A 5-Step Block Predictor and 4-Step Corrector Methods for Solving General Second Order Ordinary Differential Equations

Transcription

1 Global Joual o Pue ad ppled Matematcs ISSN Volume, Numbe (), pp 87-8 Reseac Ida Publcatos ttp://wwwpublcatocom -Step lock Pedcto ad -Step Coecto Metods o Solvg Geeal Secod Ode Oda Deetal Equatos Jmevwo G Ogoo*, Solomo debola Okuuga, Sela sop Depatmet o Matematcs, Coveat Uvest, Ota, Ogu State, Ngea *godwogoo@coveatuvestedug, selabsop@coveatuvestedug Depatmet o Matematcs, Uvest o Lagos, Lagos State, Ngea sokuuga@ulagedug bstact -step block pedcto ad -step coecto metods amed at solvg geeal secod ode oda deetal equatos dectl wll be costucted ad mplemeted o o-st poblems Ts metod, wc eteds te wok o block pedcto-coecto metods usg vaable step sze tecque possess some computatoal advatages o coosg a sutable step sze, decdg te stoppg ctea ad eo cotol I addto, some selected teoetcal popetes o te metod wll be vestgated as well as detemato o te ego o absolute stablt Numecal esults wll be gve to sow te ecec o te ew metod Kewods: pedcto-coecto metods, stoppg ctea, ego o absolute stablt, vaable step sze tecque Itoducto Ma poblems o scece ad egeeg ae educed to quatable om toug te pocess o matematcal modellg Te equatos asg ote ae epessed tems o te ukow quattes ad te devatves Suc equatos ae called deetal equatos Sce aaltcal metods ae ot adequate o dg accuate solutos to most deetal equatos, umecal metods ae equed Te deal obectve, emplog a umecal metod, s to compute a soluto o speced accuac to te deetal equato Sometmes ts s aceved b computg seveal solutos usg a metod wc as kow eo caactestcs as Jo

2 88 Jmevwo G Ogoo et al [] Ts pape cosdeed solvg dectl geeal secod ode oda deetal equatos o te om detola ad Odekule [] (,, ), ( ), ( ), a b () Numecal metods o solvg () ests lteatue aaa ad admus [] epoted tat () ca be educed to sstems o st ode equatos ad ote oe-step metods o solvg st ode equatos ae used Howeve, ake et al [] ad Mad ad Sulema [8] suggested tat educg () to te equvalet st ode sstem o twce te dmeso equatos ad te solved usg oe-step o multstep metod Ts tecque s ve well establsed but t appaetl wll cease te dmeso o te equatos ccodg to Ja ad Iega [], eplct ad mplct metods combed togete to obta a ew metods Suc metods s called te Pedcto-Coecto Metods Scolas suc as Lambet [] ad Lambet [] ave suggested tat ts tus out to be a advatage avg te pedcto ad te coecto o te same ode ga, te pedcto-coecto pa s appled te mode o coectg to covegece wc s oe o te most mpotat aspect o te pedcto-coecto metods detola ad Odekule [] ad detola et al [] sted te mao setback o te pedcto-coecto mode s te cost o developg suboute utemoe, ts suboute developed ae o lowe ode to te coecto, tus, t as geat cosequece o te accuac o te coecto esults Te lock multstep metods ae oe o te umecal metods wc ave bee suggested b seveal eseaces, see detola ad Odekule [], detola et al [], James et al [], Mad et al [7], Mad ad Sulema [8] ad Zaa et al [] Te commol block metods used to evaluate () ca be categose as oe-step block metod ad multstep block metod ga, block metods was poposed b scolas to cate o te sotcomg o pedcto-coecto metod, sce block metod povdes solutos at eac gd wt te teval o tegato wtout ovelappg teeb eadcatg te dea o suboute Scolas suc as detola ad Odekule [], Ege et al [], Ismal et al [9] ad Ke et al []poposed block multstep metods wc wee appled pedcto-coecto mode lock multstep metods ave te advatage o evaluatg smultaeousl at all pots wt te tegato teval, teeb educg te computatoal bude we evaluato s eeded at moe ta oe pot wt te gd ga, statg values ae povded b Talo sees epaso ode to compute te coecto metod Reseaces detola ad Odekule [], Ege et al [], Ismal et al [9] ad Ke et al []mplemeted block pedcto-coecto metod wc at eac pactcal applcato o te metod, te metod was ol teded to pedct ad coect te esults geeated I ts pape, te motvato s stemmedom te act tat block pedcto-coecto metods appled b deet autos eve supass ts advatage as suggested above, wc makes te block pedcto-coecto metod to be udeutlzed Hece, tee s a eed to popose a tpe o block pedcto-coecto metod te om o -step block pedcto (eplct dams-asot) ad -step coecto metods (mplct dams-moulto) mplemeted usg vaable step sze tecque

3 -Step lock Pedcto ad -Step Coecto Metods 89 Ts metod possess te ollowg advatages suc as cagg te stepsze ad detemg o a sutable step sze o te block pedcto-coecto metod, coosg te stoppg ctea ad eo cotol o mmzato Justcato We st state te teoem tat demostates te uqueess o solutos o ge ode oda deetal equatos Teoem (Estece ad Uqueess) Let (, ) be deed ad cotuous o all pots (, ) te ego D deed b a b,, wee a ad b ae te, ad let tee ests a costat L suc tat o a [ a, b] ad a two umbes ad *, (, ) (, L * ) * Ts codto s kow as Lpctz codto Te tee ests eactl oe ucto () wt te ollowg ou popetes: () () s cotuous ad deetable o [ a, b], () ( ) (, ( )), [ a, b ] () ( ) (, ( ), ), [ a, b], (v) (a) ad () See Ke et al [] ad Wed [] o detals Teoem (Weestass) Te Weestass appomato teoem states tat a cotuous ucto () ove a closed teval [ a, b] ca be appomated b a polomal (), [ a, b o degee, suc tat ( ) ( ), [ a, b] P P ] Wee s a small quatt ad s sucetl lage, see Ja et al [] Teoetcal Pocedue I ts poposed stud, we seek to eame dectl te geeal secod ode ODEs o te om (,, ), ( ), ( ), a b () Te soluto to () ma wtte as ()

4 8 Jmevwo G Ogoo et al Wee (, ( )), ad ae costat ad assume tat, Sce () ca be multpled b te same costat wtout alteg te elatosp ae abta to te etet o a multplcato costat Te abtaess as bee emoved b assumg tat Metod () s eplct ad mplct as toduced Ke et al [], Lambet [] ad Lambet [] Ts stud s ocused o te use o dam s metod o vaable step sze tecque developg a tpe o -step block pedcto ad step coecto metods o solvg geeal secod ode ODEs otwt Te metod wll be costucted based o tepolato ad collocato appoac usg powe sees as te appomate soluto o te poblem as stated Ege et al [], aes ad ude [7], Lambet [] ad Lambet [] Tus,ts powe sees soluto ca be wtte te om a ( ) () omulato o te Metod ccodg to Ismal et al [9] ad Ke et al [], a -pot block metod, te teval s dvded to subtevals o blocks wt eac teval cotag two pots, e ad te st block wle ad te secod block wee solutos to () ae to be computed Te metod wll omulate two ew evel spaced soluto values cocuetl Smlal, ts ca be eteded to a -pot oe block metod wee te backwad ad owad values ae te pots o tepolato ad collocato as well as evaluato Repesetato o -Pot lock Metod om atula [] ad Ke et al [], te k-pot block metod o () s gve b te mat te deece equato () ( ) m m Wee, ( ) m,,, m m m m ( o m,m,, ), ()

5 -Step lock Pedcto ad -Step Coecto Metods 8 ( ) ad ( ) ae matces It s assumed tat mat te deece equato s omalzed so tat ( ) s a dett mat Te block sceme s eplct te coecet mat ( ) s a ull mat Devato o -Step lock Multstep Pedcto Metod s detola ad Odekule [] ad Ege et al [], tepolatg () at o () ad collocatg () at o () gves a sstem o equatos wc ca be epessed as X U a 8, a, a, a, a, a, a,,,, T X () T U,, Solvg () ad substtutg te values o () to () gves a cotuous lea multstep metod o te om ( ) (7) Evaluatg (7) at pots o (), we obta te coveget -step block multstep pedcto metod as 99 (8) doptg atula [] ad Ke et al [], te -step block multstep pedcto metod ca be wtte mat te deece equato as () () m m m () Deetatg (7) oce ad evaluatg at te same tee dscete pots o,, o,,,, o, we obta a block o st ode devatve wc (9)

6 8 Jmevwo G Ogoo et al ca be used to deteme te devatve tem te tal value poblem (), see Ege et al [] () Devato o - Step lock Multstep Coecto Metod Itepolatg () at o () ad collocatg () at, o () gves a sstem o equatos wc ca be epessed as X U as dscussed above a 8 8, a, a, a, a, a, a,,,, T X T () U,, Solvg ()ad substtutg te values o () to () elds a cotuous lea multstep metod o te om ( ) () Evaluatg () at pots o (), we obta te coveget -step block multstep coecto metod as 9 8 () s usual, atula [] ad Ege et al [] stated tat te -step block multstep coecto metod ca be wtte mat te deece equato as () () m m m m () () ()

7 -Step lock Pedcto ad -Step Coecto Metods 8 Deetatg () oce ad evaluatg at te same tee dscete pots o,, o we ave a block o st ode devatve wc ca be used to deteme te devatve tem te tal value poblem () () Ivestgato o te Teoetcal Popetes o te Metods Ode o te Metod DetoEmbacgJa et al [], Lambet [] ad Lambet [], te lea k- step metod o (8) ad () wt assocated deece opeato L ( ); ( ) ( ) () wee () s a abta ucto, cotuousl deetable o a teval a, b I we assume tat () as a ma ge devatves as we eque, te, epadg usg Talo sees about te pot, we obta C C C q q q C q q! ( q )! ollowg Lambet [], we sa tat te metod as ode p C C C C p C p, C p C q, q,, s te te eo costat ad ( ) te pcpal local tucato eo at te pot p ( p ) p Combg Ja et al [], Lambet [] ad Lambet [], we otced tat te block multstep metod o (8) ad () as ode p,, p p C C C C C

8 8 Jmevwo G Ogoo et al Teeoe, we cocluded tat te metods (8) ad () ave ode p= ad eo costats gve b te vectos, C 7 7, 7, 7 T C 7 8,, Covegece geeg to Hae et al [8], Ke et al [] ad Lambet [], te multstep metod s coveget, te t s ecessal () stable ad () cosstet (e o ode : T (7) ad (), () (), () ()) (8) Zeo Stablt Teoem (st Root Codto) om uce [] ad Ke et al [], te multstep metods (8) ad () ae stable all te oots o te caactestc polomal () sats ad te must be a smple oot Deto s Hae et al [8] ad Moammed et al [9], te multstep metod (7) s called stable, te geeatg polomal ( ) det () E (9) satses te st oot codto, e, () Te oots o () le o o wt te ut ccle; () Te oots o te ut ccle ae smple I ode to aalze te metods o zeo-stablt, equato (8) ad () ae bot omalze ad wtte as a block metod gve b te mat te deece equatos as dscussed Moammed et al [9] () () m m () () m m () () m m () () I addto, te zeo stablt s coceed wt te stablt o te deece sstem te lmt as teds to zeo Tus, as, ( ), () z m m ()

9 -Step lock Pedcto ad -Step Coecto Metods 8 wee s te ode o te deetal equato, z s te ode o te mat ( ) ad E, see detola et al [], Moammed et al [9] ad Sa [] o detals o ou metod ( ) Solvg te mat equato above, () () ( ) det () gves ( ) () Hece, ou metod s zeo stable accodg to [8, ] Cosstec Teoem ccodg to uce [], Ke et al [] ad Lambet [], a lea multstep metod s cosstet t as ode geate ta o equal to Tus k a a, k b, () I tems o te caactestc polomal, te metod s cosstet ad ol (), () (), () () () Deto Te lea multstep metod (7) s sad to be cosstet povded ts eo ode p satses p It ca be sow tat ts mples tat te st ad secod caactestcs polomal ae ullled as see () Sce te block multstep metods (8) ad () ae cosstet as t as ode p doptg Hae et al [8] ad Lambet [], we ca deduce te covegece o te block multstep metods (8) ad () Rego o bsolute Stablt o te Metod Teoem (Secod Root Codto) om uce [], te lea multstep metod (7) s absolutel stable all te oots o te caactestc polomal ( ) ( ) z ( ) () sats

10 8 Jmevwo G Ogoo et al Deto om detola et al [], Ke et al [] ad Lambet [], te lea multstep metod (7) s sad to be absolutel stable o a gve, o tat, all te oots o () sats, s s,,, ; wee s ad Howeve, we coose ad ollow te bouda locus metod to deteme te ego o absolute stablt o te block metods ad to obta te oots o absolute stablt, we substtute te test equato ( ) det () m to te block omula to obta () () () ( ) ( ) ( ) ( ) (7) m m m Substtutg (7), we obta all te oots o te deved equato to be equal to ; ece, accodg to [] deed o teoem, te block metod s absolutel stable Teeoe, te bouda o te ego o absolute stablt s gve b ( ) ( ) (8) ( ) 7 Let e cos s, teeoe () becomes cos cos ( ) (9) cos cos cos cos 8 7 () Evaluatg () at wt [, 8 ], wc gves te teval o absolute stablt to be 9, ate evaluato at teval o ( ) Te stablt ego s sow gue ad te eclosed ego sde te bouda gue demostate te ego o absolute stablt o te poposed metod

11 -Step lock Pedcto ad -Step Coecto Metods 87 g sowg te ego o absolute stablt o te block pedcto-coecto mode, sce te oot o te stablt polomal s Note: gue s a eead dawg Implemetato o te Vaable Step-Sze Tecque doptg aes ad ude [7] ad Lambet []: () Pedcto-Coecto tecques alwas geeate two appomatos at eac step, so te ae atual caddates o eo-cotol adaptato () To demostate te eo-cotol pocedue, a vaable step-sze pedctocoecto metods usg -step eplct dams-asot metod as pedcto wle te -step mplct dams-moulto metod as coecto metods ae costucted stl, te -step pedcto Local Tucato Eo (LTE) s p ( t ) w 7 C p ()

12 88 Jmevwo G Ogoo et al Secodl, te -step coecto Local Tucato Eo (LTE) s t c w ( ) 7 C p () Wee te -step pedcto ad -step coecto metods use ts assumpto suc tat te appomatos w, w, w ae all eact, w p ad w c epesets te pedcted ad coected appomatos gve b te -step pedcto ad -step coected metods To poceed ute, we must make te assumpto tat o small valueso, we ave 7 u 7 u Te eectveess o te eo-cotol tecque depeds dectl ots assumpto O subtactg () om () ad combg te local tucato eo estmates, we ave c w w p C p Teeoe, elmatg tem volvg 7 appomato to te -step coecto local tucato eo: C c w w p 7 () () () elds all te ollowg ( ) p () Equato () s dam s estmate o coectg to covegece wc s bouded b a pescbed toleace I addto, te eo estmate () s used to decde wete to accept te esults o te cuet step o to edo te step wt a smalle step sze Te step s accepted based o a test as descbed b () as see U ad Lda [] s U ad Lda [] ad Zaa et al [], vag te step sze s cucal o te eectve peomace o a dscetzato metod Step sze adustmet o -step pedcto ad -step coecto block multstep metods usg vaable step as bee stated eale O te gve step, te use wll povde a pescbed toleace I te block multstep, vaable step-sze stateg codes, te block solutos ae accepted te local tucato eo, LTE s less ta te pescbed toleace I te eo estmate s geate ta te accepted pescbed toleace, te value o s eected, te step s epeated wt alvg te cuet step sze o otewse, te step s multpl b Te eo cotols o te code was at te st pot te block because geeal t ad gve us bette esults accodg to te ew metod utemoe, equato () guaatees te covegece cteo o te metod dug te test evaluato all, a umbe o appomato assumptos ave bee made tsdevelopmet, so pactce a ew step sze (q) s cose cosevatvel, ote as

13 -Step lock Pedcto ad -Step Coecto Metods 89 q w c w p Equato () s used decdg a ew step sze o te metod () Test Poblems Te peomace o te -step block pedcto ad -step coecto metods was caed out o o-st poblems o poblem ad te ollowg toleaces, 8,,, ad was used to compae te peomace o te ewl poposed metod wt ote estg metods as [] Poblem : Oscllato poblem Te st poblem to be cosdeed s olea ad was etacted om Vgo ad Ramos [] Ts was solved wt dea o dam s (block multstep, pedctocoecto) metods usg vaable step sze tecque alke metods pedctocoecto mode (PEC) usg vaable step sze was a multstep sceme Table dsplas esults o te compasos o te code o vaable-ode, vaable-step (VOVS), te vaable step sze Stome metod wt odes =, 8, alke s metod ad dam s metod pedcto-coecto mode wt te same ode, vaable step sze Ts poblem s epeseted te om s, (), () Poblem : Va de Pol oscllato Poblem was etacted om Vgo ad Ramos [] Howeve, alke metod o ode egt was desged ad eecuted o k-step pedcto-coecto metods usg vaable step sze epeseted multstep om Te ewl poposed block multstep, pedcto-coecto metods belogs to te aml o dams ad was ceated to solve geeal secod ode ODEs usg vaable step sze tecque Neveteless, te wellkow Vade Pol oscllato s gve b ( ), (),, [,], wee was solved wtout te dampg tem Te compute codes ae wtte Matematca ad mplemeted o wdows opeatg sstem usg Matematca 9 keel Te computatoal esults o poblem - Tables - ae computed usg te poposed metod as well as te metod Vgo ad Ramos []

14 8 Jmevwo G Ogoo et al Numecal Results Notatos TOL: Toleace Level MTD: Metod Emploed MXE: Magtude o te Mamum Eo o te Computed Soluto MPC: dam s lock Multstep Pedcto-Coecto Metods Table Numecal esults o Vgo ad Ramos [] ad MPC o solvg poblem MTH TOL Mamum Eos VOVS STOR() LK() MPC() VOVS STOR(8) LK(8) MPC() Table Numecal esults o Vgo ad Ramos [] ad MPC o solvg poblem MXE TOL MXE Dscussos ad Cocluso om Table, Vgo ad Ramos [] was mplemeted usg alke s metod o vaable step sze tecque wc s a multstep sceme Te mplemetato o a eplct ad mplct multstep (pedcto-coecto) metod as a sgle umecal soluto caot be compaed to a block pedcto-coecto metods wose soluto pots us smultaeousl geeatg te equed esults Hece, te ewl

15 -Step lock Pedcto ad -Step Coecto Metods 8 poposed -step pedcto ad -step coecto metods ae peeable applg vaable step sze tecque toduced b dam s ga, om Table, Vgo ad Ramos [] eecuted alke s metod o eplct ad mplct metods emplog vaable step sze tecque Moeove, ts caot be compaed wt te esult o te ewl poposed block multstep, -step pedcto ad -step coecto metods wc elds bette accuac tems o te mamum eo at all tested toleace levels, sce t was mplemeted usg vaable step sze tecque I addto, ts gves a bette esult at all tested toleace levels all, te mplemetato o te dam s metod o vaable step sze tecque comes wt a uge task especall decdg o a sutable ew step sze o eecutg te poblem ltoug, ts s te pce o vaable step sze tecque, but eveteless, ts elds te desed esult wt bette accuac ckowledgemet Te autos would lke to tak Coveat Uvest o te acltes ad Cete o Reseac, Iovato ad Dscove (CUCRID), Ota Ngea o acal suppot Reeeces [] O detola ad M R Odekule, Cotuous lock Hbd-Pedcto- Coecto Metod ote Soluto o, Iteatoal Joual o Matematcs ad Sot Computg, ()(): - [] O detola, M R Odekule, ad M O Udo, ou Steps Cotuous Metod o te Soluto o, meca Joual o Computatoal Matematcs, (): 9-7 [] T ake, D O woem, ad O desaa, Oe Step Metod o te Soluto o Geeal Secod Ode ODEs, Iteatoal Joual o Scece ad Tecolog, ()(): 9- [] E S uce, Te Computable Deetal Equato, Caloa State Uvest Notdge, (7), pp - [] J O Ege, S Okuuga, ad Sooluwe, Tee-Pot lock Metods o Dect Itegato o Geeal Secod Ode Deetal Equatos,dvaces Numecal alss, (): - [] S O atula, lock Metods o Secod Ode ODEs, Ite J Compute Matematcs, ()(99), pp - [7] J D aes, R L ude, Ital-Value Poblems o ODEs, Vaable Step- Sze Multstep Metods, Dubl Ct Uvest, (), pp - [8] E Hae, S P Nosett, ad G Wae, Solvg Oda Deetal Equatos I, Nost Poblems, Spge-Velag, el (987), pp 79-8 [9] Ismal, L Ke, ad M Otma, Eplct ad Implct -Pot lock Metods o Solvg Specal Secod Ode Oda Deetal Equatos Dectl, It Joual o Mat alss, ()(9): 9-

16 8 Jmevwo G Ogoo et al [] M K Ja,S K Iega, ad R K Ja, Numecal Metods o Scetc ad Egeeg Computato t ed, (7), pp 8-8 [] R K Ja ad S R K Iega, Numecal Metods, (9), pp - [] James, O desaa, ad S Josua, Cotuous lock Metod o te Soluto o Secod Ode Ital Value Poblems o ODEs, Iteatoal Joual o Pue ad ppled Matematcs, 8()(): - [] R D Jo, Numecal Metods o Deetal Equatos, New ok, (99), pp - [] L Ke, I Ismal, ad M Sulema, lock Metods o Specal Secod Ode ODEs, PD Tess(), Uvest Puta Malasa [] J D Lambet, Computatoal Metods Oda Deetal Equatos, Jo Wle & Sos, New ok, (97), pp -8 [] J D Lambet, Numecal Metods o Oda Deetal Sstems, Jo Wle & Sos, New ok, (99), pp - [7] Z Mad, N Z Mokta, ad M Sulema, Dect Two-Pot lock Oe- Step Metod o Solvg Geeal Secod-Ode Oda Deetal Equatos, Matematcal Poblems Egeeg, (): - [8] Z Mad ad M Sulema, Paallel Dect Itegato Vaable Step lock Metod o Solvg Lage Sstem o Hge Ode ODEs, dvaced Tecologes, Kakesu Jaatakumaa (Ed), (9): 8- [9] Moammed, Hamza, ad U Moammed, Sel-Statg Hbd Lea Multstep Metod o a Dect Soluto o te Geeal Secod-Ode IVPs, Joual o Matematcs, (IOSR-JM), ()(): 7- [] Sa, Devato ad pplcato o S-Pot Lea Multstep Numecal Metod o Soluto o Secod Ode IVPs, Joual o Matematcs (IOSR-JM), 7()(): -9 [] M Uad R P Lda, Compute Metods o ODEs-Deetal lgebac Equatos, Pladelpa, US, (998), pp - [] J Vgo-gua ad H Ramos, Vaable Step Sze Implemetato o Multstep Metods o (,, ), Joual o Computatoal ad ppled Matematcs, 9 (): - [] D V V Wed, Uqueess o Solutos o ODEs, meca Matematcal Motl, 7 (97): 98-9 [] aaa ad M admus, Class o Collocato Metods o Geeal Secod Ode ODEs, ca Joual o Matematcs ad Compute Scece Reseac, ()(9): 9-7 [] I Zaa, I O Kal, ad M Sulema, Vaable Step lock ackwad Deetato omula o Solvg st Ode St ODEs, Poceedgs o te Wold Cogess o Egeeg, (7): -