Systems of second order ordinary differential equations

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1 Ffth order dgolly mplct Ruge-Kutt Nystrom geerl method solvg secod Order IVPs Fudzh Isml Astrct A dgolly mplct Ruge-Kutt-Nystróm Geerl (SDIRKNG) method of ffth order wth explct frst stge for the tegrto of secod-order IVPs s preseted. A stdrd set of test prolems re tested upo d the umercl results re compred whe the sme set of test prolems re reduced to frstorder system d solved usg exstg ffth order sgly dgolly mplct Ruge-Kutt method. The tme tke to solve ech prolem over ll the stepszes re lso compred. The results suggest the superorty of the ew method. Keywords Dgolly mplct, Ruge-Kutt-Nystróm, Secodorder IVPs. I. INTRODUCTION Systems of secod order ordry dfferetl eutos rse my physcl prolems, such s celestl mechcs, strophyscs, electrocs d moleculr dymcs. The geerl form of the secod order ordry dfferetl euto c e wrtte s follows y = f ( x, y, y ), x0 x x, () wth the gve tl codtos y, y ( x 0 ) = y 0, ( x 0 ) = y 0 where y R, d f : R R R R. The fucto f s ssumed to hve dervtve of rtrry order everywhere R. Usg Ruge-Kutt (RK) type of methods euto () c e solved usg two geerl techues, the frst oe s to trsform () to frst-order prolem d the use y RK method. My clsses of RK methods hve ee developed, these clude the method costructed y Verer [] d ffth order sgly dgolly mplct Ruge-Kutt Muscrpt receved o Mrch 0, 00. (Wrte the dte o whch you sumtted your pper for revew.) Ths work ws supported y the Mlys Mstry of Scece Techology d Iovto uder Grt Scece Fud vote umer 0. Fudzh Isml s wth the Deprtmet of Mthemtcs, Uverst Putr Mlys, 00, Serdg, Selgor Mlys.( e-ml: fudzh_@yhoo.com.my, fudzh@mth.upm.edu.my). (SDIRK) method due to Cooper d Syfy [] whch c e foud Hrer d Wer []. The secod techue s to solve () drectly usg Ruge-Kutt-Nystróm Geerl (RKNG) method. Ths method geertes pproxmtos y + d y +, to y( x + ) d y ( x+ ) respectvely, for =0,,, ccordg to + y = y + hy + h k, y + = y + h k, () = = where s the umer of stges, h = x + x, d k = f ( x + ch, y + chy + h k, y + k ), =,,. We refer to () s geerlzed Ruge-Kutt-Nystróm method. Ulke ther close reltves, the Ruge-Kutt- Nystróm formuls for the specl secod order tl vlue prolem y = f ( x, y), the RKNG schemes hve ee freuetly vestgted. Zurmühl [] preseted pr of fourth order formuls reurg four stges whose c, d cocded wth the tleu of prmeters for the clsscl Ruge-Kutt method for frst-order ordry dfferetl eutos. Asorge d Torg [] performed stlty lyss of the clsscl RKNG usg sclr secod-order dfferetl eutos wth costt rel coeffcets. Further mprovemets fourth order, four stge RKNG methods, hve ee reported y Chwl d Shrm []. A umer of explct RKNG schemes, hvg orders fve, sx d seve, hve ee proposed y Fehlerg [7] d Fe [8]. I ths pper we re gog to derve ffth order sgly dgolly mplct Ruge-Kutt Nystrom method wth explct frst stge d use t to solve system of secod order IVPs. II. DERIVATION OF THE METHOD Geerlly, RKNG method c e wrtte s follows + y = y + hy + h k, y + = y + h k, = = ISSN: 79-9 ISBN:

2 k = f ( x + c h, y + hc y + h =,..., () k, y + h k ) or t c e wrtte exteded Butcher tleu s c c , where the coeffcets,, determe the method d the prmeters re reured to stsfy the followg eutos c =, ( =,..., ), () d c =, ( =,..., ). () Bsed o the work of Hrer d Wer [9], Fe [8] lsted the order codtos of RKNG method up to order sx. Here, we lsted ll the order codtos relted to y up to order fve Tle. Ad ll the order codtos up to order fve relted to y re gve Tle. c = (.7) = k (.8) c = (.9) * c = (.0) c c = (.) 0 = c = c = (.8) 0 c = (.9) 0 k = (.0) * 0 k = (.) k k kl cl = (.) 0 kl TABLE ORDER CONDITIONS UP TO ORDER FIVE FOR y (.) (.) c = (.) c c = (.) 0 c = (.7) = (.8) 0 TABLE I ORDER CONDITIONS UP TO ORDER FIVE FOR y c = (.) c = (.9) ** 0 = (.) = c (.) c = (.) cc = (.) * 0 c c = (.) c = (.) 0 k c = (.) c = (.) * c = (.) c c = 8 (.) ( c ) = (.) * 0 k = (.7) 0 c = (.) 0 Now we eed to lst dow the order codtos whch deped o c, d oly d ths set s clled set S or set elogs to y, t cossts of ll eutos Tle except those deoted y (*). The secod set of eutos whch deped o c, d or we clled t set S whch elogs to y. It cossts of ll eutos Tle except those deoted y (**). Flly ll eutos deoted y * d ** from Tle d elog to the set of eutos S whch elog to oth y d y. There re 7 eutos S, 8 eutos S d eutos S, totl umer of eutos. ISSN: 79-0 ISBN:

3 Now look t set S d use the smplfyg ssumpto c c = (). c = (.), = c, (.). c c, keep (.) d remove (.) c c = (.), = 8 c, (.) c c c, remove (.) d keep (.). = (.8), c = (.7) c c, remove (.8) d keep (.7) 8. c c, remove (.9) d keep (.8) k klcl =, (.), 0 =, (.) ck k klcl, remove (.) keep (.). Now use the smplfyg ssumpto c c =, (7) 9. c =, (.7), c =, (.) c c, remove (.7) keep (.). c c =, (.), 0 c = (.0) 0. c c =, (.), = c, (.0) c c c, remove (.), keep (.0). c = (.), 0 c c = (.) c kc keep (.) = (.7), 0. ( c ) c k c (.) k c = 0, remove (.) d k c c = (.) 0 c = 0, remove (.7) d keep c c c, remove (.) keep (.0). kck = (.), c = (.8) 0 c, remove (.) keep (.8) Thus eutos eeded to e stsfed for set S re (.), (.), (.), (.), (.0) d (.8) provded the smplfyg ssumptos re stsfed for =,,,. Now look t S, we c stll use the sme smplfyg ssumpto Usg () we hve 7. c =, (.9), 0 c =, (.8) 0. c = (..), c = (.) c c, remove (.) keep (.) ISSN: 79- ISBN:

4 .. c c = (.), = 0 c (.) 0 c c c, remove (.) keep (.) = (.8), 0 c =, (.7) c remove (.8), keep (.7) Usg ssumpto (7) we hve. c = (.7), = c, (.) 0 c c, remove (.7), keep (.). Thus for S, eutos to e stsfed re (.), (.), (.) d (.). Now look t S, whch cossts of eutos (.9), (.), (.), (.), (.0) d (.9), use the smplfyg ssumpto c c = 0 (8) c = (.9), = c (.) c = 0 c, remove (.9), keep (.). =, (.) 0 k c = (.) c, usg smplfyg ssumpto () remove (.), keep (.). cc = (.), 0 c = (.0) c c = 0 c, remove (.) = (.0), 0 k c = (.8) 0 0. c kck, remove (.0), c = (.9), = 0 c (.) 0 c = 0 c, remove (.9). The oly eutos eeded to e stsfed for ths set re (.) d smplfyg ssumpto (8) Now look t smplfyg ssumptos () d (7). For = we hve c c = c c = c = = 0 c c = c = = γ, (9) ( γ s the dgol elemet for > ) c c = c = γ thus t s ot true d (7) cot e stsfed for = thus we eed to hve = 0, c = 0, = 0 (0) From S sce we re lso usg (7) we eed to hve (from o ). = 0 d () re stsfed for =,,,, d (7) re stsfed for =,,,,. From set S other eutos eeded to e stsfed re (.), (.), (.), (.), (.0) d (.8) From S d smplfyg ssumpto (8), t s stsfed for for = sce c = 0 c c = c = β () ( β s the dgol elemet of for > From (9) d (0) we oted γ c = β = (γ ) β = d (8) re stsfed for =,,, ISSN: 79- ISBN:

5 The followg re the steps tke to ot the coeffcets of the ffth order RKNG method Step : set γ = 0., from (9) we hve c = γ 0.. = Step : From () d (7) for =, solve for d c, Step : From () d (7) for =, solve for d, usg the vlues of c = 0. d c oted step. Step : Set c = 0. 7 d c = 0. 9, from (.), (.), (.), (.), (.0) solve for =,,,, Step : Set = 0., use () d (7) for =, solve for d. Step : From eutos (0), (.8), () d (7) for =, solve for,, d Step 7; From eutos (), (7), (8) d tkg = 0,we c solve for,,, d Step 8: Usg (8) for = d the vlue of β solve for. Step 9: Set = 0., solve for usg (8) for =. Step 0: Settg =., = 0.08, = solve for, d from (8) ( =,) d (.). The coeffcets of ffth order SDIRKNG method wth explct frst stge oted re s follows: c = 0 c = 0. c = c = 0. c = 0.7 c = 0.9 = 0 = 0. = = 0. = = = 0. = 0. = = 0.09 = 0. = = = = = 0. = = 0.0 = = = = = = = = 0. = = = = 0.0 = 0. = = = 0.87 = 0.08 = 0.0 = = = 0.0 = 0.87 = = = where the vlues of d for (=()) re gve y ISSN: 79- ISBN:

6 = c., d = c III PROBLEMS TESTED The followg re some of the prolems used to vldte the ew method. Prolem. y y y = x y +, x0 r r y x, = x y +, x0 r r 0.π x 0 x, y ( x 0 ) 0, y ( x 0 ), y ( x ) = (π, = y ( x 0 ) 0, = y + y = r = d x 0 ) = y y r + x Soluto: y ( x) = cos( ), y ( x) = s( ). Source: Shrp d Fe [0]. Prolem. Source: y + 8 y + ky = 0, 0 x 0 y ( 0) =, y ( 0) =, k =. Soluto: x y( x) = ( 8x) e. (0.9.00) x ) Soluto: y ( x) = ( 8x e, The results oted from the ew method whch ws derved secto re compred wth the results whe the sme prolems re solved usg SDIRK method of order fve d fve stge due to Cooper d Syfy []. I the SDIRK method the drect pproch s used to solve the prolems y trsformg them to frst-order dfferetl euto of douled dmeso y cosderg the vector ( y, y ) s the ew vrles, Sce the method s mplct we do three tertos for the frst k d two tertos for the suseuet k. The tme tke for solvg the prolems umerclly over r h = x0 where r =,,,, s lso gve fgure. The umercl results re gve Tles -. The ottos used re s follows: STEPSIZE: The sze of the step. FCN ~ the umer of fuctos evlutos. STEP ~ the umer of steps. EMAX ~ mx y (t ) - y, (solute vlue of the true t soluto mus the computed soluto t the mesh pot ). Methods used re: : SDIRKNG method of order fve d sx stges whch ws derved ths pper : The SDIRK method order fve d fve stges due to Cooper d Syfy []. 0 0.(-0) mes 0. X ( 0 ). TABLE : NUMERICAL RESULTS FOR PROBLEM. METHOD STPSIZE FCN STEP EMAX y x ( x) ( 8x) e 8 Prolem. x = e y = y + cos( x), y = y + s( x), y (0) =, y (0) =, y (0) = y (0) = (-) x π (-0).79(-) Solutos: y x) = cos( x) s( x), y ( x) = cos( ( x ) (-0).8(-7) ISSN: 79- ISBN:

7 TABLE : NUMERICAL RESULTS FOR PROBLEM. MTD H FCN STEP EMAX Totl tme tke to solve the prolems (-) 7.89(-) (-).898(-) (-9).787(-8) Tme secods (-).7(-) TABLE : NUMERICAL RESULTS FOR PROBLEM Prolems METHOD STEPSIZE FCN STEP EMAX IV CONCLUSIONS (-) 7.787(-).0988(-9) (-).07(-) 8.009(-8).77(-0).89(-0) From the umercl results for ll the prolems tested, we c coclude tht solvg the geerl secod order eutos drectly usg the ffth order RKNG method produced smller error compred to reducto to frst order systems. From the umer of fucto evlutos we c sy tht though the RKNG method s sx stges ut the frst stge s explct thus o terto s eeded to evlute the frst k, hece the method s effectvely cot fve stges whch s comprle to the SDIRK method. I terms of tme tke to solve the prolems over ll the stepszes, reured slghtly less tme compred to, d we eleved tht f the prolems cosst of lrger system of eutos the the totl tme ged wll e more ppret. Thus, the method for the solvg the secod order IVPs drectly s more effcet th the method. ISSN: 79- ISBN:

8 REFERENCES [] J. H. Verer, Explct Ruge-Kutt methods wth estmtes of the locl tructo error. SIAM J. Numer. Al. 978, (), 7-7. [] J. G. Cooper J. G. da. Syfy, Semexplct A-stle Ruge-Kutt methods. Mth of Comp., 979 vol, pp. -. [] E. Hrer, E. d G. Wer, Solvg Ordry Dfferetl Eutos II, stff d Dfferetl Algerc Prolems, Berl: Sprger-Verlg. 99 [] R. Zurmül, Ruge-Kutt-Verfhre zur umersche Itegrto vo Dfferetlglechuge -ter Ordug. ZAMM 8, 98: 7-8. [] R. Asorge, d W. Torg, Zur Stltät des Nyströmsche Verfhres. ZAMM 0 9, : 8. [] M. M. Chwl, d S. R. Shrm. Fmles of drect fourth-order methods for the umercl soluto of geerl secod-order tl vlue prolems. ZAMM 980: 9-7. [7] E. Fehlerg,. Clsscl seveth-, sxth-, d ffth-order Ruge-Kutt- Nystróm formuls wth stepsze cotrol for geerl secod-order dfferetl eutos. NASA Techcl Report R- 97. [8] J. M. Fe, Low Order Prctcl Ruge-Kutt-Nystróm Methods. Computg y Sprger-Verlg 987, 8: [9] E. Hrer, E. d G. Wer. A theory for Nystróm methods, Numer. Mth. 97: [0] Shrp, P. W. d Fe, J. M. (99). Some Nystróm prs for the geerl secod-order tl-vlue prolem. Jourl of Comut. Ad Appl. Mth., : ISSN: 79- ISBN: