Procedia Computer Science

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1 Procedi Compuer Science 00 (0) Procedi Compuer Science The Third Informion Sysems Inernionl Conference The Exisence of Polynomil Soluion of he Nonliner Dynmicl Sysems Gunwn Nugroho *, Took Soehrno b, Took R. Biyno c Deprmen of Engineering Physics, Insiu Teknologi Sepuluh Nopember Jl Arief Rhmn Hkim, Surby, Indonesi (60) Absrc The mehod o solve he nonliner differenil euion wih vrible coefficiens is presened in his work. The mehod is bsed on he pplicion of generl Ricci euion, which is subsiued ino he considered nonliner euion nd produce polynomil euion. The polynomil order higher hn four is reduced ino he fourh order nd solved by rdicls. The soluions of he Ricci nd polynomil re hen combined performing he complee soluion which is smooh for ll ime. 0 Published by Elsevier Ld. Selecion nd/or peer-review under responsibiliy of [nme orgnizer] Keywords: Nonliner ordinry differenil euions, he Ricci euion, nlyicl soluions, dynmicl sysems. Inroducion Mny of he more relisic modeling in he dynmicl sysems re bsed on nonliner differenil euions []. The pplicion someimes involve wih vrible coefficiens []. I is well-known h he uliive nlysis of he nonliner differenil euions is sufficien o know he globl behvior on he soluions [3,4]. However, he conceps will no be very useful unil he explici soluions re produced. The explici soluions re cpble o describe he deil feures of he sysems [5]. They my lso help o exend he exisence, uniueness nd regulriy properies of he soluions which re obined from uliive nlysis [6,7]. Therefore, mehod for genering nlyicl soluions of he nonliner differenil euions is imporn from boh physicl nd mhemicl poin of views [8]. The cse of conrol sysem wih nonhomogenous physicl propery such s in vibrion wves is of priculr ineres becuse i resuled in he model wih vrible coefficiens. Since h specific problem rcs mny mhemicins nd *Corresponding uhor gunwn@ep.is.c.id, gunwnzz@yhoo.com, gunwnf3@gmil.com b ooksf@ep.is.c.id, ooksf@yhoo.com c rb@ep.is.c.id, cllrb@gmil.com, cllrb@yhoo.com

2 Gunwn Gunwn Gunwn Nugroho Nugroho Gunwn e e l./ l. / Informion Sysems Inernionl Conference 05 Auhor nme / Procedi Compuer Science00 (05) physiciss, he mehods o obin exc nd pproxime soluions re ckled sysemiclly nd some ineresing resuls my be produced [6,9,0]. In his pper, he following clss of euion will be discussed, m n m n y y y y y y y m y m n m n p m p p f, y,,..., f, y,,...,... f, y,,..., y 0 () which he mehod for obining exc soluions is conduced by using he Ricci euion. The procedure for obining he soluions is derived in uie simple wy which is bsed on he subsiuion ino he considered euion. The resuling expression is hen produced polynomil euion which hen combined wih he soluion of he Ricci o form finl soluions. The mehod h is explined in his work is new nd he resuls my hve significn conribuions in he re of differenil nd inegrl euions.. Mehod of Genering Soluions Le us consider he following nonliner differenil euions wih vrible coefficiens, y y F, y,,..., 0 () The soluion of () is consruced from he following Ricci euion, y y y 3 (b) wih,, nd 3 re vrible coefficiens. Before subsiue (b) ino (), he following relion is produced, y y y 3 y y y b y y 3 y y y y 6 y y y y... (c) (d) (e) The procedure described in (b d) will produce he following sysem, y y y 3 (f)

3 r r r r3 r Gunwn Nugroho e l. / Informion Sysems Inernionl Conference 05 Auhor nme / Procedi Compuer Science 00 (05) y y y... y y 0 (g) where r re he coefficiens from he subsiuion ino (). Firsly, he Ricci euion is rnsformed s in he following, u u 3u 0 (3) Wih u y u. b b Lemm: Consider euion (3), le A nd 3 A A b b, euion (3c) hen hs closed-form exc soluion which b, bnd A re redefined. The soluion will leve A s n rbirry funcion. b Proof: Suppose, A b rnsformed ino, u A bu b Au 0 b b Le, Au Z, he euion cn be rerrnged s, Z A Z b b Z 0 or b A b A Z b Z A A A b A A b nd 3 A A b b b A A A A A A Z 0 A b A b A Le, b b A b A b A A A A A A A A A A, he bove euion is (3b) (3c), he euion for A is hen, Muliply by he funcion nd differenie (3d) once o perform he non homogenous euion, A A A A where will be deermined ler. Suppose h, (3d) (4)

4 Gunwn Nugroho e l. / Informion Sysems Inernionl Conference 05 4 Auhor nme / Procedi Compuer Science00 (05) b3 x x A3 nd 3 A4 A3 b3 (4b) The euion is hen rewrien s, A3 3 b3 b 4 A4 3 0 (4c) b3 A A A A 3 3 If x A4 3 hen, C exp 3 d, hus euion (4c) cn A be rrnged s, b 3 H H 3 4 A b H H b 3 b3 H b 4 H A4 0 b3 (5) wih, H. The vribles in (5) is reled s in he following, A A3 b3 b nd b b b3 A3 A3 b3 A 3 A4 b3 b3 The relion for b 3 is, 3 b3 A3 A b b b3 x Subsiue ino, A3 b3 o give he expression for b 4 s, 3 r 3 A 3 (5d) x Performing (5d) ino, 3 A4 A3, o produce A 3, A4 3 3 A (5e) The nex sep is o rele he funcion A 4, s A4 3, which produce he soluion of s, Ce. Thus, by (5b) he euion for A 4 is, (5b) (5c)

5 r b 4r 3 A4 A3 or Gunwn Nugroho e l. / Informion Sysems Inernionl Conference 05 Auhor nme / Procedi Compuer Science 00 (05) A4 A3 or 3 A which hen solves s in he following, (6b) where, (6c) The euion for H becomes, Ce d 3 H H H b 3 5 b 5 0 (7) wih, A3 b3 b5 b3 A 3 A4 A3 b3 b3. Euion (7) cn be rnsformed ino, K K b 5 k b 5 0 H wih, e Kd. The bove relion hs s priculr soluion, he generl soluion is governed by K, which resuled in, l (6) l b5 l 0 or b 5 l l l l l The soluion for A is hen, (7b)

6 Gunwn Nugroho e l. / Informion Sysems Inernionl Conference 05 6 Auhor nme / Procedi Compuer Science00 (05) Kd d H e d C4 l e d C4 A exp b5d exp b5d d C3 d b5d C e e d C4 C e e d C 3 d C4 (7c) The euion (3c) becomes, Z b Z A A A 0 b A A The soluion for he Ricci euion is hen, u A Z y u Z A d d A C A A e d C C A A e d C A where A is deermined by (7c). (8) (8b) Thus, he procedure leves A s n undefined vrible, his proves lemm Since he soluion of he polynomil by rdicl is limied o he fourh order, he reducion of polynomil order should be performed. The ineresed reder will find he mehod of reducion in []. Theorem : Consider he soluion of he euion (3c) s described by (8b). By combining wih he roo of polynomil, y, he resuling expressions hus complee he soluion of he sysem defined byricci nd polynomil euions. Proof: Le y is he polynomil soluion, he combinion wih (8b) will deermined A s, d d (8c) A C3 e A e A which hen proves heorem. 3. Soluion Properies Now we re sep o nswer nd proof he uesions of exisence nd uniueness of smooh soluion. Since he coefficiens of he Ricci euion re rbirry, hey cn become powerful objecs o jusify he properies under generl iniil-boundry condiions.

7 Gunwn Nugroho e l. / Informion Sysems Inernionl Conference 05 Auhor nme / Procedi Compuer Science 00 (05) Uniueness Propery Le us consider he second order ODE which he soluion nd iniil condiion re reled s, Y y y Y 0 y 0 y 0 0. Subsiuing he soluion pirs ino (8b) will hen produce nd uniue soluion for y since i is from liner ODE. As for he polynomil soluion, under he proper selecion of coefficiens will lso produce uniue soluion. 3.. Exisence nd Regulriy Properies Apr from uniueness, he exisence nd regulriy properies depend on he chosen funcion of he vrible coefficiens of Ricci euions, i. Also he proper selecion of coefficiens will produce for ll ime. 4. Conclusions The mehod for he genering soluion of he nonliner differenil euion is proposed in his ricle. The min sregy is o subsiue he Ricci euion ino he considered euion. The resuled polynomil is hen solved by rdicls nd combined wih he soluion of he Ricci euion. I is shown h he mehod cn obin he soluions of rbirry coefficiens nd rbirry order in closed-form. The soluion is exis nd smooh for ll ime. We pln o conduc he pplicions in our fuure works. References [] Brrio RA nd Vre C, Non-Liner Sysems, Physic A 37, 006, pp [] Sdri S, Rveshi MR nd Amiri S, Efficiency Anlysis of Srigh Fin wih Vrible He Trnsfer Coefficien nd Therml Conduciviy, Journl of Mechnicl Science nd Technology 6 (4), 0, pp [3] Dohery MFnd Ollio JM, Chos in Deerminisic Sysems: Srnge Arcors, Turbulence nd Applicions in Chemicl Engineering, Chemicl Engineering Science, Vol. 43, No., 988, pp [4] Clogero F, Gomez-Ulle D, Snini PM nd Sommcl M, Towrds Theory of Chos Explined s Trvel on Riemnn Surfces, J. Phy. A: Mh. Theor. 4, 009. [5] Abdel-Gwd HI, On he Behvior of Soluions of Clss of Nonliner Pril Differenil Euions, Journl of Sisicl Physics, 999, Vol. 97, Nos. /. [6] Rmos JI, Anlyicl nd Approxime Soluions o Auonomous, Nonliner, Third-Order Ordynry Differenil Euions, Nonliner Anlysis: Rel World Applicions, 00, pp [7] Heywood JG, Ng W nd Xie W, A Numericlly Bsed Exisence Theorem for he Nvier-Sokes Euions, Journl of Mhemicl Fluid Mechnics, 999, pp [8] Glkonov VA nd Svirshchevskii AR, Exc Soluions nd Invrin Subspces of Nonliner Pril Differenil Euions in Mechnics nd Physics, Tylor & Frncis Group, Boc Ron, 007. [9] Bougoff L, On he Exc Soluions for Iniil Vlue Problems of Second Order Differenil Euions, Applied Mhemics Leers, 009, pp [0] Adomin G, Solving Fronier Problem of Physics: The Decomposiion Mehod, Kluwer Acdemic Publishers, Dordrech, The Neherlnds, 994. [] Nugroho G, Applicion of Firs Order Polynomil Differenil Euion ofr Genering Anlyicl Soluions o he Three-Dimensionl Incompressible Nvier-Sokes Euions, Europen Journl of Mhemicl Sciences, Vo., N0., 03, pp