On Some Numerical Methods for Solving Initial Value Problems in Ordinary Differential Equations

Transcription

1 IOSR Joural o Mathmatics IOSRJM ISSN: Volum, Issu Jul-Aug, PP 5- O Som Numrical Mthods or Solvig Iitial Valu Problms i Ordiar Dirtial Equatios Ogurid R. Bosd, Fadugba S. Emmaul, Okulola J. Tmitao Dpartmt o Mathmatical Scics, Ekiti Stat Uivrsit, Ado Ekiti, Ekiti Stat, Nigria. Dpartmt o Mathmatical ad Phsical Scics, A Babalola Uivrsit, Ado Ekiti, Ekiti Stat, Nigria. Abstract: This work prsts umrical mthods or solvig iitial valu problms i ordiar dirtial quatios. Eulr's mthod is prstd rom th poit o viw o Talor's algorithm which cosidrabl simpliis th rigorous aalsis whil Rug Kutta mthod attmpts to obtai gratr accurac ad at th sam tim avoid th d or highr drivativs b valuatig th giv uctio at slctd poits o ach subitrval. W discuss th stabilit ad covrgc o th two mthods udr cosidratio ad rsult obtaid is compard to th act solutio. Th rror icurrd is udrtak to dtrmi th accurac ad cosistc o th two mthods. K words: Dirtial Equatio, Error, Eulr's Mthod, Rug Kutta Mthod, Stabilit. I. Itroductio Dirtial quatios ca dscrib arl all sstm udrgo chag. Ma mathmaticias hav studid th atur o ths quatios ad ma complicatd sstms ca b dscribd quit prcisl with compact mathmatical prssios. Howvr, ma sstms ivolvig dirtial quatios ar so compl. It is i ths compl sstms whr computr simulatios ad umrical approimatios ar usul. Th tchiqus or solvig dirtial quatios basd o umrical approimatios wr dvlopd bor programmabl computrs istd. Th problm o solvig ordiar dirtial quatios is classiid ito iitial valu ad boudar valu problms, dpdig o th coditios spciid at th d poits o th domai. Thr ar umrous mthods that produc umrical approimatios to solutio o iitial valu problms i ordiar dirtial quatios such as Eulr's mthod which was th oldst ad simplst mthod origiatd b Lohard Eulr i 768, Improvd Eulr's mthod ad Rug Kutta mthods dscribd b Carl Rug ad Marti Kutta i 895 ad 95 rspctivl. Thr ar ma cllt ad haustiv tts o this subjct that ma b cosultd, such as [8], [4], [6], [5], ad [] just to mtio w. I this work w prst th practical us ad th covrgc o Eulr mthod ad Rug Kutta mthod or solvig iitial valu problms i ordiar dirtial quatios. II. Numrical Mthod Numrical mthod orms a importat part o solvig iitial valu problms i ordiar dirtial quatios, most spciall i cass whr thr is o closd orm solutio. Nt w prst two umrical mthods aml Eulr's Mthod ad Rug Kutta mthod. W prst hr th drivatio o Eulr's mthod or gratig, umricall, approimat solutios to th iitial valu problm [],,. Rug Kutta Mthod Rug Kutta mthod is a tchiqu or approimatig th solutio o ordiar dirtial quatio. This tchiqu was dvlopd aroud 9 b th mathmaticias Carl Rug ad Wilhlm Kutta. Rug Kutta mthod is popular bcaus it is icit ad usd i most computr programs or dirtial quatio. Th ollowig ar th ordrs o Rug Kutta Mthod as listd blow: Rug Kutta mthod o ordr o is calld Eulr's mthod. Rug Kutta mthod o ordr two is th sam as modiid Eulr s or Hu's Mthod. Th ourth ordr Rug Kutta mthod calld classical Rug Kutta mthod. I this papr, w shall ol cosidr th ourth ordr Rug Kutta mthod.. Drivatio o th Fourth Ordr Rug Kutta Mthod W shall driv hr th simplst o th Rug mthod. A ormula o th ollowig orm is sought: ak bk 5 Pag

2 O Som Numrical Mthods or Solvig Iitial Valu Problms i Ordiar Dirtial Equatios Whr k h,, k h h, ad a,b,, ar costats to b dtrmid so that k will agr with th Talor algorithm. Epadig i a Talor sris o ordr h, w obtai h h h... 6 h h 4 = h, h 6 It should b otd that th pasios,, ad. Th subscript mas, that all uctios ivolvd ar to b valuatd at. O th othr had, usig Talor s pasio or uctios o two variabls, w id that h k k h, k, h k hk h, all th drivativs abov ar valuatd at,. I w ow substitut this prssio or k ito ad ot that k h,, w id upo rarragmt i powrs o h ad b sttig a b, that k k k k4 6 Whr k h,, h k, h k, h k h, k ad k4 h h, k. This mthod is udoubtdl th most popular o all Rug Kutta mthods. Idd it is rqutl rrrd to as th ourth ordr Rug Kutta mthod. Ma umrical aalst rl o, bcaus it is quit stabl, accurat ad as to program.. Error Estimat or Rug Kutta Mthod For all o stp mthods lik Rug Kutta Mthod, th cocptuall-simplst diitio o local trucatio rror is that it is th rror committd i th most rct itgratio stp, o a sigl itgratio stp. W dot th th solutio to th iitial valu problm b,,. W hav otd that th trucatio rror i p ordr p Rug Kutta mthod is kp, whr k is som costat. Bouds o k or p,, 4 also ist. Th drivatio o ths bouds is ot a simpl mattr ad morovr, thir valuatio rquirs som quatitis. O o th srious draw backs o Rug Kutta mthod is rror stimatio..4 Eulr s Mthod Eulr s mthod is also calld tagt li mthod ad is th simplst umrical mthod or solvig iitial valu problm i ordiar dirtial quatio, particularl suitabl or quick programmig which was origiatd b Lohard Eulr i 768. This mthod subdividd ito thr aml, Forward Eulr s mthod. Improvd Eulr s mthod. Backward Eulr s mthod. I this work w shall ol cosidr orward Eulr s mthod..5 Drivatio o Eulr s mthod W prst blow th drivatio o Eulr s mthod or gratig, umricall, approimat solutios to th iitial valu problm, whr ad ar iitial valus or ad rspctivl. Our aim is to dtrmi approimatl th ukow uctio or. W ar told plicitl th valu o, aml, 6 Pag

3 O Som Numrical Mthods or Solvig Iitial Valu Problms i Ordiar Dirtial Equatios usig th giv dirtial quatio, w ca also dtrmi th istataous rat o chag o at poit, =, I th rat o chag o wr to rmai, or all poit, th 4 would actl,.th rat o chag o dos ot rmai, or all, but it is rasoabl to pct that it rmais clos to, or clos to. I this is th cas, th th valu o will rmai clos to, or clos to, or small umbr h, w hav h 5, 6 = h, Whr h ad is calld th stp siz. B th abov argumt, Rpatig th abov procss, w hav at poit as ollows h 8, = h, W hav Th di or,,,,4,5,..., w hav h Suppos that, or som valu o, w ar alrad computd a approimat valu or. Th Th rat o chag o or to is,,, whr,. Thus, h, Hc, h, Equatio is calld Eulr s mthod. From, w hav,,,,,,... h.6 Trucatio Errors or Eulr s Mthod Numrical stabilit ad rrors ar discussd i dpth i [] ad [7]. Thr ar two tps o rrors aris i umrical mthods aml trucatio rror which ariss primaril rom a discrtizatio procss ad roud o rror which ariss rom th iitss o umbr rprstatios i th computr. Riig a msh to rduc th trucatio rror ot causs th roud o rror to icras. To stimat th trucatio rror or Eulr s mthod, w irst rcall Talor s thorm with rmaidr, which stats that a uctio ca b padd i a sris about th poit a a a a a a! m m m m a a a m! m!... 5 Th last trm o 5 is rrrd to as th rmaidr trm. Whr a Pag

4 O Som Numrical Mthods or Solvig Iitial Valu Problms i Ordiar Dirtial Equatios I 5, lt ad a, i which h h 6 Sic satisis th ordiar dirtial quatio i, which ca b writt as, 7 Hc, h, h 8 B cosidrig 8 to Eulr s approimatio i, it is clar that Eulr s mthod is obtaid b omittig th rmaidr trm th trucatio rror i Eulr s mthod at ach stp. h i th Talor pasio o at th poit. Th omittd trm accouts or.7 Covrgc o Eulr s Mthod Th cssar ad suicit coditios or a umrical mthod to b covrgt ar stabilit ad cosistc. Stabilit dals with growth or dca o rror as umrical computatio progrsss. Now w stat th ollowig thorm to discuss th covrgc o Eulr s mthod. Thorm: I, satisis a Lipschitz coditio i ad is cotiuous i or a ad did a squc, whr,,..., k ad i, th as uiorml i whr is th solutio o th iitial valu problm. Proo: w shall start th proo o th abov thorm b drivig a boud or th rror 9 Whr ad ar calld umrical ad act valus rspctivl. W shall also show that this boud ca b mad arbitraril small. I a boud or th rror dpds ol o th kowldg o th problm but ot o its solutio, it is calld a a priori boud. I, o th othr had, kowldg o th proprtis o th solutio is rquird, its rror boud is rrrd to as a a postriori boud. To gt a a priori boud, lt us writ h, t Whr t is calld th local trucatio rror. It is th amout b which th solutio ails to satis th dirc mthod. Subtractig rom, w gt h[,, ] t Lt us writ M,, Substitutig ito, th This is th dirc quatio or. Th rror is kow, so it ca b solvd i w kow hav a boud o th Lipschitz costat M or M. Suppos w also havt t. Th w hav T To procd urthr, w d th ollowig lmma. Lmma: I satisis 4 ad h a, th M adt. W T Lb Lb T 5 Lmma: Th irst iqualit o 5 ollows b iductio. It is triviall tru or. Assumig that it is tru or, w hav rom Pag

5 O Som Numrical Mthods or Solvig Iitial Valu Problms i Ordiar Dirtial Equatios T T = T = T Hc 5 is tru or ad thus or all. Th scod iqualit i 5 ollows rom th act that, provig th lmma. Mh Ma that h a ad or Mh, 6 so To cotiu th proo o th thorm, w d to ivstigatt, th boud o th local trucatio rror. From, w hav t h, B th Ma valu thorm, w gt or, h h,, h h, h h, h h,, h h Th last trm ca b tratd b th Ma valu thorm to gt a boud Mh g h MZ, whr Z ma, th iqualit ists bcaus o th cotiuit o ad i a closd rgio. Th tratmt o th irst trm i 7 dpds o our hpothsis. I w ar prpard to assum that, also satisis a Lipschitz coditio i, w ca boud th irst trm i 7 b. Cosqutl, h L MZ T ad so rom 5, w gt t L MZ Ma Ma h M Thus th umrical solutio covrgs as h, i. 7 L h, whr L is th Lipschitz costat or III. Numrical Eprimts I ordr to coirm th applicabilit ad suitabilit o th mthods or solutio o iitial valu problms i ordiar dirtial quatios, it was computrizd i Fortra Programig laguag ad implmtd o a macro-computr adoptig doubl prcisio arithmtic. Th prormac o th mthods was chckd b comparig thir accurac ad icic. Th icic was dtrmid rom th umbr itratios couts ad umbr o uctios valuatios pr stp whil th accurac is dtrmid b th siz o th discrtizatio rror stimatd rom th dirc btw th act solutio ad th umrical approimatios. Eampl : Th irst problm cosidrd i this illustratio is th liar irst ordr iitial valu problm, with stp siz h. o th itrval whos act solutio is giv b ta. Th rsults obtaid show i Tabl ad Tabl, th compariso o th mthods to th act solutio ad th rror icurrd rspctivl. Eampl : W us Eulr s mthod to approimat th solutio o th iitial valu problm,, with stp siz h. o th itrval whos act solutio is giv b. Th rsults obtaid show i Tabl ad Tabl 4, th compariso o th mthods to th act solutio ad th rror icurrd rspctivl Pag

6 IV. O Som Numrical Mthods or Solvig Iitial Valu Problms i Ordiar Dirtial Equatios Tabl o Rsults Tabl : Th Comparativ Rsult Aalsis o Rug Kutta Mthod ad Eulr s Mthod R E Tabl : Error icurrd i Rug Kutta Mthod ad Eulr s Mthod R E Tabl : Th Comparativ Rsult Aalsis o Rug Kutta Mthod ad Eulr s Mthod R E Pag

7 O Som Numrical Mthods or Solvig Iitial Valu Problms i Ordiar Dirtial Equatios Tabl 4: Error icurrd i Rug Kutta Mthod ad Eulr s Mthod R E V. Discussio o Rsults W otic that i Tabls ad 4, th rror icurrd i Eulr's mthod is gratr tha that o Rug Kutta mthod ad th sam tim gt largr as icrass. Hc Rug Kutta mthod is mor accurat tha its coutrpart Eulr's mthod as w ca s rom Tabls ad. VI. Coclusio W hav i our disposal two umrical mthods or solvig iitial valu problms i ordiar dirtial quatios. I gral, umrical mthod has its ow advatags ad disadvatags o us: Eulr's mthod is thror bst rsrvd or simpl prrabl, rcursiv drivativs that ca b rprstd b w trms. It is simpl to implmt ad simpliis rigorous aalsis. Th major disadvatags o Eulr mthod ar th tirsom, somtims impossibl calculatio o highr drivativs ad th slow covrgc o th sris or som uctios which ivolvs trms o opposit sig whil Rug Kutta mthod is a sl-startig bcaus it dos ot us iormatio rom prviousl calculatd poits ad grall stabl but rror stimatio rmais problmatic. From th problms solvd usig FORTRAN programig laguag, it is obsrvd that a lot o usul isights ito umrical solutio o iitial valu problms hav b gaid. W coclud that Rug Kutta mthod is cosistt, covrgt, quit stabl ad mor accurat tha Eulr's mthod ad it is widl usd i solvig iitial valu problms i ordiar dirtial quatios. Rrcs [] W. E. Boc ad R. C. DiPrima, Elmtar dirtial quatio ad boudar valu problms, Joh Wil ad So,. [] L. Collatz, Numrical tratmt o dirtial quatios, Sprigr Vrlag Brli, 96. [] K. Erwi Advacd Egirig Mathmatics, Eighth Editio, Wil Publishr,. [4] A. Gilat, Matlab: A Itroductio with applicatio, Joh Wil ad Sos, 4. [5] N. Kocklr, Numrical mthods ad scitiic computig, Clardo Prss, Oord Lodo, 994. [6] J. D. Lambrt, Numrical mthod or ordiar sstms o Iitial valu problms, Joh Wil ad Sos, Nw York, 99. [7] D. C. Samul, Elmtar umrical aalsis a algorithm approach, Third Editio, Mc Graw Itratioal Book Compa, 98. [8] M. P. Stph, To comput umricall, cocpts ad stratg, Littl Brow ad Compa, Caada, Pag