On The Stability and Accuracy of Some Runge-Kutta Methods of Solving Second Order Ordinary Differential Equations
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1 Iteratioal Joural o Computatioal Egieerig Resear Vol Issue 7 O Te Stabilit ad Aura o Some Ruge-Kutta Metods o Solvig Seod Order Ordiar Dieretial Euatios S.O. Salawu R.A. Kareem ad O.T. Arowolo Departmet o Matematis Uiversit o Ilori Ilori Nigeria Departmet o Matematis Lagos State Poltei Iorodu Nigeria ABSTRACT. Tis paper sees umerial solutios to seod order dieretial euatios o te orm wit iitial value usig dieret Ruge-Kutta metods o order two. Two ases o Epliit Ruge-Kutta metod were derived ad teir stabilit was determied tis is te implemeted. Te results were ompared wit te Euler s metod or aura. KEYWORDS. Euler s Metod Ordiar Dieretial Euatio Ruge-Kutta metod Stabilit ad Talor series I. INTRODUCTION. Solutios to Dieretial euatio ave over te ears bee a ous to Applied Matematis. Te uestio te beomes ow to id te solutios to tose euatios. Te rage o Dieretial Euatios tat a be solved b straigt orward aaltial metod is relativel restrited []. Eve solutio i series ma ot alwas be satisator eiter beause o te slow overgee o te resultig series or beause o te ivolved maipulatio i repeated stages o dieretiatio [9].Ruge-Kutta metods are amog te most popular ODE solvers. Te were irst studied b Carle Ruge ad Marti Kutta aroud 9. Moder developmets are mostl due to Jo Buter i te 9s []. Te Ruge-Kutta metod is ot restrited to solvig ol irstorder dieretial euatios but Ruge-Kutta metods are also used to solve iger order ordiar dieretial euatios or oupled (simultaeous) dieretial euatios []. Te iger order euatios a be solved b osiderig a euivalet sstem o irst order euatios. However it is also possible to develop diret sigle steps metods to solve iger order euatio. Deiitio: We deie a epliit Ruge-Kutta metod wit slopes b te ollowig euatios []. b r i i i j! j i ij Were a i j r r... ij r are arbitrar. Iss 5-5 Jul Page
2 O Te Stabilit Ad Aura O Some II. DERIVATION OF SECOND ORDER RUNGE-KUTTA METHOD. To derive te Ruge-Kutta metods or seod order ordiar dieretial euatio o te orm Let us deie wit te iitial oditio..!!. r r. r r. Were b r r r ad r are arbitrar ostats to be determied. Usig Talor series epasio orm euatio. ad. gives. iv....5!!! iv....!!.7.8 iv [ ].9 We re-write euatios.7.8 ad.9 as D iv D D Were D So euatios.5 ad. beomes Iss 5-5 Jul Page
3 O Te Stabilit Ad Aura O Some D D D....!!! D D D.... Simpliig! we ave!! O D D O. Were we ave used Te substitutio o ad. i euatio. ad. ield 5 r r r D r D r O. r r r D r D r O.5 We ompare te oeiiet o euatios r r r r. ad.5 wit euatios. ad. r r Te oeiiet o i ad i orrespodig oeiiets i euatio. truatio error is O i ad.to obtai.. o euatios. ad.5 ad. will ot mat wit te or a oie o r ad r. Tus te loal O i. A solutio o euatio. ad. osidered or dieret values o r. Here we osider two ases o r. ma be Case I: r Te rom euatio. we get r b r r I te utio is idepedet o te we a ostrut a Ruge-Kutta metod i wi te loal truatio error i ad is O. Here we obtai Iss 5-5 Jul Page
4 O Te Stabilit Ad Aura O Some r r r r wit te solutio r r r r r r r Tus te Ruge-utta metod or te seod order iitial value problem Tus rom euatio wit iitial value... ad. we ave! ( ) Case II: r From euatio. we ave! r r r b 9 Tereore!!.9 9. Te two metods derived are epliit seod order Ruge-Kutta. III. STABILITY ANALYSIS. Wile umeriall solvig a iitial value problem or ordiar dieretial euatios a error is itrodued at ea itegratio step due to te iaura o te ormula. Eve we te loal error at ea step is small te total error ma beome large due to aumulatio ad ampliiatio o tese loal errors. Tis growt peomeo is alled umerial istabilit []. We sall disuss te stabilit o te Ruge-Kutta metod i.8 ad. Let us osider te dieretial euatio. Subjet to iitial oditio b Were is a real umber Iss 5-5 Jul Page
5 We sall osider te ase order ordiar dieretial euatio is o te orm O Te Stabilit Ad Aura O Some ad wile give a trival solutio. Reall tat te seod Te rom euatio. Substitute tis. i euatio.7 we ave 9 Substitutig te euatio. ito euatio.8 we ave. 8. Let us ow osider te ase we. Te solutio i tis ase is osillatig. We tereore osider te eigevalues o te matri wi is give b I te. To determie te value o ad we osider.5 We a re-write euatio.5 as. we we ave Iss 5-5 Jul Page
6 O Te Stabilit Ad Aura O Some Iss 5-5 Jul Page Substitutig euatio.7 i. we ave Were ad 578. Computig ad as utios o we id tat te roots ave uit modulus.. Tus te stabilit iterval o te Ruge-Kutta metod.8 is.. wi also sow tat is o order. For te solutios o. are epoetial i ature ad a be writte i te matri orm as e b Ad b or poit tis solutio beomes Ad te 7 lim e b Te relative error or te metod uder disussio i ase o a large umber o itegratio iterval (large small ) are to be osidered. Te maimum eigevalue o te matri.
7 O Te Stabilit Ad Aura O Some Iss 5-5 Jul Page 8 Is obtaied as... Ad tereore te relative error is give b 7 log log log log e e F We sall ow osider te stabilit o te Ruge-Kutta metod. usig dieretial euatio.. Euatio.9 beomes.8 Substitutig te euatio.9 ito euatio. we ave.9 We ow osider te ase we. Te eigevalues is give b Were ad. Computig ad as utios o te roots ave uit modulus 9 5. Tus te stabilit iterval o te Ruge-Kutta metod. is 9 5. ad is o order. For te solutios o. are epoetial i ature as above i ase I.
8 O Te Stabilit Ad Aura O Some Te maimum eigevalue o te matri. Is obtaied as 7... Te relative error is give b F log log e log e log 8 IV. IMPLEMENTATION OF THE METHOD. Cosider te iitial value problem [] Subjeted to iitial oditio.. Te omplemetar solutio is iitial oditio we ave A ad B Te geeral solutio beome A Be ad te partiular solutio gives p Usig te e Numerial solutios are preerred to te derived ases I ad ase II o epliit seod order Ruge-Kutta metod o te IVP i. obtaiig umerial solutios or values o up to ad iludig wit a step size o. as oud i Table ad Table. Table: Solutios o ase I seod ordered Ruge-Kutta metods wit. X Numerial solutio () Aaltial solutio Y() Absolute Error Iss 5-5 Jul Page 7
9 O Te Stabilit Ad Aura O Some Figure :.. Table: Solutios o ase II seod ordered Ruge-Kutta metods wit. X Numerial solutio () Aaltial solutio Y() Absolute Error Figure : Euler s metod Euler s metod is oe o ma metods or geeratig umerial solutios to dieretial euatios. Besides most o te oter metods tat migt be disussed are reiemets o Euler s metod. Tis metod is implemeted ad ompared its aura ad te error wit te metod i setio. Te Euler s metod geeralized i te orm [] Cosider te iitial value problem i euatio. Subjeted to iitial oditio. Table: Solutios o Euler s metods wit. X Numerial solutio () Aaltial solutio Y() Absolute Error Iss 5-5 Jul Page 8
10 O Te Stabilit Ad Aura O Some Figure : V. CONCLUSION. Ivestigatio arried out o some epliit seod ordered Ruge-Kutta metod i tis paper as sow tat te stabilit iterval o te Ruge-Kutta metod i ase I is. ad te stabilit iterval o te Ruge-Kutta metod i ase II is It is lear tat ase I is more stable ta ase II o te derived Ruge-Kutta metods. Te two metods are sow to be aurate eiiet ad geeral i appliatio or suiietl solutio o. Te result obtaied i te preset wor demostrate te eetiveess ad superiorit or te solutio o seod order ordiar dieretial euatio wi gave a ver ig aura we ompared wit eat solutio. REFERENCE. [] Autar Kaw Ruge-Kutta Seod Order Metod or Ordiar Dieretial Euatios. Uiversit o Sout Florida; Holisti Numerial metods Istitute. [] Bu R.C. ad Bu E.F. Itrodutio to Dieretial Euatio; Hogto Mili Compa 97 [] Buter J.C. Numerial Metods or Ordiar Dieretial Euatios; Jo Wile ad sos New Yor. [] [Buter Jo Ruge-Kutta Metods or Ordiar Dieretial Euatios; COE Worsop o Numerial Aalsis Kusu Uiversit 5. [5] Cristoper P. Grat Teor o Ordiar Dieretial Euatios; Brigam Youg Uiversit. [] Corliss G. ad Kirliger O Impliit Talor Series metods or Sti Ordiar Dieretial Euatio; Cetre or Resear o Parallel Computatio Rie Uiversit Housto 99. [7] Curtis F.G. ad Patri O.W. Applied Numerial Aalsis; Addiso Wesle Publisig Compa 989. [8] Greespa D. Teor ad Solutios o Ordiar Dieretial Euatios; Te MaMilla Compa New Yor 9. [9] Iserles A. Numerial Aalsis or Ordiar Dieretial Euatios; Cambridge Uiversit Press Cambridge 99. [] Jai M.K. Numerial Solutio o Dieretial Euatios Wile Easter Limited. [] Joe D. Homa Numerial Metods or Egieers ad Sietists; marel Deer I. [] Retewald G. Numerial Itegratio o ODEs or Iitial Value Problems. Departmet o Meaial egieerig Portlad State Uiversit.. [] Ouuga S.A. ad Aabi M.A. Computatioal Matematis; A First Course. Wim Publiatio Lagos Nigeria. [] Patil P.B. ad Verma U.P. Numerial Computatioal Metods Narosa Publisig House New Deli. Iss 5-5 Jul Page 9
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