Numerical Simulation of the Generalized Hirota-Satsuma Coupled KdV Equations by Variational Iteration Method

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.7(29) No.1,pp Numerical Simulation of the Generalized Hirota-Satsuma Coupled KdV Equations by Variational Iteration Method Laila M. B. Assas Department of Mathematics, Um-Al-qurah University, Makkah, Saudi Arabia (Received 31 December 27, accepted 12 October 28) Abstract:In this paper, the variational iteraton method is used for solving the Generalized Hirota- Satsuma Coupled KdV ( GH- S KdV ) equations. In this method general Lagrange multipliers are introduced to construct correction functionals for the models.in the current paper, we are applied this technique on interesting and important model.the results are compared with exact solution. Key words: the variational iteraton method; the Generalized Hirota- Satsuma Coupled KdV ( GH- S KdV ) equations 1 Introduction Nonlinear evolution equations have many wide array of application of many fields, which described the motion of the isolated waves, localized in a small part of space, in many fields such as physics, mechanics, biology, hydrodynamic, plasma physics, etc.. To further explain some physical phenomena, searching for exact solutions of NLPDEs is very important. Up to now, many researchers in mathematical physics have paid attention to these topics, and a lot of powerful methods have been presented such as: The modified extended tanh- function method [1], F-expansion method [2], homotopy analysis method [3], Variational iteration method ( [4]-[1]), Jacobi elliptic function method ( [11]-[13]), the projective Riccati equations method [12], Darboux transformation[14], The improved F- expansion method[15]etc..in this letter, we consider the following generalized Hirota- Satsuma Coupled KdV system [16]: u t = 1 4 u xxx + 3 u u x + 3 ( v 2 + w) x, (1) v t = 1 2 v xxx 3 u v x, (2) w t = 1 2 w xxx 3 u w x, (3) found many solutions for system ( 1, 2, 3 ) by a method based on the Riccati equation[17]. studied the explicit solutions of system ( 1, 2, 3) by using of Jacobi elliptic function( [11],[18]). investigated system ( 1, 2, 3 ) using a new transformation based on the coupled projectivericcati equations[19]. The aim of this paper is to extended the variational iteration method (VIM) proposed by He ( [2]-[26]) to solve the generalized Hirota- Satsuma Coupled KdV ( GH-S-C KdV ) equations and comparison with exact solution for different values of constants with two initial solutions. In this paper, we will survey the generalized coupled H-S KdV of the VIM. Some of the advantages of VIM are that the initial solutions can be freely chosen with some unknown parameters and that we can easily achieve the unknown parameters in the initial solutions. address: aslaila@uqu.edu.sa Copyright c World Academic Press, World Academic Union IJNS /2

2 68 International Journal of Nonlinear Science,Vol.7(29),No.1,pp Variational iteration method To illustrate its basic concepts of variational iteration method, we consider the following differential equation: Lu + Nu = g(x), where L is a linear operator, N a nonlinear operator, and g(x) an inhomogeneous term. According to the variational iteration method, we can construct a correct functional as follows: u n+1 = u n + λ{lu n(τ) + N u n (τ) g(τ)}dτ, whereλ is a general Lagrangian multiplier, which can be identify optimally via the variational theory, the subscript n denotes the nth-order approximation,δ ũ n is considered as a restricted variation ( [18]-[26]), i.e To illustrate the theory above, one example of special interest such as Generalized Hirota-Satsuma Coupled KdV equations is discussed in details and the obtained results are shown in the end of the next section. 3 Applications 3.1 Generalized Hirota-Satsuma Coupled KdV equations To illustrate its basic concepts of variational iteration method, we consider the system of Eqs. (1-3) with the initial conditions as follwing: u(x, ) = f(x); (4) v(x, ) = h(x); (5) w(x, ) = g(x) (6) To solve Eqs.(1-3) by means of He s variational iteration method, we construct a correction functional which can be written as: u n+1 (x, t) = u n (x, t) + v n+1 (x, t) = v n (x, t) + w n+1 (x, t) = w n (x, t) + λ 1 (τ) {u nt 1 4 u nxxx 3 u n u nx 3 ( v 2 n + w n ) x } dτ (7) λ 2 (τ) {v nt v nxxx + 3 u n v nx }dτ (8) λ 3 (τ) {w nt w nxxx + 3 u n w nx }dτ (9) where λ 1, λ 2 and λ 3 are general Lagrange multipliers, which can be identified optimallyvia variational theory ( [18]-[26]). After some calculation, we have the following stationary conditions: λ 1(τ) =, (1) 1 + λ 1 (τ) τ= t =, (11) λ 2(τ) =, (12) 1 + λ 2 (τ) τ= t =, (13) λ 3(τ) =, (14) 1 + λ 3 (τ) τ= t =, (15) Eqs. 1, 12 and 14 are called Lagrange- Euler equations, and Eqs.11, 13 and 15 are natural boundary conditions. The Lagrange multiplier, therefore, can be identified: λ 1 = λ 2 = λ 3 = 1, (16) IJNS for contribution: editor@nonlinearscience.org.uk

3 Laila M. B. Assas: Numerical Simulation of the Generalized Hirota-Satsuma Coupled KdV 69 As a result, we obtain the following iteration formula of the form : Case 1 u n+1 (x, t) = u n (x, t) v n+1 (x, t) = v n (x, t) {u nt 1 4 u nxxx 3 u n u nx 3 ( v 2 n + w n ) x } dτ (17) w n+1 (x, t) = w n (x, t) {v nt v nxxx + 3 u n v nx }dτ (18) {w nt w nxxx + 3 u n w nx }dτ (19) We first consider the explicit and numerical solutions of Eqs.(1-3) with the initial condition in the form u(x, ) = γ + β( α β 2 + a A b tan[ b k x ]) + α( α β 2 + a A b tan[ b k x ]) 2, (2) 1 a α v(x, ) = α β 2 + a A b tan[ b k x ], (21) w(x, ) = B o + M ( α β 2 + a A b tan[ b k x ]), (22) 4k 2 α b + 3 a β 2 12 γ α a a where A = a k ; c = 1 4 α and a, k, b, α, β, γ are arbitrary constants. We substitute the initial conditions (2-22)into the iteration formula for Eqs.(17-19) and using Mathematica consequently, the solutions of equation can be obtain u 1 (x, t) = αβ2 2 + α3 β 2 + γ + a A b 2 k 3 t sec[ b k x] 5 (.25(α 2 1)β cos[3 b k x] 4.5 a A b α sin[3 b k x]) a A b(α 2 1) β tan[ b k x] + a 2 A 2 b α tan[ b k x] a Ab k t sec[ b k x] 2 (M + β + α β α 2 β + 2 a A b (α 1) tan[ b k x]) + a Ab 2 k 3 t sec[ b k x] 4 (.75(1 α 2 )β a A bα tan[ b k x]) (23) v 1 (x, t) =.5αβ a Ab 2 k 3 t sec[ bkx] 4 + aa b tan[ bkx] + aabkt sec[ bkx] 2 (1.5αβ 2.75α 3 β 2 3γ) + aa b ( 3 + 3α 2 )β tan[ bkx] + b( 2k 2 3a 2 A 2 α) tan[ bkx] 2 (24) w 1 (x, t) = B o + M(.5αβ aab 2 k 3 t sec[ bkx] 4 + a A b tan[ bkx] + aabkt sec[ bkx] 2 (1.5αβ 2.75α 3 β 2 3γ + aa b( 3 + 3α 2 ) β tan[ bkx] + b( 2k 2 3a 2 A 2 α) tan[ bkx] 2 )) (25) and so on, in this manner the other components of the decomposition series can be easily obtained. The exact solution of this model example is given by (see ( [27],[28])) u(x, t) = α (a A b tan[ b k(x c t) ] β α 2 )2 + β ( a A b tan[ b k(x c t) ] β α 2 ) + γ (26) v(x, t) = a A b tan[ b k(x c t) ] β α 2 (27) w(x, t) = M (a A b tan[ b k(x c t) ] β α 2 ) + B o (28) IJNS homepage:

4 7 International Journal of Nonlinear Science,Vol.7(29),No.1,pp Case 2 To explore the accuracy and reliability of the variatioal iteration method for the generalized Hirota-Satsuma coupled KdV equation further, we consider the difierent initial values u(x, ) = γ + β ( αβ a µ 1 Sec[k x µ] ) + α( αβ a µ 1 Sec[k x µ] ) 2 (29) v(x, ) = αβ a µ 1 Sec[k x µ] (3) w(x, ) = ( αβ a µ 1 Sec[k x µ] ) A o + B o (31) where α, β, γ, µ, µ 1, a, k and c are arbitrary constants. For numerical purpose, the Eqs.(1, 2,3) are written an operator form as Eqs.(17-19). Similarly, we substitute the initial conditions (29, 3, 31) into Eqs.(17-19) and using Mathematica consequently, the solutions of equations(1,2,3) can be obtain in the following form u 1 (x, t) = γ + β( αβ 2 + 2a µ 1 sec[ k µ x]) + α( αβ 2 + 2a µ 1 sec[ k µ x]) a A k t µ µ 1 sec[ k µ x] tan[ k µ x] + 3 2aβ α k t µ µ 1 sec[ k µ x] tan[ k µ x] 3aβ3 α k t µ µ 1 sec[ k µ x] tan[ k µ x] + 9aβ3 α 3 k t µ µ 1 sec[ k µ x] tan[ k µ x] aβ3 α 5 k t µ µ 1 sec[ k µ x] tan[ k µ x] aβ γ k t µ µ 1 sec[ k µ x] tan[ k µ x] 2 3 2aβ γα 2 k t µ µ 1 sec[ k µ x] tan[ k µ x] 12 a 2 k t µ µ 2 1 sec[ k µ x] 2 tan[ k µ x] + 6 a 2 k tβ 2 µ µ 2 1 sec[ k µ x] 2 tan[ k µ x] 18 a 2 k tα 2 β 2 µ µ 2 1 sec[ k µ x] 2 tan[ k µ x] + 9 a 2 k tα 4 β 2 µ µ 2 1 sec[ k µ x] 2 tan[ k µ x] + 12 a 2 k t α γ µ µ 2 1 sec[ k µ x] 2 tan[ k µ x] a k 3 t β µ 3 µ 1 sec[ k µ x] 3 tan[ k µ x] a k 3 tα 2 β µ 3 µ 1 sec[ k µ x] 3 tan[ k µ x] a 3 k t α β µ µ 3 1 sec[ k µ x] 3 tan[ k µ x] 18 2a 3 k t α 3 β µ µ 3 1 sec[ k µ x] 3 tan[ k µ x] + 8 a 3 k 3 t β µ 3 µ 1 sec[ k µ x] tan[ k µ x] a k 3 t α 2 βµ 3 µ 1 sec[ k µ x] tan[ k µ x] a 2 k 3 t α µ 3 µ 2 1 sec[ k µ x] 2 tan[ k µ x] 3 (32) v 1 (x, t) = αβ 2 + 2a µ 1 sec[ k µ x] a β 2 α k t µ µ 1 sec[ k µ x] tan[ k µ x] aβ 2 α 3 k t µ µ 1 sec[ k µ x] tan[ k µ x] a γ k t µ µ 1 sec[ k µ x] tan[ k µ x] 6 a 2 β k t µ µ 2 1 sec[ k µ x] 2 tan[ k µ x] + 6a 2 β α 2 k t µ µ 2 1 sec[ k µ x] 2 tan[ k µ x] a k 3 t µ 3 µ 1 sec[ k µ x] 3 tan[ k µ x] a 3 α k t µ µ 3 1 sec[ k µ x] 3 tan[ k µ x].7717 a k 3 t µ 3 µ 1 sec[ k µ x] tan[ k µ x] 3 (33) w 1 (x, t) = A( αβ 2 + 2a µ 1 sec[ k µ x]) + B o a A k t α β 2 µ µ 1 sec[ k µ x] tan[ k µ x] a A k t α 3 β 2 µ µ 1 sec[ k µ x] tan[ k µ x] a γ k t µ µ 1 sec[ k µ x] tan[ k µ x] 6 a 2 k t µ µ 2 1 sec[ k µ x] 2 tan[ k µ x] + 6 a 2 k t α 2 β µ µ 2 1 sec[ k µ x] 2 tan[ k µ x] a A k 3 t µ 3 µ 1 sec[ k µ x] 3 tan[ k µ x] a 3 A k t α µ µ 3 1 sec[ k µ x] 3 tan[ k µ x].7717 a A k 3 t µ 3 µ 1 sec[ k µ x] tan[ k µ x] 3 (34) and so on.the exact solution of this model example is given by (see [28]) u(x, t) = γ + β ( α β 2 + 2a µ 1 sec[ k ( x c t )µ ]) + α ( α β 2 + 2a µ 1 sec[ k ( x c t )µ ]) 2 (35) v(x, t) = α β 2 + 2a µ 1 sec[ k ( x c t )µ ] (36) w(x, t) = ( α β 2 + 2a µ 1 sec[ k ( x c t )µ ]) A o + B o (37) IJNS for contribution: editor@nonlinearscience.org.uk

5 Laila M. B. Assas: Numerical Simulation of the Generalized Hirota-Satsuma Coupled KdV Numerical results To demonstrate the convergence of the VIM, the results of the numerical example are presented and only few terms are required to obtain accurate solutions. The accuracy of the VIM for the generalized Hirota- Satsuma Coupled KdV equation is controllable, and absolute errors are very small with the present choice of t, x. These results are listed in Tables 1 and 2, it is seen that the implemented method achieves a minimum accuracyof three and maximum accuracy of nine significant figures for Eqs.1, 2 and 3, for the first two approximations. Both the exact and the approximate solutions obtained for the first two approximations are plotted in Figs.1 and 2. There are no visible differences in the two solutions of each pair of diagrams. It is also evident that when more terms for the VIM are computed the numerical results get much more closer to the corresponding exact solutions with the initial conditions 2, 21, 22 and 29, 3, 31 of Eq. 17, 18, 19, respectively. Table1: Comparison between the numerical results for Eqs.17, 18, 19 subject to the initial conditions 2, 21, 22 by using the VIM with those obtained by the exact solution 26, 27, 28. Table 1: maximum pointwise error when a = 1, α = 1, M =.1, Bo =.1, k =.1, b =.2, γ =.5 and β =.3 t max u(x, t) u 2 (x, t) max v(x, t) v 2 (x, t) max w(x, t) w 2 (x, t) Table2: Comparison between the numerical results for Eqs.17, 18, 19 subject to the initial conditions 29, 3, 31 by using the VIM with those obtained by the exact solution 32, 33,34. Table 2: maximum pointwise error when a = 1, α = 1, B o =.1, k =.1, β =.3, γ =.5, c = 1, µ = 1 2 and µ 1 =.1 t max u(x, t) u 2 (x, t) max v(x, t) v 2 (x, t) max w(x, t) w 2 (x, t) Figure 1: (a) Figure 2: ( a ) (See Figs.1-6)The VIM results u(x, t), v(x, t) and w(x, t) for the first two approximations, shown in (a ), ( b ) and ( c ) in comparison with the analytical solutions, shown in (a), (b) and (c) when a = 1, α = 1, M =.1, B =.1, k =.1, b =.2, γ =.5 and β =.3, with the initial conditions (2), ( 21) and (22), respectively. IJNS homepage:

6 72 International Journal of Nonlinear Science,Vol.7(29),No.1,pp Figure 3: (b) Figure 4: b Figure 5: (c) Figure 6: c (See Figs.7-12)The VIM results u(x, t), v(x, t) and w(x, t) for the first two approximations, shown in (a ), ( b ) and ( c ) in comparison with the analytical solutions, shown in (a) (b) and (c) when a = 1, α = 1, µ = 1 2, B =.1, k =.1, A =.1, γ =.5, β =.3, c = 1, with the initial conditions (35), ( 36) and (37), respectively. and µ 1 =.1 4 Conclusions In this paper, the variational iteration method has been successfully used for finding the solutions of Generalized Hirota-Satsuma Coupled KdV equations. The solutions obtained are an infinite power series for appropriate initial conditions. The results reported here provide further evidence of the usefulness of using the variational iteration method for Generalized Hirota-Satsuma Coupled KdV equations. The numerical Figure 7: (a) Figure 8: ( a ) IJNS for contribution: editor@nonlinearscience.org.uk

7 Laila M. B. Assas: Numerical Simulation of the Generalized Hirota-Satsuma Coupled KdV 73 Figure 9: (b) Figure 1: ( b ) Figure 11: (c) Figure 12: ( c ) results obtain for nth approximation and compared with the known analytical solutions and the results show that we achieved an excellent approximation to the actual solutions of the equations by using only two iterations. The results show that the variational iteration method is a powerful mathematical tool for solving the Generalized Hirota-Satsuma Coupled KdV equations, it is also a promising method to solve other nonlinear equations. The solutions obtained are shown graphically. References [1] A.H.A. Ali: The modified extended tanh- function method for solving coupled KdV equations. Physics Letters A. 363: (27) [2] M. A. Abdou: The extended F-expansion method and its application for a class of nonlinear evolution equations. Chaos. Solitons and Fractals. 31:95-14 (27) [3] S. Abbasbandy: The application of homotopy analysis method to solve a generalized Hirota- Satsuma coupled KdV equation. Physics Letters A. 361: (27) [4] M. A. Abdou, A. A. Soliman: New applications of variational iteration method. Physica D. 211:1-8 (25) [5] M. A. Abdou, A. A. Soliman: Variational iteration method for solving Burger s and coupled Burger s equations. Journal of Computational and Applied Mathematics. 181 (2): (25) [6] M. T. Darvishi, F. Khani, A. A. Soliman: The numerical simulation for stiff systems of ordinary differential equations. Computer and Mathematics with Applications. In press [7] A. A. Soliman, M. A. Abdou: Numerical solutions of nonlinear evolution equations using variational iteration method. Journal of Computational and Applied Mathematics. 27: (27) IJNS homepage:

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