Exact and Numerical Solutions for Nonlinear. Higher Order Modified KdV Equations by Using. Variational Iteration Method

 Lionel Gibbs
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1 Adv. Studies Theor. Phys., Vol. 4, 2, o. 9, Eact ad Numerical Solutios for Noliear Higher Order Modified KdV Equatios by Usig Variatioal Iteratio Method M. Kazemiia, S. SoleimaiAmiri 2 ad S. A. Zahedi 3 Déartmet de Mathématique et de Géie, Ecole Polytechique de Motréal, Motréal, Québec, Caada 2 School of Electrical ad Comuter Egieerig, Uiversity of Tehra, Tehra, Ira 3 Deartmet of Mechaical Egieerig, Azad Uiversity, Jouybar brach, Jouybar, Ira Abstract This aer ivestigates the imlemetatio of Variatioal Iteratio Method (VIM) to ractical ad higher order oliear equatios i kid of KortewegdeVries (KdV) equatio. The obtaied solutios from third ad fourthorder modified KdV are comared with the eact ad Homotoy Perturbatio Method (HPM) solutios. Results illustrate the efficiecy ad caability of VIM to solve high order oliear roblems desite eedlessess to ay liearizatio or erturbatio rocess. Keywords: variatioal iteratio method (VIM); Homotoyerturbatio method (HPM); higher order modified oliear KdV equatio; Solitary wave solutio.. Itroductio I 895, Korteweg ad de Vries derived KdV equatio to model Russell s heomeo of Solitos []. Solitos are localized waves that roagate without chage of its shae ad velocity roerties ad stable agaist mutual collisio [2]. It was imlemeted to model oedimesioal shallow water waves with small but
2 438 M. Kazemiia et al fiite amlitudes. It has also bee used to describe a umber of imortat hysical heomea such as mageto hydrodyamics waves i warm lasma, acoustic waves i a iharmoic crystal ad ioacoustic waves [3]. Some aers, elorig various asects of the above, ca be foud i [49]. Roseau ad Hyma coducted a useful work o a geuiely oliear disersive equatio,, >, give by []; () a secial tye of the KdV equatio. The, equatio () caot be derived from a first order Lagragia ecet for =, ad did ot ossess the usual coservatio laws of eergy that KdV equatio ossessed. The stability aalysis has show that comacto solutios are stable, where the stability coditio is satisfied for arbitrary values of the oliearity arameter. The stability of the comacto solutios was ivestigated by meas of both liear stability aalysis ad by Lyauov stability criteria as well. Ulike classical Solitos, the comactos are o aalytic solutios. The oits of oaalyticity at the edge of the comacto corresod to oits of geuie oliearity for the differetial equatio ad itroduce sigularities i the associated dyamical system for the travelig waves []. A cosiderable amout of research work has bee coducted i [2 25] to ivestigate the dee qualitative chage i the geuiely oliear heomea caused by the urely oliear disersio. First, the variatioal iteratio method was roosed by He [26 28] ad was successfully alied to autoomous ordiary differetial equatio [29], to oliear artial differetial equatios with variable coefficiets [3], to Schrodiger KdV, geeralized KdV ad shallow water equatios. The variatioal iteratio method [26 32] is a owerful method to ivestigate aroimate solutios or eve closed form aalytical solutios of oliear evolutio equatios. I additio, o liearizatio or erturbatio is required by the method. Recetly, the method has bee alied to ivestigate may oliear artial differetial equatios ad autoomous ad sigular ordiary differetial equatios such that solitary wave solutios, ratioal solutios, comacto solutios ad other tyes of solutio were foud [36]. I the ast decades, directly seekig for eact solutios of oliear artial differetial equatios has become oe of the cetral themes of eretual iterest i Mathematical Physics. I our aers, we imlemeted the variatioal iteratio for the eact solutio [33] ad aroimate umerical solutio of a oliear KdVlike equatio [34] ad oliear KdV equatio [35]. Garder [32] develoed a variatioal ad its Hamiltoia formulatio to hadle this roblem ad also Garder et al. [3] itroduced various methods for solutios of the KdV equatio. The aim of this aer is to eted the variatioal iteratio method of He [2628] to derive the umerical ad eact solutios of the third ad fourthorder geeralized oliear KdV equatios:
3 Eact ad umerical solutios t t u + uu + u =, u+ 5uu + u =, (2) The aer layout is as follows: i sectio 2, the basic theory of VIM will be reseted summarily. Practical alicatios based o VIM are rereseted i sectio 3 for third ad fourthorder modified KdV equatios. Fially, the resultat oits are illustrated i coclusio. 2. Variatioal iteratio method To illustrate the basic cocets of the variatioal iteratio method, we cosider the followig differetial equatio: Lu + Nu = g, (3) where L is a liear differetial oerator, N a oliear oerator ad g a ihomogeeous term. Accordig to the variatioal iteratio method, we ca costruct a correct fuctioal as follows: ~ u = u + λ { Lu + Nu g(, } dτ, (4) +, Where λ is a geeral Lagragia multilier [37] which ca be idetified otimally by the variatioal theory [384], the subscrit deotes the thorder aroimatio, ad u is cosidered as a restricted variatio, i.e. δ u ~ =. To illustrate the above theory, we imlemet the variatioal iteratio method for fidig the eact solutio of the oliear KdV equatios. 3. Alicatios To give a clear overview of the aalysis reseted above, we have chose to reset two test roblems (), resectively. Here we shall be articularly iterested i the geeralized third ad fourthorder KdV equatios: ut + u u + u =, ut + ( + ) u u + u =, (5) Where > 2, models the disersio ad the subscrits i t ad deote artial derivatives with resect to these ideedet variables. 3.. Eamle Firstly we cosider to this equatio:
4 44 M. Kazemiia et al, Subject to the iitial coditios: (6a ) 2 = (6b) u) [ Asec h ( k )], 2( + )( + 2) 2 Where 2, km,,, are costats ad A = k, 2 m To solve Eq. (5a) by meas of the variatioal iteratio method, we costruct a correctio fuctioal which be as follows: t u+ = u + λ{( u) t + ( u ~ ) + ( u ~ ) } dτ, (7a) Where λ is the geeral Lagrage multilier [37] whose otimal value is foud usig variatioal theory, u is a iitial aroimatio which must be chose suitably ad, u ~ ~4 is the restricted variatio, i.e. δ u =. To fid the otimal value of λ we have t u u~ 3 u~ δ u+ = δu + δ λ{ + ( ) + } dτ, (7b) 3 t or t u δ u+ = δu + δ λ( ) dτ, (7c ) t which results i t δ u = + + δu ( λ) δu λ dτ =, (7d) which yields the statioary coditios λ ( τ ) =, (8a) λ ( τ ) =, (8b) + τ =t Therefore,λ ca be idetified as λ =, ad the followig variatioal iteratio formula ca be obtaied: u = u {( u) t + ( u ~ ) + ( u ~ ) } dτ +, (9) We start with a iitial aroimatio 2 u) = [ Asec h ( k )], give by Eq. (6b), by the above iteratio formula (9), we ca obtai directly the other comoets as
5 Eact ad umerical solutios 44, 8 cosh cosh 8 cosh 4 sih () sih cosh 5sih, To obtai the solutio of this equatio with iitial coditio, we simly took the equatio i a oerator form for = 4, =, ad used the iitial value to fid the zeros comoets i u, obtaiig i successio terms u, u2, u3, etc. 2 4 ut (, ) = [ Asec h( k c], () Where 2, k, m, are costatas ad obtai from this relatios 2 2( + )( + 2) 2 Ak A= k, c =, 2 2 m m This is eactly the same as obtaied by eact solutio [4]. The behavior of the solutio obtaied by the variatioal iteratio method ad the HPM solutio are show i Figs. (a) (b) Fig. The behavior of the solutio obtaied by the variatioal iteratio method (a), ad the HPM solutio (b) resectively.
6 442 M. Kazemiia et al 3.2. Eamle Here we cosider to fourthorder KdV equatio: 4 t u + 5u u + u =, u) = A sec h( k), (2) To solve Eq. (2) by meas of the variatioal iteratio method, we costruct a correctio fuctioal which reads t u u~ 4 u~ 4 4 δ u+ = δu + δ λ{ + u~ ( ) + } dτ, (3) 4 t I the same maer used i (7a) _ (8b), λ were obtaied as follows: λ ( τ ) =, (4a) + λ ( τ ) =, (4b) τ =t Therefore, λ ca be idetified as λ =, ad the followig variatioal iteratio formula ca be obtaied: 4 u+ = u {( u) t + ( u ~ ) + ( u ~ ) } dτ, (5) From this iteratio we obtai directly the comoets as:, 6 cosh 6 cosh cosh 4 sih cosh cosh 78 cosh 5, ad so o. I the same maer, the rest of comoets was obtaied usig the Male Package. 2 4 ut (, ) = A[sec h( K c], (7) Where 2, k, m, are costats ad 2 2( + )( + 2) 2 Ak A= k, c =, 2 2 m m This is eactly the same as obtaied by eact solutio [4]. The behavior of the solutio obtaied by the variatioal iteratio method ad the HPM solutio are show i Figs. 2. (6)
7 Eact ad umerical solutios Coclusio The aim of this work has bee to drive a aroimatio for solutio of Korteweg de Vries equatio (KdV for shor equatios. We have achieved this goal by alyig variatioal iteratio method ad the aroimate solutios are comared with the HPM solutios i Figs. 2. The results show that the reseted method is a owerful mathematical tool for fidig other solutios of may oliear KdV equatios with iitial coditios.the obtaied results are foud to be i good agreemet with the HPM solutio. We fid that, the variatioal iteratio method is a owerful method to ivestigate aroimate solutios or eve closed form HPM solutios of oliear evolutio equatios. I additio, o liearizatio or erturbatio is required by the method. (a) (b) Fig 2. The behavior of the solutio obtaied by the variatioal iteratio method (a), ad the HPM solutio (b) resectively I our work, we use the Male Package to calculate the series obtaied from the variatioal iteratio method Refereces [] D.J. Korteweg, G. de Vries, O the chage of form of log waves advacig i a rectagular caal, ad o a ew tye of log statioary wave, Philos. Mag. 39 (895)
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