A Self-Starting Hybrid Linear Multistep Method for a Direct Solution of the General Second-Order Initial Value Problem
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1 IOS Joural o Matematics (IOS-JM) ISSN: Volume 4 Issue 6 (Ja. - eb. ) PP 7- Sel-Startig Hbrid Liear Multistep Metod or a Direct Solutio o te Geeral Secod-Order Iitial Value Problem damu lai Moammed mia Hamza ad Umaru Moammed (Departmet o Matematics ad Statistics ederal Uiersit o ecolog Mia Nigeria.) (Departmet o Matematics ad Statistics Uiersit o Maiduguri Maiduguri Nigeria.) bstract : sel- startig brid liear multistep metod or direct solutio o te geeral secod-order iitial alue problem is cosidered. e cotiuous metod is used to obtai Multiple iite Dierece Metods (MDMs) (eac o order 7) wic are combied as simultaeous umerical itegrators to proide a direct solutio to IVPs oer sub-iterals wic do ot oerlap. e coergece o te MDMs is discussed b coeietl represetig te MDMs as a bloc metod ad eriig tat te bloc metod is zero-stable ad cosistet. e superiorit o te MDMs oer publised wor is establised umericall. Kewords: Multiple iite Dierece Metods Secod Order Boudar Value Problem Bloc Metods Multistep Metods I. Itroductio e matematical ormulatio o psical peomea i sciece ad egieerig ote leads to iitial alue problems o te orm: a a () Howeer ol a limited umber o aaltical metods are aailable or solig () directl witout reducig to a irst order sstem o iitial alue problems. Some autors ae proposed solutio to iger order iitial alue problems o ordiar dieretial euatios usig dieret approaces [-5]. I particular woemi ad Idowu [] deeloped a class o brid collocatio metod or tird order ordiar dieretial euatios. woemi[] deried a p-stable liear multistep metod or geeral tird order iitial alue problems o ordiar dieretial euatios wic is to be used i orm o predictor-corrector orms ad lie most liear multistep metods te reuire startig alues rom uge-kutta metods or a oter oe-step metods. e predictors are also deeloped i te same wa as correctors. Moreoer te bloc metods i atula [] are discrete ad are proposed or o-sti special secod order ordiar dieretial euatios i orm o a predictor- corrector itegratio process. lso lie oter liear multistep metods te are usuall applied to te iitial alue problems as a sigle ormula but te are ot sel-startig; ad te adace te umerical itegratio o te ordiar dieretial euatios i oe-step at a time wic leads to oerlappig o te piecewise polomials solutio Model. ere is te eed to deelop a metod wic is sel-startig elimiatig te use o predictors wit better accurac ad eiciec. is stud tereore propose a bloc brid multistep metod or te direct solutio o tird order iitial alue problems o ordiar dieretial euatios. ecetl seeral researces [6-] proposed LMMs or te direct solutio o te geeral secod ad tird order IVPs wic were sowed to be zero stable ad were implemeted witout te eed or eiter predictors or startig alues rom oter metods. Jator [] used te LMMs deeloped or IVPs ad additioal metods obtaied rom te same cotiuous -step LMM to sole tird order BVPs wit Diriclet ad Neuma boudar coditios ad also aaa ad Moammed [] deeloped a 5-step bloc metod or special secod order ordiar dieretial euatios. We eteded teir metods ito brid orm b addig oe o-step poit at collocatio. e 5-step bloc brid metod is zero-stable cosistet ad coerget. II. Deelopmet O Metods. I tis sectio our obectie is to derie brid liear multi-step metod (HLMM) o te orm r s Were ad are uow costats ad is ot a iteger. We ote tat = () ad do ot bot ais. I order to obtai () we proceed b seeig to approimate te eact solutio () o te orm 7 Page
2 Sel-Startig Hbrid Liear Multistep Metod or a Direct Solutio o te Geeral Secod-Orde r s l a b l are uow coeiciets to be determied ad r s are te umber o Were iterpolatio ad collocatio poits respectiel. We te costruct our cotiuous approimatio b imposig te ollowig coditios.... r (5) Euatio (4) ad (5) lead to a sstem o (rs) euatios wic is soled b ramer s rule to obtai cotiuous approimatio is costructed b substitutig te alues o maipulatio te cotiuous metod is epressed as r s Were ad () (4) l. Our l ito euatio (). ter some are cotiuous coeiciets. We ote tat sice euatio () ioles irst ad secod deriaties te irst deriatie ormula r s (7) Euatio (7) is easil obtaied rom (6) ad is te used to proide te irst ad secod deriaties or te metods b imposig te coditio z a z III. Speciicatio O e Metods Our metods are obtaied rom sectio two ad epressed i te orm o () gie b r s () i r s 7 5 i... wit te ollowig speciicatio i cotiuous orm as ollows: () = 4 (8) (9) (6) we also epress te : () 8 Page
3 Sel-Startig Hbrid Liear Multistep Metod or a Direct Solutio o te Geeral Secod-Orde e MDMs are obtaied b ealuatig () at to obtai te ollowig = () = () = (4) = (5) = (6) I particular to start te iitial alue problem or = we obtai te ollowig euatios rom (9): z = (7) It is wort otig tat te deriaties are proided as ollows: z = z = z = z 4 = z 9 = z 5 = Page
4 Sel-Startig Hbrid Liear Multistep Metod or a Direct Solutio o te Geeral Secod-Orde Page IV. alsis d Implemetatio O e Metod ollowig atula [] ad Lambert [4] we deie te local trucatio error associated wit te coetioal orm o () to be te liear dierece operator L ; (8) ssumig tat () is suicietl dieretiable we ca epad te terms i (8) as a alor series about te poit to obtai te epressio ; L (9) Were te costat coeiciets... are gie as ollows:....!... ccordig to Herici [4] we sa tat te metod (5) as order p i... P P P Our calculatios reeal tat te metods () to (6) ae order p = 7 ad error costats gie b te ector I order to aalze te metods or zero-stabilit we ormalize () to (7) ad write tem as a bloc metod gie b te matri dierece euatio B B () Were ad ad matrices ad are deied as ollows: is a idetit matri o dimesio 6 It is wort otig tat zero-stabilit is cocered wit te stabilit o te dierece sstem i te limit as teds to zero. us as te metod () teds to te dierece sstem Wose irst caracteristic polomial is gie b det 5 () ollowig atula [] te bloc metod () is zero-stable sice rom () Satis... ad or tose roots wit = te multiplicit does ot eceed. e bloc metod () is cosistet as it as order P. ccordig to Herici [4] we ca sael assert te coergece o te bloc metod ().
5 Sel-Startig Hbrid Liear Multistep Metod or a Direct Solutio o te Geeral Secod-Orde It is ital to ote tat te mai metod gie b () ca be used as a umerical itegrator directl ad sigl i te coetioal wa o oerlappig sub-iterals. Howeer our metod is implemeted more eicietl b combiig metods () to (6) eac o order see wit relatiel small error costats as simultaeous itegrators or IVPs witout looig or a oter metods to proide te startig alues. We proceed b eplicitl obtaiig iitial coditios at 5... N 5 usig te computed alues N 5 5 z5 z5 oer sub-iterals istace are simultaeousl obtaied oer te sub-iteral 4 5 ow rom te IVP or wic do ot oerlap (see []). or 5 as is are simultaeousl obtaied oer te sub-iteral 5 as 5 is ow rom te preious bloc ad so o. Hece te sub-iterals do ot oer-lap ad te solutios obtaied i tis maer are more accurate tat tose obtaied i te coetioal wa. V. Numerical Eamples I tis sectio we ae tested te perormace o te metod o tree problems b cosiderig oliear IVPs (Eamples 4.) liear o-omogeeous ODE (Eample 4.) mildl sti problem (Eample 4.). or eac eample we id absolute errors o te approimate solutio. Eample 4. We cosider te euatio Eact Solutio : l It is obious tat our metod perorms better ta tose gie i woemi [4] despite te act tat we used a larger step size =.5. Hece or tis eample our metod is clearl superior. e details o te umerical results at some selected poits are gie i able 4. able 4. woemi [] Order p = 6 =.5 woemi ad Kaode[] Order p = 8 Jator[6] Order p = 6 =.5 Our Metods Order p=7 =.5 = Eample 4. We cosider te o-omogeeous ODE gie b Eact Solutio : e cos si ltoug te umerical results o tis problem were ot compared wit aoter metod te results were compared wit te teoretical solutio as sow i able Page
6 Sel-Startig Hbrid Liear Multistep Metod or a Direct Solutio o te Geeral Secod-Orde Eact solutio () able 4. Numerical Error b Jator [7] solutio () Preset Error E E E E E-7 Eample 4.. We cosider te mildl sti IVP Eact Solutio : e ltoug te umerical results or tis problem were ot compared wit aoter metod te results were compared wit te teoretical solutio as sow i able 4.. able 4. ()-Eact -Numerical Error VI. oclusio We ae deried a ie-step cotiuous HLMM rom wic MDMs are obtaied ad applied to sole witout irst adaptig te ODE to a euialet irst order sstem or reducig it to a iitialalue problem. e MDMs are applied as simultaeous umerical itegrators oer sub-iterals wic do ot oerlap ad ece te are more accurate ta SDMs wic are geerall applied as sigle ormulas oer oerlappig iterals. We ae sow tat te metods are zero stable coerget ad wic mae tem suitable cadidates or computig solutios o wider iterals. I additio to proidig additioal metods ad deriaties te cotiuous HLMM ca be used to obtai global error estimates. Our uture researc will be ocused o adaptig te MDMs to sole tird order partial dieretial euatios. eereces [] woemi D.O.. P-stable liear multistep metod or solig geeral tird order ordiar dieretial euatios. It. J. omput Mat. 8: DOI:.8/ [] woemi D. ad IdowuO. 5. class brid collocatio metods or tird order o ordiar dieretial euatios It. J. omput. Mat. 8: DOI:.8/7659. [] atula S.O. (994). class o bloc metods or secod order IVPs. It. J. omput. Mat. 55: 9-. DOI:.8/ [4] Lambert J.D. (97). omputatioal Metods i Ordiar Dieretial Euatios (Jo Wille ad Sos New or US. ISBN: : p: 94.) [5] dee S.O. Oumai P. SiriseaU.W. ad aaa. (5). Note o startig umero metod more accuratel b a brid ormula o order our or a iitial alue problem. J.omputat. pplied Mat. 75: DOI:.6/.cam [6] Jator S.N(7). sit order liear multistep metod or te direct solutio o = ( ) Iteratioal Joural o Pure ad pplied Matematics 4 No [7] Jator S.N ad Li J (7) sel-startig liear multistep metod or a direct solutio o te geeral secod order iitial alue problem Iteratioal Joural o omputer Matematics Vol. 86 No. 5 Ma [8] Jator S.N (8) Multiple iite dierece metods or solig tird order ordiar dieretial euatios Iteratioal Joural o Pure ad pplied Matematics 4 No Page
7 Sel-Startig Hbrid Liear Multistep Metod or a Direct Solutio o te Geeral Secod-Orde [9] Moammmed U.Jia M ad Moammed.(). class o si step bloc metod or solutio o geeral secod order ordiar dieretial euatios Paciic Joural o Sciece ad ecolog. ():pp7-77. [] Moammmed U (). class o implicit ie step bloc metod or geeral secod order ordiar dieretial euatios. Joural o Nigeria Matematical Societ (JNMS). ol p 5-9 [] Jator S.N (8). O te umerical itegratio o tird order boudar alue problems b a liear multistep metod Iteratioal Joural o Pure ad pplied Matematics 46 No [] aaa ad Moammed U (). 5-step bloc metod or special secod order ordiar dieretial euatios Joural o Nigeria Matematical Societ (JNMS). ol p -6 [] atula S.O (99): Bloc Metod or Secod Order Iitial Value Problem (IVP) Iteratioal Joural o omputer Matematics Vol. 4:55-6. Eglad. [4] Herici P. (96); Discrete ariable metods or ODE`s New or US Jo Wile ad sos. Page
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