Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 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
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 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
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org. 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
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 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 0 5.0 were, N, M D D D D 6. Smlarl, te alor seres epaso o s N M 4 D 0 5. were 48
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 49 N M, D D D D 6. Substtute equatos.8,.0 a. to.5, els 5 4 4 0 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 6 6 48 4.5 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
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org.6 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
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 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.7 6.0 Aalss o te Basc Propertes 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 www.ea-jourals.org 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
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 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.
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 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
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 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.9 9.0 wc s. Pae s appromato to e sce t ca be epresse as P 7 5 6 5 4 0. 8 5 6 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 4.4 55
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 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 4.6 56
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 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 0.5000000D-0 0.059460D0 0.059850D0 0.06440D0 0.9577480D-04 0.989740D-0 0.50000000D-0 0.0580D0 0.058790D0 0.05440D0 0.6508870D-04 0.4986870D-0 0.75000000D-0 0.08650D0 0.08780D0 0.08409990D0 0.75060D-04 0.4460040D-0 0.0000000D00 0.550D0 0.5580D0 0.59590D0 0.69080D-04 0.446960D-0 0.500000D00 0.494450D0 0.49490D0 0.498750D0 0.4604790D-04 0.4998790D-0 0.5000000D00 0.85500D0 0.855080D0 0.858980D0 0.50067900D-05 0.955640D-0 0.7500000D00 0.790D0 0.6840D0 0.4080D0 0.54400D-04 0.47670D-0 0.0000000D00 0.64080D0 0.640760D0 0.644790D0 0.67990D-0 0.709670D-0 0.500000D00 0.069680D0 0.06770D0 0.07470D0 0.6600D-0 0.7890D-0 0.7500000D00 0.50760D0 0.5760D0 0.540D0 0.4999850D-0 0.6667990D-04 0.0000000D00 0.99590D0 0.9900 D0 0.99550D0 0.4908860D-0 0.67480D-04 0.500000D00 0.4495770D0 0.448940D0 0.44950D0 0.650850D-0 0.75580D-0 0.50000D00 0.50090D0 0.50540D0 0.506900D0 0.876870D-0 0.4040D-0 0.7000000D00 0.55700D0 0.556560D0 0.5565980D0 0.047050D-0 0.6057040D-0 0.4000000D00 0.649740D0 0.66950D0 0.6440D0 0.79470D-0 0.84850D-0 0.450000D00 0.6754740D0 0.67970D0 0.674900D0 0.57040D-0 0.0844470D-0 57
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 0.4500000D00 0.78770D0 0.769500D0 0.774090D0 0.8040D-0 0.6900D-0 0.4700000D00 0.804970D0 0.808050D0 0.80690D0 0.460D-0 0.66850D-0 0.4750000D00 0.874040D0 0.875740D0 0.94800D0 0.46900D-0 0.000980D-0 0.5000000D00 0.946640D0 0.9490D0 0.08640D0 0.85740D-0 0.60D-0 0.5500000D00 0.0770D0 0.08470D0 0.08640D0 0.00950D-0 0.75540D-0 0.55000000D00 0.0997590D0 0.09604D0 0.0965860D0 0.655950D-0 0.750D-0 0.57500000D00 0.890D0 0.77800D0 0.777660D0 0.4050D-0 0.64960D-0 0.60000000D00 0.66560D0 0.67570D0 0.6470D0 0.4599570D-0 0.40960D-0 0.6499990D00 0.54780D0 0.49670D0 0.500D0 0.50560D-0 0.466750D-0 0.64999980D00 0.44660D0 0.4409470D0 0.444440D0 0.56755540D-0 0.578450D-0 0.67499998D00 0.540990 0 0.5580D0 0.5640D0 0.66560D-0 0.5764960D-0 0.69999990D00 0.64580D0 0.64670D0 0.648700D0 0.68950D-0 0.688870D-0 0.7499980D00 0.74490D0 0.76680D0 0.648700D0 0.75547700D-0 0.70488450D-0 0.77499980D00 0.8509990D0 0.84740D0 0.77440D0 0.8559590D-0 0.77476500D-0 0.7999980D00 0.967750D0 0.957790D0 0.84890D0 0.8996750D-0 0.848670D-0 0.849997D00 0.07660D0 0.066840D0 0.95890D0 0.97789760D-0 0.96650D-0 0.84999970D00 0.95640D0 0.847680D0 0.067550D0 0.0870D-0 0.0088680D-0 0.87499970D00 0.8990D0 0.06940D0 0.8550D0 0.0560D-0 0.40D-0 0.89999970D00 0.446640D0 0.4880D0 0.077070D0 0.680D-0 0.44960D-0 0.9499970D00 0.5788080D0 0.564990D0 0.44890D0 0.4508490D-0 699050D-0 0.94999960D00 0.85770D0 0.6997600D0 0.565090D0 0.584960D-0 0.50540D-0 0.97499960D00 0.4005000D0 0.898860D0 0.7005770D0 0.740760D-0 0.646790D-0 0.9999960D00 0.454840D0 0.9847940D0 0.840700D0 0.8705840D-0 0.787590D-0 0.050000D0 0.4840D0 0.446050D0 0.985650D0 0.08400D-0 0.940070D-0 58
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 0.055000D0 0.4840D0 0.489440D0 0.490870D0 0.84950D-0 0.099750D-0 59
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org ABLE : NUMEICAL SOLUION OF SEMI-IMPLICI YBID UNGE-KUA OF POBLEM 4. j Yeact Error 0.0500.0046..00056 0.00087 0.05000.008657.00065 0.0040 0.07500.004880.00407 0.0008 4 0.050000.0005.0050 0.000607 5 0.06500.007.0099 0.0008 6 0.075000.004979.0056409 0.0085 7 0.087500.005605.0076857 0.00707 8 0.00000.0068495.00050 0.004448 9 0.500.007475.0767 0.0065 0 0.5000.0080975.057477 0.008898 0.7500.00877.09086 0.6 0.50000.0099708.07550 0.04657 0.6500.00946.067580 0.0806 4 0.75000.005956.00987 0.075 5 0.87500.08458.05785 0.058 6 0.00000.047.040808 0.005 7 0.500.00968.0469 0.0497 8 0.500.075.05984 0.005 9 0.7500.04485.058074 0.045556 0 0.50000.049747.644945 0.0598 0.6500.05600.0758 0.0576 0.75000.0676.078558 0.0640 0.87500.068544.086684 0.0794 4 0.00000.0748.09474 0.07857 5 0.500..09085.0586 0.08656 6 0.5000.08758.4050 0.09455 7 0.7500.068544.0647 0.066 8 0.50000.08085.09 0. 9 0.6500.08758.40409 0.695 0 0.75000.096.50999 0.60 60
Iteratoal Joural o Matematcs a Statstcs Stues ol., No., Marc 0, pp.45-6 Publse b Europea Cetre or esearc, rag a Developmet, UK www.ea-jourals.org 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 0.0000000D-0 0.9048840D00 0.0400D0 0.070980D0 0.9960D00 0.685960D00 0.99999999D-0 0.6887940D00 0.4498900D00 0.47690D00 0.800850D-0 0.6884480D-0 0.0000000D00 0.450D00 0.7640D00 0.680D00 0.97900D-0 0.479780D-0 0.0000000D00 0.76787060D-0 0.877070D-0 0.8507890D-0 0.090D-0 0.840850D-0 0.9999990D00 0.8560D-0 0.860000D-0 0.845040D-0 0.984590D-0 0.84780D-0 0.59999970D00 0.847850D00 0.8990D00 0.54400D00 0.46066940D-0 0.08680D-0 0.69999960D00 0.490D00 0.4400960D00 0.98400D00 0.9888060D-04 0.467470D-0 0.79999950D00 0.5460D00 0.58950D00 0.50609780D00 0.45070D-04 0.66790D-0 0.89999940D00 0.7900D00 0.790460D00 0.790D00 0.074670D-0 0.7987900D-0 0.0099990D0 0.00400D0 0.00980D0 0.0090D0 0.4590D-0 0.0570D-0 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 www.ea-jourals.org 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. 0 40. Fatula, S. O. 980, Numercal Itegrator or St a gl Oscllator problem Deretal Equato. Mats Computato ol. 4 pg. 74-90. 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. 95 08. 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. 6 4. Kg,. 966, uge-kutta Meto wt Costrae Mmum error Bou Mats Comp. ol 0, Pg 86-9. alsto, A. 96, uge-kutta wt mmum error bous, Mat Comp. ol. 6 pg. 4-48. Correspog autor s emal aress: pobabatola@aoo.com 6