An Improved Self-Starting Implicit Hybrid Method
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1 IOSR Joural o Matematics (IOSR-JM e-issn: 78-78, p-issn:9-76x. Volume 0, Issue Ver. II (Mar-Apr. 04, PP A Improved Sel-Startig Implicit Hbrid Metod E. O. Adeea Departmet o Matematics/Statistics, Federal Uiversit Wukari, Taraba State, Nigeria. Abstract: Te paper presets te derivatio a brid block metod or te solutios o Iitial Value Problems o Ordiar Dieretial Equatios. Tis is acieved b usig collocatio ad iterpolatio tecique to cotruct a sel-startig metod wit cotiuous coeiciets togeter wit te additioal metods rom its irst derivative wic are combied to orm a sigle block tat simultaeousl provide te approximate solutios or Iitial Value Problem. Te aalsis o te properties o te metod suc as stabilit, cosistec ad covergece are discussed ad te perormace o te metod is demostrated o two test problems to sow te accurac ad eiciec o te metod. O compariso o te results obtaied rom umerical examples wit some existig metods, we discovered tat te metod beaved avourabl well. Kewords: Cebsev polomial, Collocatio, Hbrid, Iterpolatio. Te class o problems togeter wit iitial coditios m ( x t t ( x 0 0 I. Itroductio ( x,,,,..., were t = 0, ad as received muc attetio b researcers. A lot o researc ave bee carried out to solve iger order ordiar dieretial equatios (ODEs umericall, but b irst reducig it to a sstem o irst order ODEs ad te solve b a metod desiged or a irst order ODEs. Te deiciecies o tis approac, wic iclude time wastage, computatioal burde ad ig cost o implemetatio are extesivel discussed i te literature. Various umerical scemes or solvig dieretial equatios exist i literature. Amog tese are te Ruge-kutta, Talors algoritm ad te Liear Multistep Metods (LMMs. Owig to te suitabilit o LMMs i providig solutios to ODEs, ma scolars ave developed LMMs varig rom discrete to cotiuous or te solutio o IVPs. However, it as bee reported tat cotiuous LMMs as greater advatages over te discrete metod i tat te give better error estimatio ad guaratee eas appropriatio o solutio at all iterior poits o te itegratio iterval. [] ad [] cosidered LMMs were i [], LMMs were proposed ad implemeted i a predictor-corrector sceme usig te Talor series algoritm to suppl te startig values. Altoug, te implemetatio o te metods ielded good accurac but te procedure is more costl to implemet. Ma researcers ave attempted te solutio o tis kid o problem usig LMMs witout reductio to sstem o irst order ODEs, (see [], [4], [], [6]. Covetioall, implicit LMMs, we implemeted i te predictor-corrector mode is proe to error propagatio. Tis disadvatage as led to te developmet o block metods rom liear multistep metods. Apart rom beig sel-startig, te metod does ot require te developmet o te predictors separatel, ad evaluates ewer uctios per step. Due to te elegat propert o Cebsev polomial suc as equi-oscillatio i its etire rage o deiitio ad cosequet eve distributio o error terei we sall emplo it to develop a class o iite dierece metod wic is sel startig. Its mii-max propert also make it desirable. II. Material ad Metods I tis sectio, we set out to derive te proposed cotiuous brid oe step block metod b approximatig te aaltical solutio o 0 0 m ( x,,, ( a z, ( a ( were is a cotiuous uctio, wit a Cebsev polomial i te orm ( rs ( x a T ( x ( 0 o te partitio a = x 0 < x <... < x < x + <... < x N = b o te itegratio iterval [a, b], wit a costat step size, give b = x + - x ; = 0,,..., N -. Te secod derivative o ( is give b 8 Page
2 A Improved Sel-Startig Implicit Hbrid Metod ( rs ( ( 0 x a T x ( were x[a, b], te a s are real ukow parameters to be determied ad r + s is te sum o te umber o collocatio ad iterpolatio poits. We sall iterpolate at at least two poits to be able to approximate ( ad or tis purpose, we proceed b selectig some ostep poits i suc a maer tat te zero-stabilit o te mai metod is guarateed. Te ( is iterpolated at x +s ad its secod derivative is collocated at x +r so as to obtai a sstem o equatios wic will be solved b Gaussia elimiatio metod. Te resultig a s are substituted ito ( to ield te ew cotiuous metod ater some maipulatios. Evaluatig te cotiuous metod at te desired poit gives te mai metod. Oter metods eeded to couple wit te mai metod are to be derived rom te cotiuous metod ad te solve simultaeousl to orm te block metod.. Derivatio o Two Ostep Poits Here, two ostep poits (TOP are itroduced. Tese two poits are careull selected to be ad were te collocatio poit, r = 4 ad te iterpolatio poit, s =. Applig tese i (, we ave ( x a T ( x (4 0 wit its secod derivative give b ( x a T ( x ( 0 Substitutig ( ito ( gives ( x a T ( x ( x,, (6 0 Collocatig (6 at x = x +r, r = 0,,, ad iterpolatig (4 at x = x +s, s = ad equatios writte i matrix orm AX = B as ollows: a a a a a a Solvig (7 b Gaussia elimiatio metod ields te a s as ollows: (7 lead to a sstem o 9 Page
3 A Improved Sel-Startig Implicit Hbrid Metod a0 ( a ( a ( a ( a4 ( a ( 0480 (8 Substitutig te a s, = 0(Error! Reerece source ot oud. ito (4 ields te cotiuous brid oe step metod i te orm o a cotiuous liear multistep metod described b te ormula ( x ( (9 0 were α s ad β s are cotiuous uctios ad are obtaied as parameters ( t t ( t t 4 0( t ( 7t 70t 90t 40t 4t ( t t t t t t ( t t t t t t ( t ( 7t 70t 90t 40t 4t 460 (0 were t = ad v = x - x. Evaluatig (9 at x = x ad x +, we obtai te discrete metods rom (0 as ollows: ( 0 ( 08 ad ( 0 ( 08 Te block metods are derived b evaluatig te irst derivative o (9 i order to obtai additioal equatios eeded to couple wit ( ad (. Dieretiatig (9, we obtai 60 Page
4 A Improved Sel-Startig Implicit Hbrid Metod ( x ( ( ( were ( t ( t ( t t t t t ( t t t t t ( t t t t t ( t t t t t Evaluatig (4 at x = x, x +, x + ad x +, te ollowig discrete derivative scemes are obtaied ( ( ( ( ( (4 Equatios (, ( ad ( are combied ad solved simultaeousl to obtai te ollowig explicit results. ( 7 ( 9 ( 8 (8 40 ( 40 ( Equatio (6 recovers te dicrete couterpart we power series was used as basis uctio i [8]. (6 6 Page
5 A Improved Sel-Startig Implicit Hbrid Metod III. Numerical Examples We cosider ere two test problems or te eiciec ad accurac o te metod implemeted as a block metod. Te absolute errors o te test problems are compared wit our earlier work. 4. Problems Problem, (0 0, (0 Exact Solutio : ( x e Source : Yaaa ad Badmus (009. x Probem, (0 0, (0 x e e Exact Solutio : ( x Source : Adeii et al (008. x IV. Results TABLE a: Sowig te exact solutios ad te computed results or problem X Exact Solutio Two Ostep Poits TABLE b: Compariso o absolute errors or Problem X Error i [7], p=4, k= Error i TOP, p=4, k= E e E e E E e E e E e E e E e E e E e-009 TABLE a: Sowig te exact solutios ad te computed results or problem X Ostep Exact TABLE b: Compariso o absolute errors or Problem X Error i TOP Error i [9] e e e e e e e e- 6 Page
6 A Improved Sel-Startig Implicit Hbrid Metod V. Coclusio Tis paper as demostrated te derivatio o cotiuous two-ostep brid metod or te direct itegratio o secod order ordiar dieretial equatios. It as bee observed troug compariso o te solutios o te selected test problems wit solutios obtaied i our earlier paper, [0] tat icrease i te umber o ostep poits leads to icrease i te eiciec ad accurac o te metod. Moreover, te desirable propert o a umerical solutio is to beave like te teoretical solutio o te problem as tis is vivid i te Tables sow above. I te uture paper, te scope o te paper sall be exteded to brid two step poits. Reereces [] Vigo-Aguiar, J. ad Ramos, H., Dissipative Cebsev expoetial-itted metods or umerical solutio o secod-order dieretial equatios, J. Comput. Appl. Mat. 8, 00, 87-. [] Awoemi, D.O., A class o Cotiuous Metods or geeral secod order iitial value problems i ordiar dieretial equatio. Iteratioal Joural o Computatioal Matematics, 7, 999, 9-7. [] Bu, R.A. ad Varsil er, Y.D., A umerical metod or solvig dieretial equatios o a orders. Comp. Mat. Ps. (, 99, 7-0. [4] Lambert, J.D., Numerical Metods or Ordiar Dieretial Sstems. Jo Wile, New York, 99. [] Kaode S.J., A improve umerov metod or direct solutio o geeral secod order iitial value problems o ODEs, Natioal Matematical Cetre proceedigs 00. [6] Adesaa, A.O., Aake, T.A., Bisop, S.A. ad Osilagu, J.A., Two Steps Block Metod or te solutio o geeral secod order Iitial Value Problems o Ordiar Dieretial Equatios. Joural o Natural Scieces, Egieerig ad Tecolog, J. Nat. Sci. Egr. Tec., 8(, 009, -. [7] Yaaa, Y. A. ad Badmus, A. M., A Class o Collocatio Metods or Geeral Secod Order Ordiar Dieretial Equatios. Arica Joural o Matematics ad Computer Sciece researc, (4, 009, [8] Aake T.A., Cotiuous Implicit Hbrid Oe-Step Metods or te solutios o Iitial Value Problems o geeral secod order Ordiar Dieretial Equatios. P.D Tesis (Upublised Coveat Uiversit, Ota, Nigeria, 0. [9] Adeii, R.B., Alabi, M.O. ad Folarami, R.O., A Cebsev collocatio approac or a cotiuous ormulatio o brid metods or iitial value problems i ordiar dieretial equatios. Joural o te Nigeria Associatio o Matematical Psics,, 008, [0] Adeea E.O. ad Adeii R.B., Cebsev Collocatio Approac or a Cotiuous Formulatio o Implicit Hbrid Metods or IVPs i Secod Order ODEs. Iteratioal Orgaisatio o Scietiic Researc Joural o Matematics, 6(4, 0, Page
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