Babatola, P.O Mathematical Sciences Department Federal University of Technology, PMB 704 Akure, Ondo State, Nigeria.

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1 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK NUMEICAL SOLUION OF ODINAY DIFFEENIAL EQUAIONUSING WO- SAGE SEMI-IMPLICI YBID UNGE-KUA SCEME Babatola, P.O Matematcal Sceces Departmet Feeral Uverst o ecolog, PMB 704 Akure, Oo State, Ngera. Abstract: I ts paper, aml o two-stage sem-implct br uge-kutta scemes were evelope, aale a computere to solve orar eretal equatos.er evelopmet a aalss make use o alor seres epaso, Daqust stablt test moel equato respectvel. e teoretcal results sow tat te scemes are Cosstet, Coverget a A-stable wt large terval o absolute stablt. -, 0. e results o ts sceme are compare wt result o classcal uge-kutta a atoal uge-kutta. e umercal results obtae corme tat te scemes are accurate. Kewors: A-stable, Accurate, Sem-Implct, br, Local-trucato Error..0 Itroucto Orar eretal equatos ODEs arse ma eret cotet clug geometr, mecacs, astroom, populato moelg a ma oters. ese ote lea to a ODEs o te orm,, o. o esearcers ave cotrbute to te evelopmet o te scemes tat solves tese ODEs. e cosequeces o tese cotrbuto leas to a troucto o te atoal Oe-step meto o te orm. calle Iverse Euler s sceme b Fatula 98. s prompte og-yuau 98 to trouce atoal uge-kutta Sceme o te geeral rom W K. were, 45

2 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK K K, C, aj K j.4 j g, g, aj j.5 j g,,.6 Altoug te meto s sutable, accurate a stable. It s beevle b te cult ature o te ucto evaluato o a g, te meto wc we are coserg ts works s o te orm.7 were, g, bj j.8 j Wt, g a Z.9 Wt te costrats b j j.0 a t s calle br -stage uge-kutta sceme. s meto was classe to Eplct, Sem- Implct a Implct. Babatola 00 propose te oe-stage sem-mplct sceme. I ts paper, we coser sem-mplct o two-stage or umercal soluto o ODEs. 46

3 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK 0 Developmet o te New Scemes Now settg equato.7, te two-stage sem Implct br -K sceme s o te orm were, wt., b g, b b g., g. Wt costrats a b b.4 Oe-step applcato o te sceme. wll geerate [ ].5 e parameters,,,, b, b, b are to be eterme rom te sstem o o-lear equatos geerate b aoptg te steps below: Step : ake te alor seres epaso o a,, about pot, Step : Isert te results o te epaso step to.5. Step : earrage te al epaso about te power seres o, so tat ca be epresse te orm p p P p ao a a... a p a p 0 p te orer o accurac o te sceme ca be ete. e sceme. s sa to be or orer PLambert 97, 997. I c p 0, p 0p a C p 0, P, P.7 us, to obta te values o te above coecet, we solve te set o o-lear sstem o equato.7 or p, P wt C p 0. So as to esure tat te values o te parameter el computatoal meto tat ave: 47

4 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK mmum bou o local trucato error alsto 96 mamum attaable orer o accurac Kg 966. Mmum computer storage space Gll 95. v Large terval o absolute stablt. Epag, a alor seres to get a trucato error spece as 4 4! D!! D D 5 D D D D D 0.8 Were, D D D D.9 N M 4 D were, N, M D D D D 6. Smlarl, te alor seres epaso o s N M 4 D 0 5. were 48

5 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK 49 N M, D D D D 6. Substtute equatos.8,.0 a. to.5, els a a a a a o.4 were a o 0, a D a 6 6 D D D D a D D D D D D a Imposg accurac o orer o te a o a a a 0, but a 4 0 te ollowg sstem o o-lear equatos or aml o sem-mplct br uge-kutta o orer tree. ½ b b b /6

6 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK Wt te costrats b 0 b b.7 For te parameters b, b,,,, Solvg tese equatos we obta Case : ¾, ¼, b /, b b ½,.8 4 Were, g, Case :, g.9, 6,, b b, b 6 els aoter -stage sceme o orer..0 Were, g, [ g 6,. 6 wo stage sem-implct br uge-kutta o orer our are obtae rom solvg sstem o o-lear equatos. a. 50

7 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK b b b b b b 6. Subject to te ollowg costrats b b b. to obta Case : 0., b,, b b 4 4 e,.4 were, g, 4 4, 4 g.5 Case : I 4, 4,, b b 6, b els.6 4 were 4,, g g Aalss o te Basc Propertes 5

8 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK 5 e basc propertes requre o a goo computatoal meto clue, accurac, cosstec, covergece a stablt.. Cosstec A sceme s sa to be cosstet te erece equato o te computato ormulas eactl appromate te eretal equato t te to solve Aemlu a Babatola 00. o prove tat equato. s cosstet. ecall tat. Subtract rom bot ses o equato. a urter smplcato we get., But j j j b g. ece,,, j j j j j b g b g.4 Dvg troug b a takg lmt as tes to ero. 0 lm, g.5 But,, g.6 ece, 0, lm.7

9 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK 5 ',.8 ece te meto s cosstet.. Covergece A umercal sceme suc as equato. s sa to be coverget, we apple to tal value problem., t geerate a correspog appromato, wc tes to te eact soluto as approaces t. o sow tat equato. s coverget. ecall tat.9 Wle te eact sats erece equato.9 as.0 Subtractg equato.9 rom.0. [ [ ] e. e.

10 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK Settg P Q e equato. becomes e. E ma P Q 0 Let P ma P, Q ma Q 0, e.4 0 ma, ma e E 0 0 E.5 PQ Set K PQ e equato.5 becomes E KE.6 E KE o.6a E KE.6b Substtute.6a to equato.6b E K KEo K E o K 54

11 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK E K Eo K ereore K E K Eo.7 Sce PQ K < It s eas to see tat as, E 0. s sows tat ts partcular case, te scemes coverges.. Stablt propertes Sce a Cosstet a Coverget oe-step sceme s stable, te sceme s stable. owever, to esure tat te meto s A-stable a P-stable a able to solve st tal value problem. It s aopte or soluto o te A-stablt moel test equato λ.8, o o Applg te sceme to te tal value problem.8, a recurret equato s obtae as: were P P wc s. Pae s appromato to e sce t ca be epresse as P e sceme s A-stable sce [, 0] a P-stable sce [, ] 4. Numercal Epermet I orer to corm te applcablt a sutablt o te sceme or soluto o ODEs, some sample problems were cosere, o 4. wt eact soluto e 4. wt 0..e results are sow able I., o 4. wt eact soluto e

12 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK wt 0.05.e results are sow able. 0 o 4.5 wt te aaltcal soluto gve as e results o tese attempts wt 0.0 are sow able. 0 e

13 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK ABLE : NUMEICAL SOLUION OF POBLEM 0 WI SEMI-IMPLICI YBID UNGE-KUA AND SEMI IMPLICI CLASSICAL UNGE KUA SCEME ALUES OF X YEXAC SEMI-IMPLICI SEMI IMPLICI EO OF EO OF YBID -K CLASSICAL -K EXPLICI CLASSICIAL - ALUE OF Y YBID -K K E E c D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D-0 57

14 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D-0 58

15 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK D D D D D D-0 59

16 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK ABLE : NUMEICAL SOLUION OF SEMI-IMPLICI YBID UNGE-KUA OF POBLEM 4. j Yeact Error

17 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK NUMEICAL SOLUION OF POBLEM 0 WI SEMI-IMPLICI YBID UNGE-KUA SCEME AND SEMI IMPLICI CLASSICAL UNGE KUA SCEME ALUES OF X YEXAC SEMI-IMPLICI SEMI IMPLICI EO OF SEMI- EO OF YBID -K CLASSICAL -K IMPLICI CLASSICIAL - ALUE OF Y YBID -K K E E c D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D-0 6

18 Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK Cocluso I all cases, te compute errors sow tat te scemes are ver accurate, stable a coverget. As ca be see rom te result tables, tese scemes compare avourabl wt te estg uge- Kutta scemes o te same orer. EFEENCES Aemlu,.A. & Babatola P.O. 00, Sem-Implct atoal uge Kutta ormular or appromato o St Ital alues problem ODEs, Joural o Mat Scece a Eucato ol., pg. 5. Babatola P.O. 00, A ew Iverse uge- Kutta Sceme or St ODEs Joural o NAMP, ol. 6 pg Fatula, S. O. 980, Numercal Itegrator or St a gl Oscllator problem Deretal Equato. Mats Computato ol. 4 pg Gll, S. 95, A process or step b step tegrato o Deretal Equatos, a Automatc Dgtal Computg Mace Proc. Cambrge Plos Soc. ol. 47, Pg og Yuau 980, A class o A-stable or Aα stable Eplct Scemes Computatoal a Asmptotc Metos or Bouar a Iteror Laer, Proceeg o BAILJI Coereces rt College, Dub Pg Kg,. 966, uge-kutta Meto wt Costrae Mmum error Bou Mats Comp. ol 0, Pg alsto, A. 96, uge-kutta wt mmum error bous, Mat Comp. ol. 6 pg Correspog autor s emal aress: pobabatola@aoo.com 6