Application of variational iteration method for solving the nonlinear generalized Ito system

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1 Applicaion of variaional ieraion mehod for solving he nonlinear generalized Io sysem A.M. Kawala *; Hassan A. Zedan ** *Deparmen of Mahemaics, Faculy of Science, Helwan Universiy, Cairo, Egyp **Deparmen of Mahemaics, Faculy of Science, Kafer el sheik Universiy, Cairo, Egyp Absrac: In his aricle, we implemen relaively analyical echnique called he variaional ieraion mehod VIM)for solving nonlinear generalized Io sysem. In his mehod, a correcion funcional is consruced by a general Lagrange muliplier. Two cases are given o illusrae he accuracy and effeciveness of he mehod.we compare our resuls wih resuls obained by exac soluion. This Comparison reveals ha he variaional ieraion mehod is very effecive, convenien and easier o be implemened. [A.M. Kawala; Hassan A. Zedan. Applicaion of variaional ieraion mehod for solving he nonlinear generalized Io sysem. Journal of American Science 11;71):65-659]. ISSN: ).. Keywords: Variaional ieraion mehod; Lagrange muliplier; nonlinear generalized Io sysem. 1. Inroducion In his paper, we exend he applicaion of he variaioal ieraion mehod o find approximae soluions for nonlinear generalized Io sysem. The variaional ieraion mehod, which proposed by Ji- Huan He [1-3], is considered o find analyic and approximae soluions of differenial equaions. I's effecively and easily used o solve some classes of nonlinear problems, Variaional ieraion mehod has been favorably applied o various kinds of nonlinear problems. The main propery of he mehod is in is flexibiliy and abiliy o solve nonlinear equaions accuraely and convenienly. Major applicaions o nonlinear wave equaion, nonlinear fracional differenial equaions, nonlinear oscillaions and nonlinear problems arising in various engineering applicaions are surveyed. The flexibiliy and adapaion provided by he mehod have made he mehod a srong candidae for approximae analyical soluions.. Variaional Ieraion mehod To illusrae he basic conceps of he Variaioal ieraion mehod, we consider he differenial equaion in he formal form Lu +Nu=gx), where L is a linear operaor, N a nonlinear operaor and gx) an inhomogeneous erm. According o VIM, we can consruc a correcional funcional as x) + x u n+1 x) = u n λ{ Lun ξ ) + Nun ξ )} dξ, where λ is a Lagrangeian muliplier [4], which can be deermined by using variaional heory, he subscrip n denoes he n-h order approximaion, and u ξ ) is considered as a resriced n variaional, i.e. u ξ ) =. 3. Applicaion Consider he nonlinear generalized Io sysem of parial differenial equaions [5] u = v x, 1) v = -v xxx + 3 u v x + 3 v u x ) 1 w w x + 6p x, ) w = w xxx + u w x, 3) p = p xxx + u p x. 4) To illusrae he degree of accuracy o VIM, wo cases of nonlinear generalized Io sysem exac are discussed in deails Nonlinear generalized Io sysem case 1 In his case he analyical soluion for sysem 1-4) ux,) = a k m Sn ξ ), 5a) vx,) = 3 k m 1 + m ) 1 k m 1 + m ) a k m 5b) k m a c1 ) + k m 1 + m ) 6 k m a ) Sn ξ) wx,) = c + c 1 Sn ξ ), 5c) px,) = e + c c 1 Sn ξ ). 5d) where a, k, m, c, c 1 and e are consan, and ξ = k x +- k 3 1+ m )+3ka ) + ξ, and ξ is consan. we sar wih iniial approximaion u = ux,), v = vx,), w = wx,) and p = px,) given by u x,) = a k m Sn k x + ξ ), 6a) n 65

2 v x,) = k m 1 + m ) 1k m k m + 9k m a c ) Sn kx + ξ ) 1 w x,) = c + c 1 Sn kx + ξ ), 6c) p x,) = e + c c 1 Sn kx + ξ ). 6d) 1 + m To solve he sysem 1-4) by means of variaional ieraion mehod, we can consruc he correc funcional as follows: ) a { v v u v v u w w p dτ,6b) n + nxxx+ 3 n nx + 3 n nx) + 1 n nx 6 nx)}, 9b) v n+1 x,) = v n x,) + w n+1 x,) = w n x,) - p n+1 x,) = p n x,) - 9d) wih n. { w w 3u w } dτ, 9c) n nxxx n nx { p p 3u p } dτ. n nxxx n nx u n+1 x,) = u n x,) + λ { u v dτ, 7a) 1 n nx} v n+1 x,) = v n x,) + λ { vn + vnxxx+ 3u nvnx + 3v nunx) + 1w nwnx + 6pnx)} dτ 7b) w n+1 x,) = w n x,) + λ w w 3u w dτ {, 7c) 3 n nxxx n nx} p n+1 x,) = p n x,) + λ { p p 3u p dτ. 7d) 3 n nxxx n nx} where λ 1, λ, 3 o deermined, and w w n nx, p nx, w n nx λ and λ4 are Lagrange mulipliers are v nx, v n nx u, v u n nx, u and u p n nx resriced variaions, i.e. = are denoes δ v nx, u v n nx = δ, δ v u n nx =, δ w w = n nx, δ p = nx, and δ u p n nx =. Is saionary condiions can be obained as follows : 1 + λ 1 τ= =, 1 + λ τ= =, 1 + λ 3 τ= =, 1 + λ 4 τ= =, 1 λ = 1 λ = 3 λ = -1, λ = -1, λ = λ 3 = -1, λ 4 = λ 4 = -1. 8a) 8b) 8c) 8d) subsiuing 8a - 8d) in 7a - 7d), and he following variaional ieraion formula can be obained u n+1 x,) = u n x,) - { u v } dτ, 9a) n nx By he above ieraion formulas 9a - 9d ),we can obain direcly he order componens as u 1 x,) = a +4 k3a k m + k 4 m + k 4 m 4 ) Cn kx + ξ ) Dn kx + ξ ) Sn kx + ξ ) - k m Sn kx + ξ ), v 1 x,) =1 k m { c 1-9 a k m +1 a k 4 m - 3 k 6 m + 1 a k 4 m 4-6k 6 m 4-3 k 6 m k 7 m 5-3 a + k 1+m )Cn 3 kx + ξ ) Dn kx + ξ ) Sn kx + ξ ) + 4k 4 m 4-3 a + k 1+ m ))Sn kx + ξ ) -8 k 5 m 4 Cn kx + ξ ) Dn kx + ξ ) Sn kx + ξ ) 9 a - 3 a k + 9 k 4-3 a k m +18 k 4 m + 9 k 4 m a k -k m ) Dn kx + ξ ) 8 k m m)-3 a + k 1 + m )Sn kx + ξ ) } w 1 x,) = c - c 1 k 3 m Cn 3 kx + ξ ) Dn kx + ξ ) + c 1 Sn kx + ξ ) + c 1 k Cn kx + ξ ) Dn kx + ξ ) 3 a - k Dn kx + ξ ) + k - 3 m) m Sn kx + ξ ), p 1 x,) = e + c c 1 Sn kx + ξ ) - c c 1 k 3 Cn kx + ξ ) Dn kx + ξ ) m Cn kx + ξ ) + Dn kx + ξ ) -4m Sn kx + ξ ) )+ 6 c c 1 k Cn kx + ξ ) Dn kx + ξ ) a -k m Sn kx + ξ ). and so on, in he same manner using Mahemaica Package, we can evaluae he numerical soluions o he res componens of ieraion formulas 9a - 9d ) wih n h approximaions n =3 ). The obained numerical resuls are summarized in Table

3 x= - x= -1 x= x= 1 x= Table 1. Comparison of he exac and numerical soluions for Io sysem Error u Error v Error w Error p E E E E E E E E E E E E E E-1.35E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-1-8.5E E E E E E E E E E E-8 From hese resuls we conclude ha he variaional ieraion mehod for Io sysem, gives high degree of accuracy in comparison wih analyical soluion 5a-5d). Now we sudy he diagrams obained by VIM and analyical soluions o show he relaion beween u,v,w,p and x wih differen values for, and he relaion beween u,v,w,p and wih differen values for x. 3.. Nonlinear generalized Io sysem case In his case he analyical soluion for sysem 1-4) ux,) = a k m Cs ξ ), 1a) vx,) = 3 k m ) 1 k m ) a + 9 k a c 1 ) 1b) k + k m ) 6 k a ) Cs ξ) 4 wx,) = c + c 1 Cs ξ ), 1c) px,) = e + c c 1 Cs ξ ). 1d) where a, k, m, c, c 1 and e are consan, and ξ = k x +- k 3 1+ m )+3ka ) + ξ, and ξ is consan. we sar wih iniial approximaion u = ux,), v = vx,), w = wx,) and p = px,) given by u x,) = a k m Sn k x + ξ ), 11a) v x,) = 3 k m ) 1 k m ) a + 9 k a c 1 ) k 11b) 4 + k m ) 6 k a) Cs k x +ξ ) w x,) = c + c 1 Sn kx + ξ ), 11c) p x,) = e + c c 1 Sn kx + ξ ). 11d) By using ieraion formulas 9a - 9d ),we can obain direcly case order componens as u 1 x,) = a - k Cs k x + ξ )+4k 3 3 a k -+m ) Csk x + ξ ) Dsk x + ξ ) Nsk x + ξ ). v 1 x,) = 1/ k ) {c 1-33 a k - 4 a k m ) + k 6 - +m ) ) - 43 a k 4 - k m ) Cs k x + ξ )+ 65

4 18 k 7 3 a - k -+m ) Cs 3 k x + ξ )Dsk x + ξ )Nsk x + ξ ) + 8 k 5 Csk x + ξ )Dsk x + ξ ) Nsk x + ξ ) 9 a + 6 a k + 36 k 4 3 a k m - 36 k 4 m + 9 k 4 m 4 83 a k - k m )Ds k x + ξ ) - 83 a k - k m )Ns k x + ξ )} w 1 x,) = c + c 1 Csk x + ξ ) + c 1 k 3 Cs k x + ξ )Dsk x + ξ ) Nsk x + ξ ) - c 1 k 3 Ds 3 k x + ξ ) Nsk x + ξ )- c 1 k Dsk x + ξ ) Nsk x + ξ ) 3 a + k Ns k x + ξ )) p 1 x,) = e + c c 1 Csk x + ξ )+ 4 c c 1 k 3 Cs k x + ξ )Dsk x + ξ ) Nsk x + ξ ) c c 1 k 3 Ds 3 k x + ξ ) Nsk x + ξ ) - c c 1 k Dsk x + ξ ) Nsk x + ξ )3 a + k Ns k x + ξ )) and so on, in he same manner we can evaluae he res componens of ieraion formulas 9a - 9d ) wih n h approximaions n =3 ).The obained numerical resuls are summarized in Table. The behavior of he soluion obained by VIM and analyic soluion are shown in Figs. 3a - 3d ) and 4a - 4d). Now we sudy he diagrams obained by VIM and analyical soluions o show he relaion beween u,v,w,p and x wih differen values for, and he relaion beween u,v,w,p and wih differen values for x. The behavior of he 3rd ieraion obained by VIM Fig[4a - 4d] and he analyic soluion Fig[3a - 3d]. Then, he surfaces respecively show he soluion ux,),vx,), wx,), px,). Fig show relaion beween u,v,w,p and x wih consan values for. 653

5 Fig show relaion beween u,v,w,p and wih consan values for x. 654

6 The behavior of he 3rd ieraion obained by VIM Fig[1a - 1d] and he analyic soluion Fig[a - d]. Then, he surfaces respecively show he soluion ux,),vx,), wx,), px,) 655

7 4- Conclusion In his paper, he variaional ieraion mehod has been successfully used o find approximae soluion for he nonlinear generalized Io sysem of parial differenial equaion. The numerical resuls obain using n approximaions n=3), compared wih analyic soluion show he high degree of accuracy. Table. comparison of he nd case exac and numerical soluions for Io sysem x= - Error u Error v Error w Error p.5.61e E E E E E E E E-1.563E E E-8 x= E E E E E E E E E E E E-7 x= E E E E E E E E E-5 x= E E-7 1.4E-7 1.4E E E E E E E E E-6 x= E E E E E E E E E E E E-8 x= E E E E E E E E E E E E-7 656

8 Fig show relaion beween u,v,w,p and x wih consan values for. 657

9 Fig show relaion beween u,v,w,p and wih consan values for x. 658

10 The behavior of he 3rd ieraion obained by VIM Fig[4a - 4d] and he analyic soluion Fig[3a - 3d]. Then, he surfaces respecively show he soluion ux,),vx,), wx,), px,) * Corresponding auhor: A.M. Kawala Deparmen of Mahemaics, Faculy of Science, Helwan Universiy, Cairo, Egyp kawala_6_1@yahoo.com References 1. J.H. He, A New Approach o Nonlinear Parial Differenial Equaion, Shanghi Universiy, Shanghi Insiue of Applied Mah. and Mechanics.. J.H. He, Variaioal ieraion mehod for delay differenial equaion, commun.nonlinear Sci. Numer. Simula. 4) 1997) M.A. Abdu, A.A. Soliman, Variaioal ieraion mehod for solving Burger s and coupled Burger sequaions, J. Comp. Appl. Mech ) M.Inokui, H. Sckine, T. Mura, General Use of he Lagrange Muliplier in Non-Linear Mahemaical Physics, in Variaioal mehod in he Mechanics of Solids, Pergamon Press, New York, 1978,pp A.A.Darwish, Hassan A.Zedan, A.M.Kawala, A.M.El Hadid " Invarian soluions and soliary wave soluions for some Parial Differenial Equaions" M.SC.7. 1//1 659