Research Article Implementing a Type of Block Predictor-corrector Mode for Solving General Second Order Ordinary Differential Equations

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1 Researc Joural o Appled Sceces Egeerg ad ecolog (7): DOI:96/raset745 ISSN: ; e-issn: Mawell Scetc Publcato orp Submtted: September 9 5 Accepted: October 5 Publsed: Aprl 5 6 Researc Artcle Implemetg a pe o Bloc Predctor-corrector Mode or Solvg Geeral Secod Order Ordar Deretal Euatos Jmevwo G Ogoo Solomo A Ouuga ad Ncolas Amewa Omoregbe Departmet o Matematcs ollege o Scece ad ecolog oveat Uverst Ota Ogu State Ngera Departmet o Matematcs Uverst o Lagos Lagos Ngera Abstract: e paper s geared towards mplemetg a tpe o bloc predctor-corrector mode capable o tegratggeeral secod order ordar deretal euatos usg varable step sze s tecue wll be carred out o ost problems e mode wc emaated rom Mle s estmate as ma computato advatages suc as cagg ad desgg a sutable step sze correctg to covergece error cotrol/mmzato wt better accurac compare to oter metods wt ed step sze Moreover te approac wll adopt te estmates o te prcpal local trucato error o a par o eplct (predctor) ad mplct (corrector) Adams aml wc are mplemeted P(E) m mode Numercal eamples are gve to eame te ecec o te metod ad compared wt subsstg metods Kewords: Ad prase bloc predctor-corrector mode correctg to covergece ost problems prcpal local trucato error varable step sze tecue INRODUION Rsg rom te advet o computg maces ad programmg laguages te umercal soluto o Ital Value Problems (IVPs) or Ordar Deretal Euatos (ODEs) as bee te topc to eplore b umercal aalsts mostl procedures or te umercal soluto o te geeral secod-order ODEs o te orm as see Ke et al (): ad ( ) R R m R m (a) α ( a) β [ a b] : () e geeral soluto to () ca be coded as: α β - () were te step sze s α α β are uow costats wc are uuel speced suc tat te ormula s o order as dscussed Aewa et al () We assume tat R s sucetl deretable o [a b] ad satses a global Lpctz codto e tere s a costat L suc tat: ( ) L ( ) R Uder ts presumpto E () assured te estece ad uueess deed o [a b] as dscussed Lambert (97) ad Xe ad a (4) were a ad b are te ad (I) [ () () () ] or () ad [ ] Aga Weerstrass appromato teorem stads as a ustcato or () See (Ja et al 7) or detals E () arses rom ma pscal peomea a broad compass o applcatos Largel te eld scece ad egeerg suc as te electrc crcuts damped ad udamped mass-sprg sstems orced oscllatos ad oter areas o practcal applcatos as troduced b Mad et al () Autors suc as Aae et al () Ismal et al (9) ad Mad ad Sulema (9) dcated tat () ca be reduced to a euvalet rst-order sstem o two tmes te dmeso ad evaluated utlzg te estg oe-step metod le Ruge-Kutta metod or bloc multstep metod s metod as bee descrbed to crease te dmeso o te problem computatoal eect ad ver dcult Bloc multstep metods are oe o te umercal metods wc ave bee suggested b several researcers (Adesaa et al ; James et al ; Ke et al ; Mad et al ; Mad ad Sulema 9; Zara et al 7) e commo bloc metods used to solve te problems ca be arraged to categores as oe-step bloc metod ad orrespodg Autor: Jmevwo G Ogoo Departmet o Matematcs ollege o Scece ad ecolog oveat Uverst Ota Ogu State Ngera s wor s lcesed uder a reatve ommos Attrbuto 4 Iteratoal Lcese (URL: ttp://creatvecommosorg/lceses/b/4/) 76

2 Res J Appl Sc Eg ecol (7): multstep bloc metod Neverteless scolars ave proposed a alteratve metod to solve at oce () as dscussed Adesaa et al ( ) Aae et al () Ege et al () Ismal et al (9) Ke et al () ad Mad et al () However (Adesaa et al ; Ege et al ; Ismal et al 9; Ke et al ) proposed bloc multstep metods wc were emploed predctor-corrector mode Bloc multstep metods ave te vatage o evaluatg smultaeousl at all pots wt te tegrato terval tereb reducg te computatoal burde we evaluato s reured at more ta oe pot wt te grd Aga alor seres epaso s used to provde te tal values order to compute te corrector Researcers suc as Adesaa et al () Ege et al () Ismal et al (9) ad Ke et al () ovated bloc predctor-corrector metod wc at eac practcal applcato o te metod te metod was ol teded to predct ad correct te results geerated I ts stud te motvato s stemmed b te act tat tere are ver ew wor bee doe solvg ost ODEs usg bloc predctor-corrector mode eort wll be geared towards developg a tpe o bloc predctor-corrector mode usg varable step sze tecue oterwse called Mle s estmate s metod ave several beets as earler stated te abstract Deto : Accordg to Aewa et al () A bloc-b-bloc metod s a metod or computg vectors Y Y seuece Let te r-vector (r s te umber o pots wt te bloc) Y µ µ ad G µ or mr m be gve as: Y ( ) r w ( r w + e te l -bloc r-pot metods or () are gve b: ) soluto Y to te soluto o () at pots as Aewa et al () e ma am o ts stud s to troduce a tpe o bloc predctorcorrector mode usg varable steps sze tecue or matematcall tegratg geeral secod order ODEs drectl DERIVAION O HE MEHOD Adoptg (Aewa et al ) ts secto te target s to derve te prcpal bloc predctorcorrector mode o te orm () We move aead b seeg a estmate o te eact soluto () b assumg a cotuous soluto Y() o te orm: Y( ) + mϑ ( ) () Suc tat [a b] m are uow coecets ad ϑ () are polomal bass uctos o degree + - were s te umber o terpolato pots ad te collocato pots are respectvel cose to sats ad e teger deotes te step umber o te metod We tus costruct a -step bloc metod bloc metod wt ϑ () b mposg te ollowg codtos: - ( ) m - - m( - - )( ) Z - (4) (5) were s e appromato or te eact soluto ( ) ( ) s te grd de ad + It sould be oted tat E (4) ad (5) leads to a sstem o + euatos o te AX B were ( ) Y A Y + w ( ) B w w were A () B () ( -( ( A are r b r matrces as ( ( ( troduced b atula (99) us rom te above deto a bloc metod as ( -) ( ( X te vatage tat eac applcato te soluto s [ ] appromated at more ta oe pot smultaeousl U [ ] (6) e umber o pots depeds o te maer o costructo o te bloc metod ereore applg Solvg E (6) usg Matematca we get te tese metods ca gve ucer ad aster solutos to coecets o m ad substtutg te values o m to te problem wc ca be maaged to produce a (4) ad ater some algebrac computato te bloc desred accurac See (Mad ad Sulema 7; predctor-corrector mode s obta as: Meraoo et al ) e bloc algortm proposed ts stud s based (7) + α - β - β o terpolato ad collocato e cotuous represetato o te algortm geerates a ma dscrete collocato metod to reder te appromate were α ad β are cotuous coecets

3 ANALYSIS O SOME HEOREIAL PROPERIES Order o accurac o te metod: oormg to Aewa et al () ad Lambert (97) we spec te assocated lear multstep metod (7) ad te derece operator as: L[ ( ); ] [ ( + ) + ( + ) ] α Res J Appl Sc Eg ecol (7): β (8) Presumg tat () s sucetl ad cotuousl deretable o a terval [a b] ad tat () as as ma ger dervatves as demaded te we wrte te codtos (8) as a alor seres epresso o ( ) ad ( ) " ( ) as: ( )! ( ) ( ) ( ) ad ( ) ( ( )! + ) (9) ( ) Substtutg (8) ad (9) to (7) we obta te ollowg epresso: L[ ( ); ] c ( p+ ) () ( ) + ( ) + + p+ ( ) + p + c c () Hece we otced tat te Predctor-orrector Mode (P(E) m ) o (7) as order p p+ p I are gve as ollows: c α + α + α + + α c α α α c ( ) - ( )! α + α + α + + α β + β + β + + β - c ( ) - (! α + α + + β α + ( - )! β β ereore te metod (7) as order p ad error costats gve b te vector p+ ocurrg wt Lambert (97) we sa tat te metod () as order p : p + L[ ( ); ] O( ) p p+ p+ ) () ereore p+ s te error costat ad p+ p+ (p+) (X ) s te prcpal local trucato error at te pot Stablt aalss o te metod: o eame te metod or stablt (7) s ormalsed ad composed as a bloc metod gve b te matr te derece euatos as preseted Aewa et al () Ke et al () Moammed et al () ad Awar () () A Y A Y B () () + + m m- m B () ( ) m- () 78 were Y m r Y m r+ r+ m r m e matrces A () A () B () B () are r b r matrces wt real etres wle Y m Y m- m m- are r-vectors speced above ollowg (Ke et al ; Lambert 97) we stc to te boudar locus metod to decde o te rego o absolute stablt o te bloc predctorcorrector mode ad to obta te roots o absolute stablt Substtutg te test euato -λ ad λ to te bloc metod () to obta: r+ r+ () () () () [ r( + )-( )] ρ( ) det A B λ A - B λ r () Replacg () we obta all te roots o te derved euato to be r ereore accordg to Lambert (97) te predctor-corrector mode s absolutel stable So as cosdered Adesaa et al () Lambert (97) ad Awar () te boudar o te rego o absolute stablt ca be obtaed b llg (7) to: ρ( r) ρ( ( r) σ ( r) σ ( e e θ θ ) ) (4) θ Ad permt r e cosθ + sθ te ater reducto togeter wt smplg (4) wt [ 8 ] Accordgl te boudar o te rego o absolute stablt rests o te real as g s ree ad drawg IMPLEMENAION O HE MEHOD Embracg (ares ad Burde ; Lambert 97) aterward ts s mplemeted te P(E) m mode te t becomes ver pertet te eplct (predctor) ad te mplct (corrector) metods are dvduall o te same order ad ts prereuste maes t essetal or te step umber o te eplct (predctor) metod to be greater ta tat o te mplct (corrector) metod oseuetl te mode P(E) m ca be ormall decded as ollows or m : P(E) m : [] - - [ m] [ m] α + β [ s] [ ] s ( ) [ s+ ] [ m] [ s] + + α β β [ m-] } s m- (5)

4 Res J Appl Sc Eg ecol (7): te step wt a smaller step sze e step s accepted based o a test as prescrbed b (8) as Ascer ad Petzold (998) Euato (8) s te covergece crtera oterwse called Mle s estmate or correctg to covergece urtermore E (8) esures te covergece crtero o te mode durg te test evaluato NUMERIAL EXPERIMENS g : Sowg te rego o absolute stablt o te bloc predctor-corrector mode sce te root o te stablt polomal s r Note tat as m te result o calculatg wt te above mode wll cle to tose gve b te mode o correctg to covergece Moreover predctor-corrector par based o () ca be appled e mode P(E) m speced b (5) were s te step sze Sce te predctor ad corrector bot ave te same order p Mle s devce s applcable ad relevat Accordg to Dormad (996) ad Lambert (97) Mle s devce suggests tat t s possble to estmate te prcpal local trucato error o te eplctmplct (predctor-corrector mode) metod wtout estmatg ger dervatves o () Assume tat p p * were p * ad p represets te order o te eplct (predctor) ad mplct (corrector) metod wt te same order Now or a metod o order p te prcpal local trucato errors ca be well deed as: ( p+ ) p+ ( ) ( ) O( ) (6) W * p+ - + p+ e perormace o te bloc predctor-corrector mode (P(E) m ) was carred out o ost ad mldl st problems or problem tested 5 ad 5 te ollowg toleraces (covergece crtera) ad - were used to compare te perormace o te ewl proposed metod wt oter estg metods as James et al () ad Ke et al () ested problems: ( ) () ( ) e eact soluto s gve b: ( ) + + I ested problem: rgoometr problem (Nost): ω + ( )s ( ) +ω w ω [] e eact soluto s gve b: ( ) cos( ω ) + s( ω) + s Also: p p+ ( p+ ) p+ ( ) ( ) O( ) (7) were W ad are called te predcted ad corrected appromatos gve b metod o order p wle * p+ ad p+ are depedet o Neglectg terms o degree p+ ad above t s eas to mae estmates o te prcpal local trucato error o te mode as: ( ( ) p+ * W p+ p+ p+ p+ ) - p + (8) Notg te act tat p+ * p+ ad W urtermore te estmate o te prcpal local trucato error (8) s used to determe weter to accept te results o te curret step or to recostruct ost ODEs 79 e rst tested problem 5 to be dscussed was etracted rom James et al () Moreover our steps cotuous metod or te soluto o " ( ) was developed ad mplemeted usg ed step sze us te ewl proposed metod s ormulated to solve ost usg varable step sze tecue Secodl tested Problem 5 was etracted rom Ke et al () However te bloc metods or specal secod order ODEs was desged ad eecuted usg ed step sze tecue Moreover te mplemetato o te eplct -pot -bloc metod ad mplemetato o te eplct -pot -bloc metod was carred out usg lear derece operator as well as ter comparso e ewl proposed metod belogs to te aml o Adams oterwse called Mle s estmate ad was created to solve

5 able : omparg te umercal results o James et al () ad ewl proposed metod (BP) or solvg problem 5 (James et al ) ewl proposed metod (BP) X Mamum errors olerace levels Mamum errors 999 (-5) (-) 849 (-4) 47 (-) (-4) 4 67 (-) (-) -6 6 (-6) 6 6 (-) 7 5 (-) (-) (-) 9 76 (-) (-) 7 (-) able : omparg te umercal results o Ke et al () ad ewl proposed metod (BP) or solvg problem 5 OL MH MAXE - EPIB (-) EPIB 877 (-) BP 567 (-4) -4 EPIB (-4) BP 5 (-5) -5 EPIB 48 (-5) BP 9489 (-6) -6 EPIB 7664 (-6) BP 859 (-8) -8 EPIB 7668 (-8) EPIB 4856 (-8) BP 565 (-) - EPIB 5846 (-) BP 565 (-) ONLUSION Res J Appl Sc Eg ecol (7): e mplemetato o te tpe o bloc predctorcorrector mode (P(E) m ) was carred out o geeral secod order ODEs ad eecuted o ost problems ested Problems 5 ad 5 are good eamples o ost ODEs See (Lambert 97) or detals e ewl proposed metod s a tpe o bloc predctorcorrector mode (P(E) m ) oterwse called Mle s estmates rom able (James et al ) was mplemeted usg ed step sze wc does ot allow or step sze cages correctg to covergece error cotrol/mmzato Aga rom able (Ke et al ) was also eecuted usg ed step sze wc as usual mae room or step sze varato Neverteless ts caot be compared wt te result o te ewl proposed metod (BP) wc elds better accurac terms o te mamum error at all tested tolerace levels sce t was mplemeted usg varable step sze tecue I addto ts gves a better result at all tested tolerace levels Hece te ewl proposed metod (BP) s preerable applg varable step sze tecue troduced b Mle s e rego o absolute stablt or te tpe o bloc predctor-corrector mode we r eplas tat te metod was mplemeted o ost ODEs e bloc predctor-corrector mode s wrtte computer laguage usg Matematca ad mplemeted o wdows operatg sstem usg Matematca 9 Kerel e computatoal results or 7 tested problems 5-5 able ad are computed usg te bloc predctor-corrector mode as well as te metod James et al () ad Ke et al () NOAIONS OL : olerace level MD : Metod emploed MAXE : Magtude o te mamum error o te computed soluto BP : Bloc Predctor-orrector Mode (P(E) m ) AKNOWLEDGMEN e autors would le to ta te oveat Uverst or provdg acal support troug grats durg te stud perod REERENES Adesaa AO MR Odeule AO Adeee otuous bloc brd-predctor-corrector metod or te soluto o ( ) It J Mat Sot omput : 5-4 Adesaa AO MR Odeule ad MO Udo our steps cotuous metod or te soluto o ( ) Am J omput Mat : Aewa AO SN Jator ad NM Yao otuous bloc bacward deretato ormula or solvg st ordar deretal euatos omput Mat Appl 65: Aae A DO Awoem ad AO Adesaa A oe step metod or te soluto o geeral secod order ordar deretal euatos It J Sc ecol : 59-6 Ascer UM ad LR Petzold 998 omputer Metods or Ordar Deretal Euatos ad Deretal-algebrac Euatos Socet or Idustral ad Appled Matematcs Pladelpa Awar YS Dervato ad applcato o spot lear multstep umercal metod or soluto o secod order tal value problems IOSR J Mat 7: -9 Dormad JR 996 Numercal Metods or Deretal Euatos R Press New Yor Ege JO SA Ouuga ad AB Sooluwe -pot bloc metods or drect tegrato o geeral secod-order ordar deretal euatos Adv Numer Aal : 4 ares JD ad RL Burde Ital-value problems or ODEs: Varable step-sze multstep metods Dubl t Uverst Broos/ole atula SO 99 Bloc metods or secod order IVPs It J omput Mat 4: 55-6 Ismal YL Ke ad M Otma 9 Eplct ad mplct -pot bloc metods or solvg specal secod order ordar deretal euatos drectl It J Mat Aal : 9-54

6 Res J Appl Sc Eg ecol (7): Ja MK SRK Iegar ad RK Ja 7 Numercal Metods or Scetc ad Egeerg omputato 5t Ed New Age Iteratoal New Del James AA AO Adesaa ad SO Josua otuous bloc metod or te soluto o secod order tal value problems o ordar deretal euato It J Pure Appl Mat 8: Ke YL I Ismal ad M Sulema Bloc Metods or Specal Secod Order ODEs Lambert Academc Publsg Uverst Putra Malasa Lambert JD 97 omputatoal Metods Ordar Deretal Euatos Jo Wle & Sos Ic cester Mad ZA ad MB Sulema 7 Implemetato o our-pot ull mplct bloc metod or solvg ordar deretal euatos Appl Mat omput 84: 54-5 Mad ZA ad M Sulema 9 Parallel Drect Itegrato Varable Step Bloc Metod or Solvg Large Sstem o Hger Order ordar deretal euatos It J omput Mat Sc (): -7 Mad ZA NZ Motar ad M Sulema Drect two-pot bloc oe-step metod or solvg geeral secod-order ordar deretal euatos Mat Probl Eg : -6 Meraoo S ZA Mad ad M Sulema Varable step mplct bloc multstep metod or solvg rst-order ODEs J omput Appl Mat : Moammed AA A Hamza ad U Moammed A sel-startg brd lear multstep metod or a drect soluto o te geeral secodorder IVPs J Mat 4: 7- Xe L ad H a 4 otuous parallel bloc metods ad ter applcatos Appl Mat omput 4: 56-7 Zara BI IO Karl ad S Moamed 7 Varable step bloc bacward deretato ormula or solvg rst order st ODEs Proceedg o te World ogress o Egeerg : -6 7