Diagonally Implicit Runge-Kutta Nystrom General Method Order Five for Solving Second Order IVPs

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1 WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml Dgoll Implt Ruge-Kutt Nstrom Geerl Method Order Fve for Solvg Seod Order IVPs FUDZIAH ISMAIL Deprtmet of Mthemts Uverst Putr Mls Serdg Selgor MALAYSIA fudzh@mth.upm.edu.m or fudzh_@hoo.om.m Astrt: - A dgoll mplt Ruge-Kutt-Nstróm Geerl (SDIRKNG) method of ffth order wth explt frst stge for the tegrto of seod-order IVPs s preseted. A stdrd set of test prolems re tested upo d the umerl results re ompred whe the sme set of test prolems re redued to frstorder sstem d solved usg exstg ffth order sgl dgoll mplt Ruge-Kutt method. The tme tke to solve eh prolem over ll the stepszes re lso ompred. The results suggest the superort of the ew method. Ke-Words: - Dgoll mplt Ruge-Kutt-Nstróm Seod-order IVPs. Itroduto M phsl prolems e formulted the form of ordr dfferetl equtos. These dfferetl equtos e lssfed s oudr vlue prolem d tl vlue prolems. Work o oudr vlue prolem e see Gordez et. l []. Sstems of seod order ordr dfferetl equtos rse m phsl prolems suh s elestl mehs strophss eletros d moleulr dms. The geerl form of the seod order ordr dfferetl equto e wrtte s follows f ( x ) x x x () wth the gve tl odtos ( x ) ( x ) where R d f : R R R R. The futo f s ssumed to hve dervtve of rtrr order everwhere R. Usg Ruge-Kutt (RK) tpe of methods equto () e solved usg two geerl tehques the frst oe s to trsform () to frst-order prolem d the use RK method. M lsses of RK methods hve ee developed these lude the method ostruted Isml et. l [] D et. l [] Verer [] d ffth order sgl dgoll mplt Ruge-Kutt (SDIRK) method due to Cooper d Sf [5] whh e foud Hrer d Wer []. The seod tehque s to solve () dretl usg Ruge-Kutt-Nstróm Geerl (RKNG) method. Ths method geertes pproxmtos + d + to ( x + ) d ( x+ ) respetvel for ordg to q + + h + h k q + + h k where q s the umer of stges h x + x d k f ( x q. + h + h + h () k + We refer to () s geerlzed Ruge-Kutt- Nstróm method. Ulke ther lose reltves the Ruge-Kutt-Nstróm formuls for the spel seod order tl vlue prolem f ( x ) the k ) ISSN: Issue 7 Volume 9 Jul

2 WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml RKNG shemes hve ee frequetl vestgted. Zurmühl [7] preseted pr of fourth order formuls requrg four stges whose d oded wth the tleu of prmeters for the lssl Ruge-Kutt method for frst-order ordr dfferetl equtos. Asorge d Torg [8] performed stlt lss of the lssl RKNG usg slr seod-order dfferetl equtos wth ostt rel oeffets. I determg the stlt rego of the method tehque used to solve the stlt poloml lso e oted from Mures [9]. Further mprovemets fourth order four stge RKNG methods hve ee reported Chwl d Shrm []. A umer of explt RKNG shemes hvg orders fve sx d seve hve ee proposed Fehlerg [] d Fe []. I ths pper we re gog to derve ffth order sgl dgoll mplt Ruge-Kutt Nstrom method wth explt frst stge d use t to solve sstem of seod order IVPs. Dervto of the Method Geerll RKNG method e wrtte s follows q + + h + h k q + + h k k f ( x + h... q () + h + h k + h k or t e wrtte exteded Buther tleu s q q... qq... q... qq q... q ) where the oeffets determe the method d the prmeters re requred to stsf the followg equtos d (... q) () (... q). (5) Bsed o the work of Hrer d Wer [] Fe [] lsted the order odtos of RKNG method up to order sx. Here we lsted ll the order odtos relted to up to order fve Tle. Ad ll the order odtos up to order fve relted to re gve Tle. TABLE I ORDER CONDITIONS UP TO ORDER FIVE FOR (.) (.) (.) (.) * k (.) 5 (.) (.) (.5) * (.) (.5) k * (.) ( ) (.7) 8 ISSN: Issue 7 Volume 9 Jul

3 WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml (.7) (.8) The seod set of equtos whh deped o d or we lled t set S whh elogs to. It ossts of ll equtos Tle exept those deoted (**). (.8) k k (.9) (.9)* k (.)* 5 (.) (.) (.) k k kl l (.) kl TABLE ORDER CONDITIONS UP TO ORDER FIVE FOR (.) (.) Fll ll equtos deoted * d ** from Tle d elog to the set of equtos S whh elog to oth d. There re 7 equtos S 8 equtos S d equtos S totl umer of equtos. Now look t set S d use the smplfg ssumpto () Cert order equtos e removed s follows:. (.) (.) keep (.) d remove (.) (.) (.7). (.) (.5) 8 (.) (.8) remove (.) d (.) (.9) ** keep (.5). kk (.8) (.7) (.5) Now we eed to lst dow the order odtos whh deped o d ol d ths set s lled set S or set elogs to t ossts of ll equtos Tle exept those deoted (*).. remove (.8) d keep (.7) (.) 5 (.) ISSN: Issue 7 Volume 9 Jul

4 WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml remove (.) keep remove (.) keep (.). Now use the smplfg ssumpto 5. (.) (.) (.) 5 remove (.) d keep (.) (.7). ( ) (7) 9. (.7) (.5). remove (.7) keep (.5) (.) 5 (.) 5 remove (.) keep (.) 7. (.) k keep (.) k (.9) remove (.7) d. (.) (.8) (.8) remove (.) keep 8. (.8) remove (.9) d keep (.8) k kl l (.) (.) k k kl l Thus equtos eeded to e stsfed for set S re (.) (.) (.) (.5) (.) d (.8) provded the smplfg ssumptos re stsfed for q. Now look t S we stll use the sme smplfg ssumpto Usg () we hve. (..) (.) remove (.) keep (.) ISSN: Issue 7 Volume 9 Jul

5 WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml. (.) (.5) (.5) remove (.) keep 7. keep (.5). k (.) (.5). (.8) k k usg smplfg 5. (.7) remove (.8) keep (.7) Usg ssumpto (7) we hve (.7) (.5) remove (.7) keep (.5). Thus for S equtos to e stsfed re (.) (.) (.) d (.5) ssumpto () remove (.) keep (.5). k (.) 5 (.) remove (.) (.) (.8) kk remove (.) (.9) (.5) Now look t S whh ossts of equtos (.9) (.) (.5) (.) (.) d (.9) use the smplfg ssumpto (8) remove (.9). The ol equtos eeded to e stsfed for ths set re (.5) d smplfg ssumpto (8) Now look t smplfg ssumptos () d (7).. (.9) (.5) remove (.9) For we hve γ (9) ISSN: Issue 7 Volume 9 Jul

6 WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml ( γ s the dgol elemet γ for > ) thus t s ot true d (7) ot e stsfed for thus we eed to hve () From S se we re lso usg (7) we eed to hve (from o 5). d () re stsfed for 5 d (7) re stsfed for 5. From set S other equtos eeded to e stsfed re (.) (.) (.) (.5) (.) d (.8) From S d smplfg ssumpto (8) t s stsfed for for se β () (β s the dgol elemet of for > From (9) d () we oted γ β (γ ) β d (8) re stsfed for The followg re the steps tke to ot the oeffets of the ffth order RKNG method Step : set γ. 5 γ.5. from (9) we hve Step : From () d (7) for solve for d Step : From () d (7) for solve for d usg the vlues of. 5 d oted step. Step : Set d. 9 from (.) (.) (.) (.5) (.) solve for 5 Step 5: Set 5. use () d (7) for 5 solve for 5 d 5. Step : From equtos () (.8) () d (7) for solve for d 5 Step 7; From equtos () (7) (8) d tkg we solve for 5 d Step 8: Usg (8) for d the vlue of β solve for. Step 9: Set. solve for usg (8) for. Step: Settg solve for 5 d from (8) ( 5) d (.5). The oeffets of ffth order SDIRKNG method wth explt frst stge oted re s follows: ISSN: Issue 7 Volume 9 Jul

7 WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml where the vlues of d for (()) re gve Numerl Results. d The followg re some of the prolems used to vldte the ew method. Prolem. () () e () x () e e Soluto: ( x) e x Soure: Edwrds Jr d Pe []. Prolem. x + x x r r x + x x r r e e ( x) e x..5π x ( x ) ( x ) ( x ) (π ) ISSN: Issue 7 Volume 9 Jul

8 WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml ( x ) + r d x r + x Soluto: ( x) os( ) ( x) s( ). Soure: Shrp d Fe [5]. Prolem k x ( ) ( ) k. Soluto: x ( x) ( 8x) e. Soure: /hpter5.pdf (.9.). Prolem. + s( ) () () x π There s o true soluto ut the vlue t Soure: Chwl d Ro [] Prolem.5 + os( x) x π Solutos: + s( x) () () () () π s The results oted from the ew method whh ws derved seto re ompred wth the results whe the sme prolems re solved usg SDIRK method of order fve d fve stge due to Cooper d Sf [5]. I the SDIRK method the dret pproh s used to solve the prolems trsformg them to frst-order dfferetl equto of douled dmeso osderg the vetor ( ) s the ew vrles Se the method s mplt we do three tertos for the frst k d two tertos for the susequet k. The tme tke for solvg the prolems umerll r over h x where r 5 s lso gve fgure. The umerl results re gve Tles -7. The ottos used re s follows: MTHD: Method used. H~ The sze of the step. FCN ~ the umer of futos evlutos. STEP ~ the umer of steps. ERR ~ mx (t ) - true soluto mus the mesh pot ). Methods used re: (solute vlue of the t omputed soluto t the : SDIRKNG method of order fve d sx stges whh ws derved ths pper. : The SDIRK method order fve d fve stges due to Cooper d Sf [5] d.(-) mes. X ( ). x) os( x) s( x) ( x) os( ( x ) ISSN: Issue 7 Volume 9 Jul

9 WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml TABLE : NUMERICAL RESULTS FOR PROBLEM. TABLE5: NUMERICAL RESULTS FOR PROBLEM. MTHD H FCN STEP ERR MTHD H FCN STEP ERR..7(-5)..8(-).97(-5) 7.5(-)..97(-8)..7(-).95(-8) 5.8(-5)..(-)..(-9).79(-) 5.78(-8)..88(-)..(-).9(-) 5.7(-) TABLE : NUMERICAL RESULTS FOR PROBLEM. TABLE: NUMERICAL RESULTS FOR PROBLEM. MTHD H FCN STEP ERR MTHD H FCN STEP ERR (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-5) (-7) (-5) ISSN: Issue 7 Volume 9 Jul

10 WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml TABLE7: NUMERICAL RESULTS FOR PROBLEM.5 MTHD H FCN STEP ERR (-5) 7.78(-) (-9) Coluso For most of the prolems tested we olude tht solvg the geerl seod order equtos dretl usg the ffth order RKNG method produed smller error ompred to reduto to frst order sstems. From the umer of futo evlutos we s tht though the RKNG method s sx stges ut the frst stge s explt thus o terto s eeded to evlute the frst k hee the method s effetvel ot fve stges whh s omprle to the SDIRK method (-5).(-) 8.5(-8) I terms of tme tke to solve the prolems over ll the stepszes requred slghtl less tme ompred to d we eleved tht f the prolems osst of lrger sstem of equtos the the totl tme ged wll e more ppret (-) Thus the method for the solvg the seod order IVPs dretl s more effet th the method (-) Referees: Tme seods Fgure.: Totl tme tke to solve the prolems 8 [] D. G. Gordez H. V. Meldze d T.D. Dvtshvl. O Oe Geerlzto of Boudr Vlue Prolem for Ordr Dfferetl Equtos o Grphs the Threedmesol Spe WSEAS Trsto o Mthemts vol 8 o 8 9: 57-. [] F. Isml N. I. Che Jws M. Sulem d A. Jfr Solvg Ler Ordr Dfferetl Equtos usg Sgl Dgoll Implt Ruge-Kutt ffth order fve stge method. WSEAS Trsto o Mthemts vol 8 o 8 9: 9-. [] U. K. S. D F. Isml M. Sulem d M. Othm Appled Mthemts New York: Sprger 9. [] J. H. Verer Explt Ruge-Kutt methods wth estmtes of the lol truto error. SIAM J. Numer. Al () Prolems [5] J. G. Cooper J. G. da. Sf Semexplt A-stle Ruge-Kutt methods. Mth of Comp. 979 vol pp ISSN: Issue 7 Volume 9 Jul

11 WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml [] E. Hrer E. d G. Wer Solvg Ordr Dfferetl Equtos II stff d Dfferetl Alger Prolems Berl:Sprger-Verlg. 99. [7] R. Zurmül Ruge-Kutt-Verfhre zur umershe Itegrto vo Dfferetlglehuge -ter Ordug. ZAMM 8 98: 7-8. [8] R. Asorge d W. Torg Zur Stltät des Nströmshe Verfhres. ZAMM : 9 : [9] C. A. Mures The Poloml Roots Reprtto d Mmum Roots Seprto WSEAS Trsto o Mthemts vol 8 o7 8: [] M. M. Chwl d S. R. Shrm. Fmles of dret fourth-order methods for the umerl soluto of geerl seod-order tl vlue prolems. ZAMM 98: [] J. M. Fe Low Order Prtl Ruge-Kutt- Nstróm Methods. Computg Sprger- Verlg 987 8: [] E. Hrer E. d G. Wer. A theor for Nstróm methods Numer. Mth. 5 97: 8-. [] Edwrds Jr C. H. d Pe D. E. (99). Elemetr Dfferetl Equtos wth Boudr Vlue Prolem. Prete-Hll Eglewood Clffs New Jerse. [5] Shrp P. W. d Fe J. M. (99). Some Nstróm prs for the geerl seodorder tl-vlue prolem. Jourl of Comut. Ad Appl. Mth. : [] Chwl M.M d Ro P. S. (985). Hghur P-stle methods for f ( t ) IMA J. Numer Al. (5) 5-. [] E. Fehlerg. Clssl seveth- sxth- d ffth-order Ruge-Kutt-Nstróm formuls wth stepsze otrol for geerl seod-order dfferetl equtos. NASA Tehl Report R- 97. ISSN: Issue 7 Volume 9 Jul