Euler s Method for Solving Initial Value Problems in Ordinary Differential Equations.

Transcription

1 Eulr s Mthod for Solvig Iitial Valu Problms i Ordiar Diffrtial Equatios. Suda Fadugba, M.Sc. * ; Bosd Ogurid, Ph.D. ; ad Tao Okulola, M.Sc. 3 Dpartmt of Mathmatical ad Phsical Scics, Af Babalola Uivrsit, Ado Ekiti, Nigria. Dpartmt of Mathmatical Scics, Ekiti Stat Uivrsit, Ado Ekiti, Nigria. 3 Dpartmt of Mathmatical ad Phsical Scics, Af Babalola Uivrsit, Ado Ekiti, Nigiria. mmasfad6@ahoo.com* ABSTRACT This work prsts Eulr s mthod for solvig iitial valu problms i ordiar diffrtial quatios. This mthod is prstd from th poit of viw of Talor s algorithm which cosidrabl simplifis th rigorous aalsis. W discuss th stabilit ad covrgc of th mthod udr cosidratio ad th rsult obtaid is compard to th act solutio. Th rror icurrd is udrtak to dtrmi th accurac ad cosistc of Eulr s mthod. Kwords: diffrtial quatio, Eulr s mthod, rror, covrgc, stabilit INTRODUCTION Diffrtial quatios ca dscrib arl all sstms udrgo chag. Th ar ubiquitous i scic ad girig as wll as coomics, social scic, biolog, busiss, tc. Ma mathmaticias hav studid th atur of ths quatios ad ma complicatd sstms ca b dscribd quit prcisl with compact mathmatical prssios. Howvr, ma sstms ivolvig diffrtial quatios ar so compl or th sstms that th dscrib ar so larg that a purl aaltical solutio to th quatio is ot tractabl. It is i ths compl sstms whr computr simulatios ad umrical approimatios ar usful. Th tchiqus for solvig diffrtial quatios basd o umrical approimatios wr dvlopd bfor programmabl computrs istd. Th problm of solvig ordiar diffrtial quatios is classifid ito two aml iitial valu problms ad boudar valu problms, dpdig o th coditios at th d poits of th domai ar spcifid. All th coditios of iitial valu problm ar spcifid at th iitial poit. Thr ar umrous mthods that produc umrical approimatios to solutio of iitial valu problm i ordiar diffrtial quatio such as Eulr s mthod which was th oldst ad simplst such mthod origiatd b Lohard Eulr i 768, Improvd Eulr mthod, Rug Kutta mthods dscribd b Carl Rug ad Marti Kutta i 85 ad 5, rspctivl. Thr ar ma cllt ad haustiv tts o this subjct that ma b cosultd, such as Boc ad DiPrima, Erwi 3, Stph 83, Collatz 6, ad Gilat 4 just to mtio fw. I this work w prst th practical us ad th covrgc of Eulr mthod for solvig iitial valu problm i ordiar diffrtial quatio. NUMERICAL METHOD Th umrical mthod forms a importat part of solvig iitial valu problm i ordiar diffrtial quatio, most spciall i cass whr thr is o closd form aaltic formula or difficult to obtai act solutio. Nt, w shall prst Eulr s mthod for solvig iitial valu problms i ordiar diffrtial quatios. Eulr s Mthod Eulr s mthod is also calld tagt li mthod ad is th simplst umrical mthod for solvig iitial valu problm i ordiar diffrtial quatio, particularl suitabl for quick programmig which was origiatd b Lohard Eulr i 768. This mthod subdividd ito thr aml: Th Pacific Joural of Scic ad Tcholog 5 Volum 3. Numbr. Novmbr Fall

2 Forward Eulr s mthod. Improvd Eulr s mthod. Backward Eulr s mthod. I this work w shall ol cosidr forward Eulr s mthod. Drivatio of Eulr s Mthod W prst blow th drivatio of Eulr s mthod for gratig, umricall, approimat solutios to th iitial valu problm: f, Whr ad ar iitial valus for ad, rspctivl. Our aim is to dtrmi approimatl th ukow fuctio for. W ar told plicitl th valu of, aml, usig th giv diffrtial quatio, w ca also dtrmi actl th istataous rat of chag of at poit f, = f, 3 If th rat of chag of wr to rmai f, for all poit, th would actl f,. Th rat of chag of dos ot rmai f, for all, but it is rasoabl to pct that it rmais clos to f, for clos to. If this is th cas, th th valu of will rmai clos to f,, for for clos to small umbr h, w hav: h 4 f, = hf, h 5 Whr ad is calld th stp siz. B th abov argumt: 6 Rpatig th abov procss, w hav at poit as follows: h 7 f, hf, = W hav: 8 Th dfi for,,,3,4,5,..., w hav h Suppos that, for som valu of, w ar alrad computd a approimat valu for. Th th rat of chag of for clos to is f, f, f, f,. r Thus, Hc, hf, hf, wh Equatio is calld Eulr s mthod. From, w hav: h f,,,,,3,... Trucatio Errors For Eulr s Mthod 3 Numrical stabilit ad rrors ar discussd i dpth i Lambrt 73 ad Kocklr 4. Thr ar two tps of rrors aris i umrical mthods aml trucatio rror which ariss primaril from a discrtizatio procss ad roud Th Pacific Joural of Scic ad Tcholog 53 Volum 3. Numbr. Novmbr Fall

3 off rror which ariss from th fiitss of umbr rprstatios i th computr. Rfiig a msh to rduc th trucatio rror oft causs th roud off rror to icras. To stimat th trucatio rror for Eulr s mthod, w first rcall Talor s thorm with rmaidr, which stats that a fuctio f ca b padd i a sris about th poit f f m f a a a m! a f a a f a a! m f m a m! m 4... Th last trm of 4 is rfrrd to as th rmaidr trm. Whr a. I 4, lt ad a h, i which: h. Whr 5 Sic satisfis th ordiar diffrtial quatio i, which ca b writt as: f, 6 Whr is th act solutio at. Hc, hf, h 7 B cosidrig 7 to Eulr s approimatio i, it is clar that Eulr s mthod is obtaid b omittig th rmaidr trm h i th Talor pasio of at th poit. Th omittd trm accouts for th trucatio rror i Eulr s mthod at ach stp. Covrgc of Eulr s Mthod Th cssar ad sufficit coditios for a umrical mthod to b covrgt ar stabilit ad cosistc. Stabilit dals with growth or dca of rror as umrical computatio progrsss. Now w stat th followig thorm to discuss th covrgc of Eulr s mthod. Thorm: If f, satisfis a Lipschitz coditio i ad is cotiuous i for a ad dfid a squc, whr,,...,k ad if, th as uiforml i whr is th solutio of th iitial valu problm ad. Proof: w shall start th proof of th abov thorm b drivig a boud for th rror: 8 Whr ad ar calld umrical ad act valus rspctivl. W shall also show that this boud ca b mad arbitraril small. If a boud for th rror dpds ol o th kowldg of th problm but ot o its solutio, it is calld a a priori boud. If, o th othr had, kowldg of th proprtis of th solutio is rquird, its rror boud is rfrrd to as a a postriori boud. To gt a a priori boud, lt us writ: t hf, Whr t is calld th local trucatio rror. It is th amout b which th solutio fails to satisf th diffrc mthod. Subtractig from, w gt: h[ f, f, ] Lt us writ: t M f, f, Th Pacific Joural of Scic ad Tcholog 54 Volum 3. Numbr. Novmbr Fall

4 Substitutig ito, th: This is th diffrc quatio for. Th rror is kow, so it ca b solvd if w kow M ad t. W hav a boud of th Lipschitz costat M for M. Suppos w also hav T t. Th w hav: T 3 To procd furthr, w d th followig lmma. Lmma: If th: T satisfis 3 ad h a T Lb Lb, 4 Lmma: Th first iqualit of 4 follows b iductio. It is triviall tru for. Assumig that it is tru for, w hav from 3: T T =T 5 Hc 4 is tru for ad thus for all. Th scod iqualit i 5 follows from th fact that Mh h a ad for, so Mh Ma that, provig th lmma. To cotiu th proof of th thorm, w d to ivstigatt, th boud o th local trucatio rror. From, w hav: t hf, B th Ma valu thorm, w gt for h f, h, f, h f h, h f h, h f h h, f, h 6 Th last trm ca b tratd b th Ma valu thorm to gt a boud Mh g h Z ma MZ, whr, th iqualit ists bcaus of th cotiuit of ad f i a closd rgio. Th tratmt of th first trm i 6 dpds o our hpothsis. If w ar prpard to assum that f, also satisfis a Lipschitz coditio i, w ca boud th first trm i 6 b L h, whr L is th Lipschitz costat for f. Cosqutl, from 4, w gt: t h L MZ T ad L MZ Ma Ma h 7 M Thus th umrical solutio covrgs as h, if. Algorithm for Eulr s Mthod Samul, 8 Th tpical stps of Eulr s mthod ar giv blow: Stp : dfi f, Stp : iput iitial valus ad Stp 3: iput stp sizs h ad umbr of stps Stp 4: calculat ad : N h for Stp 5: output ad hf, so Th Pacific Joural of Scic ad Tcholog 55 Volum 3. Numbr. Novmbr Fall

5 Stp 6: d NUMERICAL EXAMPLES Now, w prst som umrical ampls as follows: 7 c is allowd to cool Eampl : A ball at dow i air at a ambit tmpratur of 7 c. Assumig hat is lost ol du to radiatio. Th diffrtial quatio for th tmpratur of th ball is giv b: 6. 4 f 4, k 4 t, t t h k Th act solutio of th ordiar diffrtial quatio is giv b th solutio of a o liar quatio as: , 7 dt d Fid th tmpratur at siz of h 4s. Stp : d.67 dt t 48s Assum a stp, t, 3 c.53l.85 ta t Th solutio to this oliar quatio at t 48s is k. Th ffct of stp siz o Eulr s mthod is show i Tabl ad Figur. k f t,.67 hf t, hf t, 4 8., Tabl : Effct of Stp Siz Tmpratur at 48scods as a fuctio of stp siz, h. Stp Siz, h 48 E t t f, k Thrfor, 4 t, t t h k Eampl : W us Eulr s mthod to approimat th solutio of th iitial valu problm,, with stp siz h. o th itrval whos act solutio is giv b. Th rsults obtaid show i Tabl, th compariso of th mthod to th act solutio ad th rror icurrd i Eulr s mthod. Stp : For, t 4, 6. Figur : Effct of Stp Siz o Eulr s Mthod hf t, Th Pacific Joural of Scic ad Tcholog 56 Volum 3. Numbr. Novmbr Fall

6 CONCLUSION Tabl : Th Comparativ Rsult Aalsis ad Error gratd from Eulr s Mthod Eampl 3: W shall approimat th solutio of, th iitial valu problm usig Eulr s mthod with stp siz h. o th itrval whos act solutio is giv b ta. Th rsults obtaid show i Tabl 3, th compariso of th mthod to th act solutio ad th rror icurrd i Eulr s mthod. DISCUSSION OF RESULTS W otic that i ampl, th accurac of th approimatios gts wors as w furthr awa from th iitial valu ad i ampls ad 3, th rror gt largr as icrass. Tabl 3: Th Comparativ Rsult Aalsis ad Error gratd from Eulr s Mthod. I gral, ach umrical mthod has its ow advatags ad disadvatags of us: Eulr s mthod is thrfor bst rsrvd for simpl prfrabl, rcursiv drivativs that ca b rprstd b fw trms. It is simpl to implmt ad simplifis rigorous aalsis. Th major disadvatags of this mthod ar th tirsom, somtims impossibl calculatio of highr drivativs ad th slow covrgc of th sris for som fuctios which ivolvs trms of opposit sig. REFERENCES. Boc, W.E. ad R.C. DiPrima.. Elmtar Diffrtial Equatio ad Boudar Valu Problms. Joh Wil ad Sos: Nw York, NY.. Collatz, L. 6. Numrical Tratmt of Diffrtial Equatios. Sprigr Vrlag: Brli, Grma. 3. Erwi, K. 3. Advacd Egirig Mathmatics. Eighth Editio. Wil Publishrs: Nw York, NY. 4. Gilat, A. 4. Matlab: A Itroductio with Applicatio. Joh Wil ad Sos: Nw York, NY. 5. Kocklr, N. 4. Numrical Mthods ad Scitific Computig. Clardo Prss: Oford, UK. 6. Lambrt, J.D.. Numrical Mthod for Ordiar Sstms of Iitial Valu Problms. Joh Wil ad Sos: Nw York, NY. 7. Samul, D.C. 8. Elmtar Numrical Aalsis: A Algorithm Approach. Third Editio. Mc Graw Itratioal Book Compa: Nw York, NY. Th Pacific Joural of Scic ad Tcholog 57 Volum 3. Numbr. Novmbr Fall

7 8. Stph, M.P. 83. To Comput Numricall, Cocpts ad Stratg. Littl Brow ad Compa. Ottawa, Caada. ABOUT THE AUTHORS Suda Fadugba, is a Lcturr i th Dpartmt of Mathmatical ad Phsical Scics, Af Babalola Uivrsit, Ado Ekiti, Nigria. H is a rgistrd mmbr of Joural of Mathmatical Fiac. H holds a Mastr of Scic M.Sc. i Mathmatics from th Uivrsit of Ibada, Nigria. His rsarch itrsts ar i Numrical Aalsis ad Fiacial Mathmatics. SUGGESTED CITATION Fadugba, S., B. Ogurid, ad T. Okulola.. Eulr s Mthod for Solvig Iitial Valu Problms i Ordiar Diffrtial Equatios. Pacific Joural of Scic ad Tcholog. 3:5-58. Pacific Joural of Scic ad Tcholog Dr. Mrs. Bosd Ogurid, is a Lcturr I i th Dpartmt of Mathmatical Scics, Ekiti Stat Uivrsit, Ado Ekiti, Nigria. Sh holds a Ph.D. dgr i Mathmatics. Hr rsarch itrsts ar i Ordiar Diffrtial Equatios ad Numrical Aalsis. Tao Okulola, is a Lcturr i th Dpartmt of Mathmatical ad Phsical Scics, Af Babalola Uivrsit, Ado Ekiti, Nigria. H holds a Mastr of Scic i Mathmatics from Uivrsit of Ibada, Nigria. His rsarch itrst is i Numrical Aalsis. Th Pacific Joural of Scic ad Tcholog 58 Volum 3. Numbr. Novmbr Fall