Implicit Runge-Kutta method for Van der pol problem

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Appled d Computtol Mthemts 5; 4(-: - Publshed ole Jul, 4 (http://www.seepublshggroup.om//m do:.48/.m.s.54.

1 Appled d Computtol Mthemts 5; 4(-: - Publshed ole Jul, 4 ( do:.48/.m.s.54. ISSN: 8-55 (Prt; ISSN: 8-5 (Ole Implt Ruge-Kutt method for V der pol problem Jfr Bzr *, Mesm Nvd Deprtmet of Appled Mthemts, Fult of Mthemtl sees, Uverst of Gul, P.O Box: 45-94, Rsht, Ir Emls ddress: fr.bzr@gml.om (J. Bzr, bzr@gul..r (J. Bzr, mvd@gml.om (M. Nvd To te ths rtle: Jfr Bzr, Mesm Nvd. Implt Ruge-Kutt Method for V Der Pol Problem. Appled d Computtol Mthemts. Spel Issue: New orettos Appled d Computtol Mthemts. Vol. 4, No. -, 5, pp. -. do:.48/.m.s.54. Abstrt: I ths musrpt the mplt Ruge-Kutt (IRK method, wth three slopes of order fve hs bee expled, d s ppled to V der pol stff dfferetl equto. Truto error, of order fve, hs bee estmted. Stblt of the proedure for the V der pol equto, s lzed b the Lpuov method. To llustrte the struture of the method, Algorthm s preseted to solve ths stff problem. Results ofrm the vldt d the blt of ths pproh. Kewords: Implt Method, Tlor Seres, Legedre Orthogol Poloml, V Der Pol Equto, Lpuov Futo. Itroduto A Implt Ruge-Kutt method for solvg dfferetl equto = f ( t, wth ν slopes s defed b the followg equto: d Where ν w K = = ν K = hf ( t h, K = = ν =,,..., ν Ad,, ν, w, w,..., w ν re prmeters tht wll be determed. The futo K s K = h f ( t h, K K K, defed b set of ν mplt equto.( see [] d [] = wk wk wk, ( I eletro, the V der Pol oslltor s o- =,,. Where,,, w,, re twelve rbtrr prmeters, whh should be determed. The Tlor seres gves oservtve oslltor wth o-ler dmpg. Ths problem ws orgll trodued b V der pol (9 the stud of eletro rut b the followg seod order stff dfferetl equto u u u u '' ε ( ' =. Where u s futo of the tme t, d ε s postve slr prmeter dtg the olert d the stregth of the dmpg. (see [4] p- & [] p-. The Implt Ruge-Kutt Method I ths mplt method let's ν =, the we hve the followg equto Where 4 v ( t = ( t h ( t h ( t h ( t h ( t... ( 4

2 Appled d Computtol Mthemts 5; 4(-: - 7 ( t = f ( t, ( t = ( f t ff = ( f f = Df t ( t = [( f ff f f f ( f ff ] = D f f Df tt t t t f ff f f f f f f ff f f v ( = [( ttt tt t ( tt t f ff f ff f = D f f D f Df Df f ( t ( t ] Df Bsed o expso of two-vrble futo, the equto K ( hges to: ( These equtos re mplt d we ot esl obt the explt expresso fork, K, d K. I order to determek, K, dk expltl, we ssume the followg form Where A, B, E, D, df re ukow to be determed. Substtutg fork, K, d K from ( 4 to (, d o equtg the terms wth detl powers of h Tlors seres, we obt the followg results: A = f B = f ( A A A f = ( f f f = Df t t E = ( f Df D f D = [ ( ( ( ] f Df ( Df Df ( f D f D f F = [ [( ( ( ( ( f Df Df Df ( fd f D f ] [( ( ( ( ( f Df Df Df (4

3 8 Jfr Bzr d Mesm Nvd: Implt Ruge-Kutt Method for V Der Pol Problem ( fd f D f ] [( ( ( ( f Df Df Df f D f D f f [ (( fdf Df (( fdf Df (( fdf Df ] ft ( fd f ( fttdf ( f ftdf 4 4 f ( fdf D f,, 4 ( ( ]] (5 Numerl method equto ( wth the help of (4 m be wrtte s = h ( w A w A w A h ( w B w B w B h ( w E w E w E h ( w D w D w D... 4 ( Where A, B, E, D, d F re gve b.(5 B equtg the oeffets of the terms wth the detl powers of h ( d (, the followg equtos re obted. whh re the sme s the sstem of equtos Buther trodued ( [] & [] (7 : w = (7 b : w = (7 : w = (7 d : w = 4, = (7 e : w (7 f : w 8, = ( 7 g : w 8, = ( 7 h : w, = ( 7 : w k k =. 4,, k = w,, be determed from (7 7 d, d from (7e 7 d three more the followg equtos,,, =,,, wll be lulted. Buther (94 trodued RK method bsed o the Rdu d Lobtto qudrture formuls. I ths proedure the oeffets re tke the Rdu's roots of the Legedre poloml of degree three: d dx ( ( 4 4 x x = =. The soluto of the sstem (7, fter substtuto of the vlues of results the followg oeffet of Rdu formul of order fve: C W = A = Substtutos of these vlues (, leds to:

4 Appled d Computtol Mthemts 5; 4(-: - 9 (8. Numerl Exmple To llustrte the method, let's ppl IRK o the followg stff problem, whh s kow s V der pol equto: ε '' ( ' =, ( '( =. I the frst step b osderg the ew depedet vrbles, V der pol equto s wrtte, equvletl, s the followg sstem of two frst order dfferetl equto: u u ' = f ( u, v = v ε ( u, u( = v ' g ( u, v u, v ( ε = = = For pplg tertve formul ( 8, to the sstem ( 9 the prmeters K, K should be omputed from the u v IRK_NEWTON ALG( ɛ, step sze h, u, v (9 followg formuls: Ku = hf ( t h, u, Ku, v, Kv Kv = hg ( t h, u, Ku, v, Kv Where,,. The u = u K K K 9 v = v K K K 9 u u u v v v ( Pseudo ode Algorthm: Let's expl the bove method wth followg Pseudo ode of Newto tertve proedure for solvg exmple ( 9 :( tke,, The results of pplg ths lgorthm to the V der pol

5 Jfr Bzr d Mesm Nvd: Implt Ruge-Kutt Method for V Der Pol Problem equto, re plotted for ε =, h =., the followg Fg. must be dded to the omputed qutt order tht the result be extl equl to the qutt tht we re lookg for. Ths mes: ( true omputed qutt T = ( ext soluto The the ext vlue of ( t wll stsf ( t = ( t hϕ ( t, ( t, h T Where ϕ( t, ( t, h s futo of the rgumet t,, d h, d s lled the remet futo, d T s the lol truto error. Let's tke omputed qutt of (8 we get: ( t = t s Fg.. The soluto of Vs der pol equto 4. Truto Error Alss Defto:[] truto error s the qutt T whh f ( t, = t 5 ( t = ( t K K K T t = ( t h = t h( t h h( t h 5 h( t h T T = h 9 The trdtol vlue of the truto error s usull lled Lpuov futo, defed o the set ( osdered s: T = C h ( ζ. omprg wth the S p = { X R : X < P}, d V ( X the the bove vlue of truto error results C =. zero soluto of X = F ( X s stble [9]. 7 Theorem. Let's the postve defte futo So the truto error s of O ( h,.e. the method V ( X exsts s suh V ( X o the ope set Ω, (8 s of order fve. (see [] d [] X = F X, F s futo F : Ω R,M ludes 5. Stblt Alss of the V Der Pol Sstem Defto. Stblt d smptot stblt: The X = F X, s stble, f soluto of sstem of equto s ( for ll t t we get: [9] ε > δ > ( x (t x (t < δ x ( t x ( t < ε ε d the soluto s smptot stble, f t s stble d lso δ > ( x (t x (t < δ x (t x ( t, s t Defto. Ivrt set: A set suh s M R s vrt set of sstem of equto X = F ( X, f from X M olude tht X ( t, X M, for ll t R [9]. Theorem l. If the slr postve defte futo V ( X, ( ll vrt subsets of E λ, where = { : = } d λ = { X R V X λ} E X R V : (. The soluto X ( t, X C λ X = F X, overges to them []. Corollr. Let's the ssumptos of the theorem hold. If zero s ol vrt pot of E, the the zero soluto of X = F ( X s smptotll stble []. of ( Stblt of the sstem of equto (9 s proved the followg: Let's V ( u, v = ( u v, whh s postve defte o R d u V ( u, v = εu (, the o the strp { ( u, v R } : u, v Ω= < < < <, we hve. It s obvous tht for E = {( u, v Ω : u = }, sstem ( 9 s overted to the followg sstem: V ( X

6 Appled d Computtol Mthemts 5; 4(-: - u ' = v v ' = Regrdg the lst orollr the set ludg zero s the ol vrt subset of., s the smptot E Ad ( stble pot of (9. To determe the smptot stblt re, Let's osder the set of urves V ( u, v = λ, where λ, ( u, v Ω. Ths set s obvousl losed d the urves re smmetr wth respet to the u xs. The futo u s deresg o the tervl (, ] d reses o the < u <. For the ostt λ, the urve V ( u, v = λ uts the borders t oe of the pots (, or (,. So the best vlue for the prmeter λ s equl to ( (, d smptot λ = m (, = stblt re ossts of the potes the losed rle C = { ( u, v Ω : u v λ }. The lmt le of the V der pol equto s out of the rle u v = (see [] p-.. Colusos Se there re three dfferet tops studed, oluso s lso dvded dfferet prts; A-From truto error seto, we olude tht order of ths mplt method s fve, d ths mes tht ths umerl method s prese for polomls of degree less th sx. B- The dsdvtge of the Ruge-Kutt methods s tht the volve osderbl more omputtos, but hve the dvtge of self strtg. wth h =. ol used utl C- Method ( ε =, d for ε >, the order of the method should resed or relted step sze deresed. D- From stblt lss seto, we olude tht the method ppled to the V der pol equto s stble, Ad ths mes the formul of the umerl method s sestve to smll hge the lol errors. Referees [] Buther J.C. Numerl Methods for Ordr Dfferetl Equtos. Joh Wle,. [] Frk R, Shed J, Uberhuber C.W: Order results for mplt Ruge-Kutt methods ppled to stff sstems. SIAM J. Numer. Al.,, (985. [] Hrer E, Lubh C, Rohe M: Error of Ruge-Kutt methods for stff problems studed v dfferetl lgebr equtos. BIT 8, 78-7 (988. [4] Hrer E, Lubh C, Rohe M: The Numerl Soluto of Dfferetl-Algebr Sstems b Ruge-Kutt Methods. Sprger verlg (989. [5] Hrer. E, Wer. G & Nørsett S.P. " Solvg Ordr Dfferetl Equtos I, ostff problems ", Sprger Seres Computtol Mthemts 4, DOI.7/ , Sprger-Verlg Berl Hedelberg. [] J M.J. Numerl Soluto of Dfferetl Equtos. Joh Wle & Sos (As Pte Ltd(979. [7] Klm R. E. & Bertrm J. F: "Cotrol Sstem Alss d Desg v the Seod Method of Lpuov", J. Bs Egrgvol.88 9 pp.7; 94. [8] Lefshetz.s. Dfferetl equto: Geometr theor, d edto. Itersee, New York, 9. [9] Lkshmkthm v, Leel s: Dfferetl d tegrl equltes: theor d ppltos, volume I, Alem Press.(99 [] Rm Mh Ro.M. A ote o tegrl eqult, J. Id Mth, So. 7, 7-9, 9. [] Rm Moh Ro.M. Ordr dfferetl equtos : theor d ppltos, Lodo : E. Arold, 98, ISBN :

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

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