. (). A Improved Numerous Metod or Direct Solutio o Geerl Secod Order Iitil Vlue Problems o Ordir Dieretil Equtios Semir Nt mt ceter Abu Nigeri 9- []. Y Y.A d Bdmus A.M (). A Clss O Colloctio Metods For Geerl Secod Order Ordir Dieretil Equtios Aric Jourl o Mtemtics d Computer Sciece Reserc () 9-7. IJST IJST Publictios UK. All rigts reserved. 9