Application of second derivative Runge-Kutta collocation methods to stiff systems of initial value problems

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eoado Joal o See ISSN 8- Ie 8 Jaa-Je. - Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada Glb AKUBU * Deame o aema ad Sa Ue o ad PB 9 ad Boo Sae Nea Deame o aemaal See Abbaka Taawa Balewa Ue PB 8 Ba Ba Sae Nea E-mal: dadaakb@mal.om &malle@mal.om *Coeod ao: Poe: 889 & 88 Aba I ae we ode e oo o ml eod deae Re-Ka olloao meod deed o e oo meal olo o em o ode al ale oblem oda deeal eqao. Tee meod ae obaed baed o e mle olloao eqe w ae ow o be oee w moed eo o abole abl. Alo e mlemeao o e eod deae Re-Ka olloao meod ema eae de o e ml ae o e meod e adaae aed make em able o ol em w eeale o lae modl l loe o e maa a. Some abole abl aae ad ode o aa o e meod ae ded. Fall we ow wo oble wa o mleme e meod ad omae em o ome meal eamle od e leae o demoae e ode o aa ad elabl o e meod. Kewod Blok bd ema; Coo ema; le olloao omla; Seod-deae Re-Ka meod; S em ://l.aademde.o/

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Iodo Samala ARKUS ad Dada G. AKUBU I e a ew deade ma ao ooed dee meal eao meod o oba moe aae aomae olo o oda deeal eqao ODE eeall em o al ale oblem oda deeal eqao. Howee eee meal eeme ae ow a ml eao ae e meal eo oe e el meod we aled o ole em o ODE [-]. I ala ml Re-Ka meod la a moa ole e meal olo o em o ODE beae e ae ood abl oee ad ode o oeee. I ae we dee ml eod-deae Re-Ka SDRK olloao meod deed o e oo meal olo o em o al ale oblem ODE o e om d d [ T] Hee e kow o a ma :[ o T] R d. ad e -ad de o :[ T] R d R d w amed o be el moo R d e e al ale ad d/d e deeal oeao. e > be a oa e ze ad we dee e d b..n wee NT- ad a e o oml aed o o e eao eal deed b < < <..< T Te ma eao o ode e eod deae em o dee a e o meod w ae able o e oo meal olo o deeal em w Jaoba a lae eeale l loe o e maa a. Howee o ome moa lae o oblem eea o e ake o ee o allow eod-deae eao meod o be ed ee [-8] ad ae e meal meod ed o be m le aao. Fe we wll eame deal e oblem o mleme e eoddeae Re-Ka eao meod w ed me e. We ow a ee o eomo a ee a be aeed e meod ae mlemeed a aoae wa ee ae ll ome moa aal oblem o be oeome o eamle e allao o e eod deae em e meod w o lle e a e deae em [9]. Tee ae eeal eme ao wo deed meod a a adle o- eod ad ollao oblem ad [-9] b lle eem o ae bee doe de ml eod-deae Re-Ka olloao meod w mmal o

eoado Joal o See ISSN 8- Ie 8 Jaa-Je. - ealao oml aae ode ad mamal a ee. Hee ee moae o ode e oo o e ml eod deae Re-Ka olloao meod ae. Deo.: A meal meod ad o be A-able π/ eo o abole abl oa e e wede W wee W { λ : < π a λ } ee []. < Teoem.: I ae z odo w oa e e al ale oblem oee a qe olo o e eal [ T] ee []. Deo.: A olo o. ad o be able e a > ee δ> a a oe olo o. w ae a ˆ a alo ae ˆ δ.a.b o all >a. Te olo amoall able addo o.b a. aeal ad meod Te ml eod deae Re-Ka olloao meod I eo o obee o debe e oo o e ml eoddeae Re-Ka olloao meod baed o e mle olloao eqe. I ead we eek a aomae olo o e ea olo o. b e eola o e om. w we ool deeable. We e e m o be eqal o o a o be able o deeme { }.. I omlao deoe e mbe o eolao o ed ad > >ae d olloao o. Ieola. a e o { } ad olloa ad a e o { } we oba e ollow em o eqao

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada G. AKUBU K. K.. K. I a eqao. o. a be eeed e ma-eo om a D.7 wee e -qae ma D - eo ad ae deed a ollow: D D D D D O O O.8 T T K K K K wee D - ad D --.8 eee e ad eod deae eeel ad oeod o e deeao w ee o. Smla o e Vademode ma e ma D.7 o-la. Coeqel eqao.7 a e qe olo e b C wee CD-.9 Reaa eqao.7 o.9 we oba e mle olloao omla o e e [] w wa a eealzao o [] ad ee we eed o eod deae a ollow.. wee:

eoado Joal o See ISSN 8- Ie 8 Jaa-Je. - 7 d d Hee ae oo oee o e meod w ae o be deemed. Te ae amed olomal o e om. Te meal oa oee. ae o be deemed. I a e aboe oee a be obaed om e omoe o e ma D -. Ta e de. old. Aal ealao o e mae C ad D ae aed o w a ome aleba em o eamle ale o deeme e oa oee ad.. D C O O O. To oba e oo eme we e. o. o ae. φ. wee: φ

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada G. AKUBU Fom. we ae T K. Reall a a. ede o T K. T ead. ll we oba e ooed oo eme a ollow: T T C K K K K. wee T deoe aoe o e ma C. ad e eo -. Remak. We all D e mle olloao ad eolao ma w a a e mle e. Fom.8 e olm o D w e e oo oee a be oba om e oeod olm o C. A a be ee e ee o C ae e oa oee o e olomal e. w ae o be deemed. Te ma C e olo eo o ad D emed e daa w amed o be o-la o e eee o e ee ma C. I e eod-deae meod we ee a o ol e o ealaed a ome emedae o b addo e o D D wee D e deeal oeao ee []. Hee addo o e omao o e -ale a e eal ae e adad Re-Ka meod e moded meod ole om -ale wee ad ae a deed.. Aod o [] ee meod a be aal e o o eala ae omaable o oe eala ad a ee be moe ee a e adad Re- Ka meod e mbe o o ealao ewe. I oee o ewe e oee o e de omla. ealaed a ome ea o e blok ma om a ˆ G I A F I A e N N.7a 8

eoado Joal o See ISSN 8- Ie 8 Jaa-Je. - T ˆT I F b I G b N N.7b wee A[a ] Â[â ] dae e deedee o e ae o e deae od a bˆ e oe ae ad b[b ] ˆ b ae eo o qadae we ow ow e al el deed o e deae omed a e ao ae I e de ma o ze eqal o e deeal eqao em o be oled ad N e dmeo o e em. Alo e Koeke od o wo mae. Fo ml we we e meod.7 a ollow: AG ˆ AF ˆT b G T b F ad e blok eo R N ae deed b F G.8a.8b.9 Te oee o e ml eod-deae Re-Ka olloao meod a be oeel eeeed moe omal a eeded aoed Be Tablea o e om A T b Aˆ T bˆ. wee [] e abae eo w dae e oo w e e o e ae ale. Seao o e ml eod deae Re-Ka olloao meod A o-ode ml eod deae Re-Ka olloao meod I eo we deelo e eeal om o odo o e oee o e ml eod-deae Re-Ka olloao meod. To oba e oee o e meod e ma D.8 ake e om 9

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada G. AKUBU D. Ie e ma D. oe we oba e oo eme o e om. a ollow. wee 7 7 7 7 8 9 9 To e e ale o. we d e zeo o e N deee Jaob olomal ee [] deed b N N N P γ.a wee N N N K γ γ.b w γ. Ne b b ⅓ o e oo eme. ad ealae a ad we oba e ollow blok bd dee eme w a be aled mlaeol a blok meod o dee o deed 7 9.a

eoado Joal o See ISSN 8- Ie 8 Jaa-Je. - [ 99 ] [ ].b I we oe e blok bd dee eme.a-b o ml eoddeae Re-Ka olloao meod ad we e meod e om o.8 we ae e ollow 9 F 7 F G Te eal ae ale a e e ae allaed a 8 8 G F F G 9 7 G F F G w e ae deae a ollow: F F F. G. Te oee o e ml eod-deae Re-Ka olloao meod. a be oeel eeeed moe omal a eeded aoed Be Tablea o e om. a ollow: A T b A T b 9 8 9 7 8 7 8 A -ode ml eod deae Re-Ka olloao meod Hee we dee ml eod-deae Re-Ka olloao meod o e ode o e meal olo o em. Te ma D.8 ake e ollow om

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada G. AKUBU D. wee ad ae e zeo o. Jaob olomal. o deee w ae ald e eal [ ]. Ie e ma. oe we oba e oo eme a ollow:.7 wee 8 9 9 79 8 7 8 7 9 79 8 8 9 8 8 9 7 7 9 9 7 7 8 8 8 Eala e oo eme.7 a e o { } we oba e blok bd dee eme w a alo be aled mlaeol a blok meod o dee o deed

eoado Joal o See ISSN 8- Ie 8 Jaa-Je. - [ 8 8 ] [ 7 7 ] [ 8 88 87 7 8 ] 7 [ 87 77 99 9 7 ] 7 88 87 8 7 8 7 [ 99 9 87 77 7 ]. 7.8 W e meod e omalm o.8 we ae F F 7 7 7 7 F 7 G 8 Te eal ae ale a e e ae allaed a 7 8 9 F 77 7 79 87 G 87 7 79 87 F 87 7 F 99 7 7 G G 8 7 G 7 9 F 9 87 9 87 7 7 F 97 G F 97 99 7 7 77 7 G G G 7 7 8 7 7 7 7 7 F F F G G G 7 7 7 8 8 wee e ae deae ae a ollow: F F.9 F F We we e oee o e meod.9 a eeded aoed Be Tablea. a ollow: 7 9 88 87 7 8887 7 7 9 9 87 9 87 78 8 99 9 7 99 9 7 78 8 7 7 7 7

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada G. AKUBU 8 8 8 8 7 7 7 7 Aal o e oee o e ml SDRK olloao meod Ode Coe Zeo-Sabl ad Coeee o e Iml SDRKC W e ml-e olloao omla. we aoae e lea deee oeao l deed b: [ ;] l. wee a aba o ool deeable o [ T]. Follow [7] we a we e em. a a Talo ee eao abo e o o oba e eeo l [ ; ] C C C C. wee e oa oee C ae e a ollow: C C C! C!!! Aod o [7] e mle olloao omla. a ode K l[ ; ] Ο C C C C. Teeoe C e eo oa ad C e al loal ao eo a e o ee [8]. Hee om o allao e ode ad eo oa o e oed meod ae eeed Table. I lea om e Table a e ml eod-deae Re-Ka olloao meod ae o -ode. Te ae malle eo oa ad ee moe aae a e oeoal Rada-Re-Ka meod o e ame ode o oeee.

eoado Joal o See ISSN 8- Ie 8 Jaa-Je. - Table. Ode ad eo oa o e ml SDRK olloao meod eod Ode Eo oa eod. P C 9.87 - eod.9 P C 7.7 - Deo.: Coe A lea mle meod ad o be oe e ode o e meod eae o eqal o oe a > ee [7]. ρ adρ σ wee ρz ad σz ae eeel e ad d aae olomal. Fom Table ad deo. we a ae a e ml eod-deae Re-Ka olloao meod ae oe. Deo.: Zeo-abl A lea mle meod ad o be zeo-able e oo: ρ λ de k A λ K k λ k λ ee [7]. Baed o deo. e ewl oed ml eod-deae Re-Ka olloao meod ae zeo-able. Deo.: Coeee Te eea ad e odo o a lea mle meod o be oee ae a m be oe ad zeo-able ee []. Hee om deo. ad. e ml eod-deae Re-Ka olloao meod ae oee. Reo o abole abl o e ml SDRK olloao meod To d e abl oee o e ml eod-deae Re-Ka olloao meod we eomlae em a eeal lea meod ee [9]. Hee we e e oao oded b [] w a eeal lea meod eeeed b e aoed ma oa A U B V [ ] A U N.a B V

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada G. AKUBU wee B A A μ μ e I U T T B A B ν θ μ μ I I e I V ad e[ ] R. Hee.a ake e om ] [ ] [ ] [ ] [ V B U A.b wee deoe qae a o om ea e ad o e e e ad deoe ae ale ed e omao o e e. Te oee o ee mae dae e elao bewee e ao meal qae a ae e omao o abl eo. Te eleme o e mae A U B ad V ae bed o e abl ma. I e ee o [] we al. o e lea e eqao λ> ad λ C b o e eod-deae -ode meod we e λ w lead o e eee elao [] z [] N- zλ wee e abl ma z deed b U za zb V z. We alo dee e abl olomal ηz b e elao de z I z η η ρ. ad e abole abl eo R o e meod e b η η ρ R z C : To ome e eo o abole abl we be e eleme o e mae A U B ad V o e abl o. ad all o e abl olomal.

eoado Joal o See ISSN 8- Ie 8 Jaa-Je. - o e meod w loed o ode e eqed a o e abole abl eo o e meod a ow Fe. Uable Reo - - - - 7 8 eod. A -able Uable Reo - - - - 8 eod.9 A -able Fe. Reo o abole abl o e ml SDRK olloao meod Te meod oed ae ae A-able e e eo o o e omle lae ode e eloed Fe w ol e mall oo de e le ad-al lae. Nmeal eeme O am eo o omae e eomae o e ew meod w oe meod o e oeoal e Rada meod w ae wdel kow o be amo e 7

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada G. AKUBU mo ee meod o e meal olo o oda deeal eqao. Teeoe we eo e meal el obaed om e alao o e ew ml eod-deae Re-Ka olloao meod. We ee e omed el e omalm o []. I e eeao we e e o deoe e mbe o o ealao ad E o dae e ea olo. Eamle. Code e em o ode deeal eqao e Te ea olo e e. e. T em a eeale o lae modl l loe o e maa a -±. We ole e em e ae o w.ad e el obaed ae ow Table. I eamle we omae e el om e wo ew meod de b de e Table. Te olo e obaed ae omaed w e ea E olo Fe. T o a omao ee e el om e d deae meod o [] Table ae 7 ad el o [] Table ae. Table. Abole eo e meal eao o eamle eod. eod.9.8887 -.88979 -.9787 -.779 -.9987779-7.8977 -.8797-9.99978-9.87978 -.7779 -.789888-8 7.9777-9.7879 -.997977799-8.789998 -.78 -.898 -.89977 -.888 -.779787-8

eoado Joal o See ISSN 8- Ie 8 Jaa-Je. -.9.8 E E.9.8 E E.7.7............ a b Fe. Gaal lo o eamle Seod Deae Re-Ka Colloao meod: a Solo e o eamle meod. w e ; b Solo e o eamle meod.9 w e Eamle : Te eod eamle a lea em o eqao ee [] 998 999 999o Te ea olo e b e e o We ole em e ae. I eamle we alo omae e el om e wo ew meod Table ad e ee e obaed ae omaed w e ea olo Fe. Table. Abole eo e meal eao o eamle eod. eod.9 7.77-7.7997789 -.9 -.7889889 -. -.778778 -. - 7.779888 -.8998-7.77979 -.97877 -.878 -. -.88798 -. -.897 -.78 -.887889 -.78-9

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada G. AKUBU a 7 8 9 - -.... E E b 7 8 9 - -.... E E Fe. Gaal lo o eamle eod deae Re-Ka olloao meod: a Solo e o eamle meod. w e ;b Solo e o eamle meod.9 w e Eamle : Te d e eamle a oblem ake om []... 8. 8.... Te ea olo e 8 o8 e. e o8 e. e 8 e. Te eeale o e em ae λ - ad λ.±8. Te omed olo o e eamle w e eal o ae eeed Table ad e aal lo ae ow Fe. Eamle : We ode aoe lea oblem w alal eeed o b [7] a a obleome oblem o ome e meod beae e eeale o oblem ae l loe o e maa a wee ome eao wee kow o be ee. Te eeale o e Jaoba ae λ -± λ - λ - λ -. ad λ -....

eoado Joal o See ISSN 8- Ie 8 Jaa-Je. - Table. Abole eo e meal eao o eamle Rada IIA meod eod.9.778 -.887 -.7888 -.898 -.79977 -.887 -.9879 -.7888 -.879 -.7888 -.9788799 -.887979 -.998899-8.887897 -.7899 -.99978 -.9979 -.77 -.7-7.77 -.979 -.88798-8.78998 -.8998 -.98777988 -.99978-9.998-8.9798 -.887 -.7788 -.... -. -. - E - E E E E E -. -........7.8.9 a b.......7.8.9 Fe. Gaal lo o eamle Rada IIA meod [] ad SDRKC meod: a Solo e o eamle Rada IIA meod [] w e b Solo e o eamle meod.9 w e We ole oblem e ae. Te omed olo ae ow Table wle e aal lo ae dlaed Fe. To ol e o omoe { } ae ow e Table o ale. Eamle : HIRES Te H adae eoe oblem o o a em o 8 olea oda deeal eqao w oae om la olo ad debe ow l oled mooee. I ela e adae eoe o oo mooee o e ba o oome b mea o a emal eao ol e eaa e om o al ale oblem ee [89].

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada G. AKUBU Table. Abole eo e meal eao o eamle Rada IIA meod eod.9.8778-8.8 -.99797-8.88897 -.9 -. -.88998-7.8787998 -.7888-8.8777979 -.7 -.97877 -. -.8998 -.9899-8.887997 -.99877-8.78879 -.877787878 -.977-7.7777 -.9 -.88989 -.77 -.8787 -.998787 -.7 -.9889997-7.8998 -.. -. E E E E E E.. -. E E E E E E - - -. a.......7.8.9 -........7.8.9 b Fe. Gaal lo o eamle Rada IIA meod [] ad SDRKC meod: a Solo e o eamle Rada IIA meod [] w e b Solo e o eamle meod.9 w e.7.7 8 7 8 7 8... 8.7..7 8 8.7 8..9.8 8.. 7...7.7 7..9 7 7.7. 8

eoado Joal o See ISSN 8- Ie 8 Jaa-Je. - We ole e oblem e eal ad omae e aal lo w e ODE ode o alab Fe.....8... 7 8 ode ode ode ode ode ode 7ode 8ode....8... 7 8 ode ode ode ode ode ode 7ode 8ode -. a.......7.8.9 -........7.8.9 b Fe. Gaal lo o eamle SDRKC ad e ODE ode o alab: a Solo e o eamle meod. w e b Solo e o eamle meod.9 w e Cold emak I olo e meod o deed ae o a ae bee o ode a la o ml eod-deae Re-Ka olloao meod able o e aomae meal eao o em alal em o oda deeal eqao. Te deed meod oed a ee wa o d meal olo o em o al ale oblem we e eod deae em ae ea o ealae. We ee wo ew meod o ode o ad a ae eded o aae eao o em o eqao. We ae alo omaed e meal olo obaed w e oeoal Rada Re-Ka meod w ae wdel eozed o be amo e mo ee meod o em [ ]. Te olo e ae alo omaed w e ea olo aall Fe. Akowledeme T wok wa oed b Tea Edao T Fd TETF Re. No. TETF /DAST &D.D/./NO-CA/BAS&BNAS. Teeoe e ao aell akowleded e aal o o e TETF.

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada G. AKUBU Te ao ae alo ael o e aomo eeee o /e el omme ad eo Reeee. Be J. C. Coee o d o Re-Ka eao oee J. Aal. a. Soe 9. 8 -.. E W. H. Seod-deae mle meod o oda deeal eqao SIA J. Nme. Aal. 97. -.. Hae E. Nøe S. P. Wae G. Sol Oda Deeal Eqao II: S ad Deeal Aleba Poblem Seod Reed Edo Bel Gema Se 99.. Vo-Aa J. Ramo H. A aml o A-able Re-Ka olloao meod o e ode o al ale oblem IA J. Nme Aal. 7 7.798-87.. Ca J. R. H ode meod o e meal eao o oda deeal eqao Nme. a. 978. 8-9.. T. Re-Ka e eao omla ld e ealao o e eod deae Pa I Pbl. RIS Koo Ue Jaa 98 8. -. 7. T. akb D. G. Two-Se Faml o ook-aead lea mle meod o ODE Te ee ad Eee Reew Doa Ue Jaa. 8-88. 8. Uabe. A ml oe-e meod o -ode aa o e meal eao o oda deeal eqao J. Nme. a. 97. -. 9. akb D. G. Kwam A.. Iml wo-deae Re-Ka olloao meod o em o al ale oblem J. Nea a. Soe. 8-.. Dalq G. A eal abl oblem o lea mle meod BIT Nme. a. 9. 7-.. Be J. C. Ode ad abl o Re-Ka meod Na Ue Co... Oma P. Awoem D. O. Jao S. N. Sea U. W. New lea mle meod w oo oee o ode al ale oblem ODE J. Nea a. Soe 99. 7-.

eoado Joal o See ISSN 8- Ie 8 Jaa-Je. -. e I. Nøe S. P. Seoeee o mle olloao a. Com. 989. -8.. Kale K. H. Wae G. Re-Ka oee w mlle ode J. Com 97 99. 9-.. Ca R. P. K. Ta A.. J. O el wo- deae Re-Ka meod Nme. Alo.. 7-9.. Vllade J. ele.. Solo o Deeal Eqao odel b Polomal Aomao Pee-Hall INC Ealewood Cl New Jee 978.. 7. ambe J. D. Comaoal eod Oda Deeal Eqao Jo Wll ad So New ok 97. 8. T. A moded eo o Uabe ml le-e meod J. Com. Al. a. 987. -. 9. Bae K. Be J. C. Sabl ea o ml Re-Ka meod SIA J. Nme. Aal. 979. -7.. Be J. C. Nmeal eod o Oda Deeal Eqao d ed. Jo Wle & So d 8... Collom J. P. Jakewz Z. Coo o wo e Re-Ka TSRK meod w lae eo o abole abl J. Com. ad Al. a. 7. -7.. Be J. C. Hoa G. Seod deae meod w Re-Ka abl Nme. Alo.. -9.. Al K. E. Hoa G. Td deae mle meod o em Ie. J. o Nolea See. -.. Akewa O. A. Akkawe B. da S. B. A aml o oo d deae blok meod o ol em o ode oda deeal eqao J. Nea a. Soe. -8.. ambe J. D. Nmeal eod o Oda Deeal Sem Jo Wll & So New ok 99.. Be J. C. Ieao oee baed o Rada qadae omla a. Com. 9 8. -. 7. Fala S. O. Nmeal eao o ad l ollao deeal eqao J. a. Com. 98. 7-9.

Alao o eod deae Re-Ka olloao meod o em o al ale oblem Samala ARKUS ad Dada G. AKUBU 8. Gozalez-Po S. Gozalez-Coeo C. ooj. I. Ieae eme o Ga meod J. Com. a. Al. 99 77. 7 8. 9. Säe E. Feb B. A ew aoa o ela e H Iadae Reoe o oomooee o e ba o oome J. a. Bo. 97. -.

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