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

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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 fourth-order 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 [38-4], the subscrit deotes the th-order 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 fourth-order 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 fourth-order 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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