Applied Mahemaical Sciences, Vol. 2, 28, no. 1, 471-477 Applicaion of He s Variaional Ieraion Mehod for Solving Sevenh Order Sawada-Koera Equaions Hossein Jafari a,1, Allahbakhsh Yazdani a, Javad Vahidi b, Davood D. Ganji c a Deparmen of Mahemaics, Universiy of Mazandaran, Babolsar, Iran jafari h@mah.com, yazdani@umz.ac.ir b Deparmen of Compuer, Shomal Universiy, Amol, Iran j.vahidi@shomal.ac.ir c Deparmen of Mechanical Engineering, Universiy of Mazandaran, Babol, Iran mirgang@ni.ac.ir Absrac In his paper, He s variaional ieraion mehod (VIM) has been used o obain soluions of he sevenh-order Sawada-Koera equaion (ssk) and a La s sevenh order KdV equaions(lskdv).the numerical soluions are compared wih he Adomian decomposiion mehod(adm) and he known analyical soluions.the work confirms he power of he VIM in reducing he size of calculaions w.r.. ADM. Some illusraive eamples have been presened. Mahemaics Subjec Classificaion: 47J3,49S5, 35G25 Keywords: Variaional ieraion mehod; The sevenh order Sawada- Koera equaion; La s sevenh order KdV equaion; Adomian decomposiion mehod 1 Inroducion Analyical mehods commonly used o solve nonlinear equaions are very resriced and numerical echniques involving discreizaion of he variables on he oher hand gives rise o rounding off errors. Recenly inroduced variaional ieraion mehod by He[5, 6, 7, 8], which gives rapidly convergen successive approimaions of he eac soluion if such a soluion eiss, has proven successful in deriving analyical soluions 1 auhor for correspondence
472 H. Jafari, A. Yazdani, J. Vahidi and D. D. Ganji of linear and nonlinear differenial equaions. This mehod is preferable over numerical mehods as i is free from rounding off errors and neiher requires large compuer power/memory. He [6, 7, 13] has applied his mehod for obaining analyical soluions of auonomous ordinary differenial equaion, nonlinear parial differenial equaions wih variable coefficiens and inegrodifferenial equaions. The variaional ieraion mehod was successfully applied o Burger s and coupled Burger s equaions [1], o Schruodinger-KdV, generalized KdV and shallow waer equaions [2], o linear Helmholz parial differenial equaion [11]. Linear and nonlinear wave equaions, KdV, K(2,2), Burgers, and cubic Boussinesq equaions have been solved by Wazwaz [14, 15] using he variaional ieraion mehod. In he presen paper we employ VIM mehod for solving following equaions. u + (63u 4 + 63(2u 2 u + uu 2 ) + 21(uu + u 2 + u u )+u ) =, (1) u + (35u 4 + 7(u 2 u + uu 2 ) + 7(2uu +3u 2 +4u u )+u ) =, (2) Eq. (1) is known as he sevenh order Sawada -Koera equaion [4, 9] and Eq. (2) is known as he La s sevenh-order KdV equaion [4, 12] respecively. Furher we compare he resul wih given soluions using ADM [4, 3]. The paper has been organized as follows. Secion II, gives a brief review of VIM. Secion III, consiss of main resuls of he paper, in which variaional ieraion mehod of he ssk and LsKdV equaions has been developed. In Secion IV, illusraive eamples are given. Conclusions are presened in Secion V. 2 He s variaional ieraion mehod For he purpose of illusraion of he mehodology o he proposed mehod, using variaional ieraion mehod, we begin by considering a differenial equaion in he formal form, Lu + Nu = g(, ), (3) where L is a linear operaor, N a nonlinear operaor and g(, ) is he source inhomogeneous erm. According o he variaional ieraion mehod, we can consruc a correcion funcional as follow u n+1 (, ) =u n (, )+ λ(ξ)(lu n (ξ)+nũ(ξ) g(ξ)) dξ, n, (4) where λ is a general Lagrangian muliplier [1], which can be idenified opimally via he variaional heory, he subscrip n denoes he nh order approimaion, ũ n is considered as a resriced variaion [7, 8, 1] i.e., δũ n =.
Applicaion of He s VIM 473 So, we firs deermine he Lagrange muliplier λ ha will be idenified opimally via inegraion by pars. The successive approimaions u n+1 (, ), n of he soluion u(, ) will be readily obained upon using he obained Lagrange muliplier and by using any selecive funcion u. Consequenly, he soluion u(, ) = lim n u n(, ). (5) 3 Applying VIM for ssk and LsKdV Now for applying VIM, firs we rewrie Eq. (1) in he following form L (u) + (63N 1 (u) + 63(2N 2 (u)+n 3 (u)) + 21(N 4 (u)) + L (u)) =, (6) where he noaions N 1 (u) =u 4,N 2 (u) =u 2 u,n 3 (u) =uu 2, N 4 (u) =uu + u 2 + u u, symbolize he nonlinear erms, respecively. The noaion L = and L = 6 symbolize he linear differenial operaors. 6 The correcion funcional for Eq.(6) reads u n+1 (, ) = u n (, )+ λ(ξ) ξ (u n)+(n(ũ n )) dξ, n, (7) where, N(u) = (63N 1 (u) + 63(2N 2 (u)+n 3 (u)) + 21(N 4 (u)) + L (u). Taking variaion wih respec o he independen variable u n, noicing ha δn(ũ n )= δu n+1 (, ) = δu n (, )+δ λ(ξ) ξ (u n)+(n(ũ n )) dξ = δu n (, )+λδu n ξ= λ (ξ) δu n dξ =, (8) This yields he saionary condiions 1+λ(ξ) =, λ (ξ) ξ= =. (9) This in urn gives λ(ξ) = 1. Subsiuing his value of he Lagrange muliplier ino he funcional (7) gives he ieraion formula u n+1 (, ) = u n (, ) ξ (u n)+(n(u n )) dξ, n. (1) Using he zeroh approimaion u (, ) ino (1) we obain he successive approimaions. In he same manner for LsKdV (2) we go he following ieraion formula u n+1 (, ) = u n (, ) ξ (u n)+(f (u n )) dξ, n, (11) where F (u) =35u 4 + 7(u 2 u + uu 2 ) + 7(2uu +3u 2 +4u u )+u. Finally, we approimae he soluion u(, ) = lim n u n(, ).
474 H. Jafari, A. Yazdani, J. Vahidi and D. D. Ganji 4 Illusraive Eamples To demonsrae he effeciveness of he mehod we consider here Eqs.(1) and (2) wih given iniial condiion. Eample 4.1 )[4] Consider he ssk equaion (1) wih he iniial condiion u(, ) = 4k2 3 (2 3 anh2 (k)). (12) Subsiuing (12) ino Eq.(1) we obain he following successive approimaions u (, ) = 4 3 k2 ( 2 3 anh 2 (k) ), u 1 (, ) = u (, )+ 1 9 k8 sech 2 (k) [2176 896 cosh(2k)], u 2 (, ) = u 1 (, )+ 1 27 sech8 (k)k 14 2 ([6328576 6566144 cosh(2k)+ 177248 cosh(4k) + 24832 cosh(6k) 12544 cosh(8k)] + 1 3 k6 [254481232 2746548224 cosh(2k) + 3578 cosh(4k)+ 41746432 cosh(6k) 5619712 cosh(8k)] + 1 9 k12 2 [598192256 2385514 cosh(2k) 5559342368 cosh(4k)+ 17983784 cosh(6k) 1258815488 cosh(8k)] 1 135 k18 3 [ 238459436464 + 29135957556944 cosh(2k) 828341695232 cosh(4k) + 1956737552 cosh(6k) 563949338624 cosh(8k)]) and so on. In Fig.1 and Fig. 2 we draw u 3 (, ) and u(, ) = 4k2 (2 3 3 anh 2 (k( 256k6 ))) which is he eac soluion [12] for k =.1 and 1 < 3 <1..1.1 U.2 U.2 -.1.1 -.1.1-1 -5 5 -.1 1-.2-1 -5 5 -.1 1-.2 Fig. 1. Approimae Soluion u 3 (, ) Fig. 2. Eac Soluion u(, )
Applicaion of He s VIM 475 Remark 1: ssk equaion has been solved by ADM by El-Sayed and Kaya [4]. I should be remarked ha he graph drawn here using VIM is agreemen wih ha drawn using ADM [4] bu jus afer 3 ieraion. Remark 2: In [4], for solving his equaions using ADM hey compue Adomian polynomials for N 1 (u), N 2 (u), N 3 (u) and N 4 (u). Eample 4.2 )[4] Consider he LsKdV equaion wih given iniial condiion, u(, ) = 2k 2 sech 2 (k). (13) Subsiuing (13) ino Eq.(11) we obain he following successive approimaions u (, ) = 2k 2 sech 2 (k), u 1 (, ) = u (, ) 128k 8 sech 2 (k), u 2 (, ) = u 1 (, ) 128 3 sech8 (k)k 14 2 (444192 3 k 18 + 17232 2 k 12 + 3464k 6 +( 34464 2 k 12 7168k 6 + 183) cosh(2k)+ 36(448k 6 11) cosh(4k) 3 cosh(6k) 24) and so on. Using he above erms, in Fig.3, u 3 (, ) is drawn for k =.1 and [ 1, 1]. In Fig.4, eac soluion u(, ) =2k 2 sech 2 (k) [12] is drawn..2.15 U.1.5-1 -5 5 -.1 1-.2.1.2.2.15 U.1.5-1 -5 5 -.1 1-.2.1.2 Fig. 3.Approimae Soluion u 3 (, ) Fig. 4. Eac Soluionu(, ) Remark 3: I should be remarked ha he graph drawn here using VIM is in ecellen agreemen wih ha drawn using ADM [4]bu jus afer 3 ieraion. Remark 4: In [4], for every nonlinear pars of F (u) hey have calculaed Adomian polynomials. 5 Conclusion Variaional ieraion mehod is a powerful ool which is capable of handling linear/nonlinear parial differenial equaions. The mehod has been successfully
476 H. Jafari, A. Yazdani, J. Vahidi and D. D. Ganji applied o ssk and LsKdV equaions. Also, comparisons were made beween He s variaional ieraion mehod and Adomian decomposiion mehod (ADM) for ssk and LsKdV equaions. The VIM reduces he volume of calculaions wihou requiring o compue he Adomian polynomials. However, ADM requires he use of Adomian polynomials for nonlinear erms, and his needs more work. For nonlinear equaions ha arise frequenly o epress nonlinear phenomenon, He s variaional ieraion mehod faciliaes he compuaional work and gives he soluion rapidly if compared wih Adomian mehod. Mahemaica has been used for compuaions in his paper. References [1] M.A. Abdou and A.A. Soliman, Variaional ieraion mehod for solving Burger s and coupled Burger s equaions, J. Compu. Appl. Mah., 181 ( 25) 245 251. [2] M.A. Abdou, A.A. Soliman, New applicaions of variaional ieraion mehod, Physica D, 211 (25) 1 8. [3] G. Adomian, Solving Fronier Problems of Physics: The Decomposiion Mehod, Kluwer, 1994. [4] S. M. El-Sayed and D. Kaya, An applicaion of he ADM o seven order Sawada-Koera equaions, Appl. Mah. Compu., 157 (24) 93 11. [5] J.H. He, Variaional ieraion mehod -Some recen resuls and new inerpreaions, J. Compu. Appl. Mah., 27 (1) (27) 3 17. [6] J.H. He, Variaional principles for some nonlinear parial differenial equaions wih variable coefficiens, Chaos, Solions and Fracals, 19 (24) 847 851. [7] J. H. He, Variaional ieraion mehod for auonomous ordinary differenial sysems, Appl. Mah. Compu., 114 (2) 115 123. [8] J. H. He, Some asympoic mehods for srongly nonlinear equaions, In. J. Mod. Phys. B, 2 (1) (26) 1141 1199. [9] W. Hereman, P.P. Banerjee, A. Korpel, G. Assano, A. van Immerzeele and A. Meerpoel, Eac soliary wave soluions of nonlinear evoluion and wave equaions using a direc algebraic mehod, J. Phys. A: Mah. Gen., 19 (1986) 67-?628.
Applicaion of He s VIM 477 [1] M. Inokui, e al., General use of he Lagrange muliplier in non-linear mahemaical physics, in: S. Nema-Nasser (Ed.), Variaional Mehod in he Mechanics of Solids, Pergamon Press, Oford, 1978, pp. 156-?162. [11] S. Momani and S. Abuasad, Applicaion of He s variaional ieraion mehod o Helmholz equaion, Chaocs, Solions and Fracals, 27 (26) 1119 1123. [12] E. J. Parkes and B.R. Duffy, An auomaed anh-funcion mehod for finding soliary wave soluions o non-linear evoluion equaions, Compu. Phys. Commun., 98 (1996) 288 3. [13] S. Q. Wang and J. H. He, Variaional ieraion mehod for solving inegrodifferenial equaions, Physics Leers A, (27)[In Press]. [14] A. M. Wazwaz, The variaional ieraion mehod: A reliable analyic ool for solving linear and nonlinear wave equaions, Compuers and Mahemaics wih Applicaions, (27)[In Press]. [15] A. M. Wazwaz, The variaional ieraion mehod for raional soluions for KdV, K(2,2), Burgers, and cubic Boussinesq equaions, J. Compu. Appl. Mah., 27(1) (27) 18 23. Received: Sepember 24, 27