He s semi-inverse method for soliton solutions of Boussinesq system

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1 ISSN , Engl, UK World Journal of Modelling Simulation Vol. 9 (2013) No. 1, pp He s semi-inverse method for soliton solutions of Boussinesq system H. Kheir, A. Jabbari, A. Yildirim, A. K. Alomari Department of Mathematis, Faulty of Siene, Ege University, Bornova-Izmir 35100, Turkey (Reeived Deember , Aepted November ) Abstrat. In this paper, we apply He s semi-inverse method to establish a variational theory for the Boussinesq system. Based on this formulation, a solitary solution an be easily obtained using Ritz method. Moreover, the results are also ompared with He s homotopy perturbation method, Liao s homotopy analysis method homotopy padé method. The results reveal that the proposed method is very effetive simple. Keywords: He s semi-inverse method, Boussinesq system, Homotopy analysis method; Homotopy perturbation method; Homotopy padé tehnique 1 Introdution There have been various approahes to searh for soliton solutions for nonlinear wave equations. These methods inlude the inverse sattering method [19], Hirota s bilinear method [22], Bäklund transformation [20], tanh-oth method [4], Jaobi ellipti funtion method [26], (G /G)-expansion method [3, 10], homotopy perturbation method (HPM) [12], Luapanov s artifiial small parameter method, δ-expansion method, Adomian deomposition method, variational iterative method [27], homotopy analysis method (HAM), homotopy padé method (HPadéM) [1, 2, 5 7, 11, 23 25] so on. In the past few deades, qualitative analysis together with ingenious mathematial tehniques for hling various nonlinear problems has been studied. Among them, variational approahes, suh as the semiinverse method is a powerful effetive method to searh for variational priniples for physial problems provides physial insight into the nature of the solution of problem [16, 17, 29 35]. In this paper we use He s semi-inverse method [16, 17, 29 35] to establish a variational formula for Boussinesq system. The Boussinesq system [18, 21] u t + v x = 0, v t + a(u 2 ) x bu xxx = 0, (1) is used to model two-way propagation of ertain water waves in a uniform horizontal hannel filled with an irrotational invisid liquid [21, 28]. System (1) was solved using different methods. was widely disussed by many authors using different methods. Wazwaz disussed this system by the tanh method the sine-osine method [9]. 2 He s semi-inverse method We suppose that the given nonlinear partial differential equation for u(x, t) to be in the form P (u, u x, u t, u xx, u xt, u tt, ) = 0, (2) Corresponding author. address: ahmetyildirim@gmail.om. Published by World Aademi Press, World Aademi Union

2 4 H. Kheir & A. Jabbari & et al.: He s semi-inverse method for soliton solutions where P is a polynomial in its arguments. A. Jabbari et.al. in [3] have been written the He s semi-inverse method in the following steps: step 1. Seek solitary wave solutions of Eq. (2) by taking u(x, t) = U(ξ), ξ = x t, transform Eq. (2) to the ordinary differential equation Q(U, U, U, ) = 0, (3) where prime denotes the derivative with respet to ξ. step 2. If possible, integrate Eq. (3) term by term one or more times. This yields onstant(s) of integration. For simpliity, the integration onstant(s) an be set to zero. step 3. Aording to the He s semi-inverse method, we onstrut the following trial-funtional J(U) = L dξ, (4) where L is an unknown funtion of U its derivatives. There exist alternative approahes to the onstrution of the trial-funtionals, see Refs. [13 15]. step 4. By Ritz method, we an obtain different forms of solitary wave solutions, suh as U(ξ) = p sh 2 (qξ), U(ξ) = p seh 2 (qξ),u(ξ) = p tanh 2 (qξ), U(ξ) = p oth 2 (qξ) so on. For example in this paper we searh a solitary wave solution in the form U(ξ) = p seh 2 (qξ), (5) where p q are onstants to be further determined. substituting Eq. (5) into Eq. (4) making J stationary with respet to p q, we have J = 0, p (6) J = 0. q (7) Solving simultaneously the Eq. (6) Eq. (7) we obtain p q. Hene, the solitary wave solution Eq. (5) is well determined. 3 Homotopy analysis method The HAM, first proposed by Liao in his Ph.D dissertation [23], is an elegant method whih has proved its effetiveness effiieny in solving many types of nonlinear equations [1, 2]. Liao in his book [24] proved that HAM is a generalization of some previously used tehniques suh as the δ-expansion method, artifiial small parameter method Adomian deomposition method. Moreover, unlike previous analyti tehniques, the HAM provides a onvenient way to adjust ontrol the region rate of onvergene [25]. It should be noted that the HPM is a partiular ase of the HAM [23]. There exist some tehniques to aelerate the onvergene of a given series of solutions. Among them, the so-alled Padé method is widely applied. For further details, the reader is referred to Liao [24]. For onveniene of the readers, we will first present a brief desription of the stard HAM. To ahieve our goal, let us assume the nonlinear system of differential equations be in the form of N j [u 1 (x, t), u 2 (x, t),, u m (x, t)] = 0, j = 1, 2,, n, (8) where N j are nonlinear operators, t is an independent variable, u i (t) are unknown funtions. By means of generalizing the stard homotopy method, Liao onstrut the zeroth-order deformation equation as follows (1 q)l j [φ i (x, t, q) u i,0 (x, t)] = q H(t)N j [φ 1 (x, t, q), φ 2 (x, t, q),, φ m (x, t, q)], (9) i = 1, 2,, m; j = 1, 2,, n, WJMS for ontribution: submit@wjms.org.uk

3 World Journal of Modelling Simulation, Vol. 9 (2013) No. 1, pp where q [0, 1] is an embedding parameter, L j are linear operators, u i,0 (x, t) are initial guesses of u i (x, t), φ i (x, t; q) are unknown funtions, H(x, t) are auxiliary parameter auxiliary funtion respetively. It is important to note that, one has great freedom to hoose auxiliary objets suh as L j in HAM; This freedom plays an important role in establishing the keystone of validity flexibility of HAM as shown in this paper. Obviously, when q = 0 q = 1, both φ i (x, t, 0) = u i,0 (x, t) φ i (x, t, 1) = u i (x, t), i = 1, 2,, m, (10) hold. Thus as q inreases from 0 to 1, the solutions of φ i (x, t; q) hange from the initial guesses u i,0 (x, t) to the solutions u i (x, t). Exping φ i (x, t; q) in Taylor series with respet to q, one has where φ i (x, t, q) = u i,0 (x, t) + + k=1 u i,k (x, t)q k, i = 1, 2,, m, (11) u i,k (x, t) = 1 k φ i (x, t, q) q=0 k! q k, i = 1, 2,, m. (12) If the auxiliary linear operators, the initial guesses, the auxiliary parameter, the auxiliary funtion are so properly hosen, then the series Eq. (11) onverges at q = 1, then one has φ i (x, t, 1) = u i,0 (x, t) + + k=1 u i,k (x, t), i = 1, 2,, m, (13) whih must be one of the solutions of the original nonlinear equations, as proved by Liao. Define the vetors u i,n (t) = {u i,0 (x, t), u i,1 (x, t),, u i,n (x, t)}, i = 1, 2,, m. (14) Differentiating Eq. (9), k times with respet to the embedding parameter q then setting q = 0 finally dividing them by k!, we have the so-alled kth-order deformation equation L j [u i,k (x, t) χ k u i,k 1 (x, t)] = R j,k ( u i,k 1 (x, t)), i = 1, 2,, m; j = 1, 2,, n, (15) subjet to the initial onditions L j (0) = 0, where R j,k ( u i,k 1 (x, t)) = 1 k 1 N j [φ 1 (x, t, q), φ 2 (x, t, q), φ m (x, t, q)] q=0 (k 1)! q k 1 (16) χ m = { 0 m 1, 1 m > 1. (17) It should be emphasized that u i,k (x, t) is governed by the linear Eq. (15) Eq. (16) with the linear boundary onditions that ome from the original problem. These equations an be easily solved by symboli omputation softwares suh as Maple Mathematia. 4 Homotopy padé method Traditionally the [m, n] Padé for u(x, t) is in the form m k=0 F k(x)t k 1 + n k=1 F m+1+k(x)t k or m k=0 G k(t)x k 1 + n k=1 G k+m+1(t)x k, WJMS for subsription: info@wjms.org.uk

4 6 H. Kheir & A. Jabbari & et al.: He s semi-inverse method for soliton solutions where F k (r) G k (t) are funtions. In Homotopy Padé approximation, we employ the traditional Padé tehnique to the series Eq. (11) for the embedding parameter q to gain the [m, n] Padé approximation in the form of m k=0 w k(x, t)q k 1 + n k=1 w m+k+1(x, t)q k, (18) where w k (t, x) is a funtion for i = 0, 1,, m, m + 2,, m + n + 1, w i (x, t) is determined by produt of the denominator of the above expression in the m+n i=0 u i(x, t)q i equating the powers of q i, i = 0, 1,, m+n. Thus we have m+n+1 equations m+n+1 unknowns w i (x, t), i = 0, 1,, m, m+ 2,, m + n + 1. By setting q = 1 in Eq. (18) the so-alled [m, n] Homotopy Padé approximation in the following form is yield. m k=0 w k(x, t) 1 + n k=1 w m+k+1(x, t). (19) It is found that the [m, n] Homotopy Padé approximation often onverges faster than the orresponding traditional [m, n] Padé approximation in many ases the [m, m] Homotopy Padé approximation is independent of the auxiliary parameter. In these ases, even if the orresponding solution series diverge, utilizing the Homotopy-Padé tehnique will result in a onvergent series [8]. However, up to now, It has not seen a mathematial proof about it in general ases in literature [24]. The HPadéM an greatly enlarge the onvergene region of the solution series. Besides, the results solutions of HPadéM often onverge faster than solutions alulated by HAM. 5 Appliations In this setion we apply the proposed methods to solve the Boussinesq system. 5.1 He s semi-inverse method In order to seek a travelling wave solution of Boussinesq system, we introdue a transformation where is arbitrary onstant. Substituting Eq. (20) into Eq. (1) yields u(x, t) = u(ξ), v(x, t) = v(ξ), xi = x t, (20) u + v = 0, v + a(u 2 ) bu = 0, a, b 0, (21) where the prime expresses the derivative with respet to ξ. Integrating the resulting system, negleting the onstant of integration we find As a result we obtain the ODE v = u, v + au 2 bu = 0. (22) 2 u au 2 + bu = 0. (23) By He s semi-inverse method [16], we an obtain at the following variational formulation J = By Ritz-like method, we searh for a solitary wave solution in the form 0 [ u 2 a 3 u3 b 2 (u ) 2 ]dξ. (24) WJMS for ontribution: submit@wjms.org.uk

5 World Journal of Modelling Simulation, Vol. 9 (2013) No. 1, pp where p q are unknown onstant to be further determined. Substituting Eq. (25) into Eq. (24), we have J = 0 u(ξ) = p seh 2 (qξ), (25) [ 2 p 2 2 seh4 (qξ) ap3 3 seh6 (qξ) 2bp 2 q 2 seh 4 (qξ) tanh 2 (qξ)]dξ = 2 p 2 3q Making J stationary with p q results in 8ap3 45q 4bp2 q 15. (26) From Eq. (27) Eq. (28), we get J p = 22 p 3q 8ap2 15q 8bpq = 0, (27) 15 J q = 2 p 2 3q 2 + 8ap3 45q 2 4bp2 = 0. (28) 15 The solitary solution is, therefore, obtained as follows p = 32, a 0 (29) 2a q = 2, b < 0. (30) b u(x, t) = 32 2a seh2 ( 2 33 (x t)), v(x, t) = b 2a seh2 ( 2 (x t)), (31) b where a 0 b < 0. Similarly, other types of solutions an be found. 5.2 Homotopy analysis method Let us onsider the system Eq. (1) with the initial onditions u 0 (x, t) = u(x, 0) = 32 2a seh2 ( 2 b (x t)), v 0(x, t) = v(x, 0) = 33 2a seh2 ( 2 (x t)). (32) b The system Eq. (1) with the initial onditions Eq. (32) have the exat solutions Eq. (31). To solve system Eq. (1) by means of the HAM, we hoose the auxiliary linear operators as follows: L 1 [φ 1 (x, t; q)] = φ 1(x, t; q), L 2 [φ 2 (x, t; q)] = φ 2(x, t; q) with the property L 1 [ 1 ] = 0, L 2 [ 2 ] = 0, where 1 2 are onstants, φ 1 φ 2 are real funtions. For simpliity, using equations in Eq. (1), we define the nonlinear operator as N 1 [φ 1 (x, t; q), φ 2 (x, t; q)] = φ 1(x, t; q) N 2 [φ 1 (x, t; q), φ 2 (x, t; q)] = φ 2(x, t; q) + φ 2(x, t; q), + a φ2 1 (x, t; q) b 3 φ 1 (x, t; q) 3. With the aid of the above definition, we onstrut the zeroth-order deformation equations (1 q)l 1 [φ 1 (x, t; q) u 0 (x, t)] = q 1 H 1 (x, t)n 1 [φ 1 (x, t; q), φ 2 (x, t; q)], (33) (1 q)l 2 [φ 2 (x, t; q) v 0 (x, t)] = q 2 H 2 (x, t)n 2 [φ 1 (x, t; q), φ 2 (x, t; q)]. (34) WJMS for subsription: info@wjms.org.uk

6 8 H. Kheir & A. Jabbari & et al.: He s semi-inverse method for soliton solutions Obviously, in ase q = 0 in Eq. (33) Eq. (34), we have with q = 1, we obtain phi 1 (x, t; 0) = u 0 (x, t), phi 2 (x, t; 0) = v 0 (x, t) (35) φ 1 (x, t; 1) = u(x, t), phi 2 (x, t; 1) = v(x, t). (36) Therefore, as the embedding parameter q inreases from 0 to 1, φ 1 (x, t; q) φ 2 (x, t; q) vary from the initial guess u 0 (x, t) v 0 (x, t) to the solution u(x, t) v(x, t), respetively. For simpliity, we suppose 1 = 2 = H 2 (x, t) = H 1 (x, t) = 1. Differentiating equations Eq. (33) Eq. (34)m times with respet to the embedding parameter q, the mth-order deformation equations read subjet to initial onditions where L 1 [u m (x, t) χ m u m 1 (x, t)] = R m,1 (u m 1, v m 1 ), (37) L 2 [v m (x, t) χ m v m 1 (x, t)] = R m,2 (u m 1, v m 1 ) (38) R m,1 (u m 1, v m 1 ) = u m 1 R m,2 (u m 1, v m 1 ) = v m 1 u m (x, 0) = 0, v m (x, 0) = 0, (39) + v m 1, (40) m 1 u m 1 j + 2a (u j ) b 3 u m 1 3 (41) χ m is defined by Eq. (17). The solutions of the mth-order deformation equations Eq. (37) Eq. (38) for m 1 beome j=0 u m (x, t) = χ m u m 1 (x, t) + L 1 1 (R m,1(u m 1, v m 1 )), (42) v m (x, t) = χ m v m 1 (x, t) + L 1 2 (R m,2(u m 1, v m 1 )). (43) Now, from Eq. (42), Eq. (43) Eq. (32), we an suessively obtain u 1 (x, t) = 3 4 t 2a sinh( b osh 3 (, v 1(x, t) = 3 5 t 2a sinh( b osh 3 (, u 2 (x, t) = 3 4 t 8a ( b) (4bsinh( 3 osh 3 ( v 2 (x, t) = 3 5 t 8a ( b) (4bsinh( 3 osh 3 ( 2 b 2 b 3 2 bt 2 2 bt ) x) osh 4 + ( 2 x) osh 2 ( b + 4 bsinh( osh 3 ( 3 2 bt 2 2 bt ) x) osh 4 + ( 2 x) osh 2 ( b + 4 bsinh( osh 3 ( 2 b 2 b so on. Then the series solutions obtained by the HAM an be written in the form u(x, t) = = u 0 (x, t) + u 1 (x, t) + u 2 (x, t) +, (44) v(x, t) = = v 0 (x, t) + v 1 (x, t) + v 2 (x, t) +. (45) x) ), x) ), WJMS for ontribution: submit@wjms.org.uk

7 World Journal of Modelling Simulation, Vol. 9 (2013) No. 1, pp Convergene of ham solution Theorem 1. If the series u(x, t) = u 0 (x, t) + u m (x, t) v(x, t) = v 0 (x, t) + v m (x, t), onverge, where u m (x, t) v m (x, t) are governed by the Eq. (42) (43) under the definitions Eq. (40) Eq. (41), then they must be the exat solutions of Eq. (1) with initial onditions Eq. (32). Proof. If the series Eq. (44) Eq. (44) are onvergent, then we an write Thus, S 1 = u m, S 2 = m=0 v m, m=0 lim u n = 0, lim v n = 0, are hold. Due to Eq. (33), Eq. (37) Eq. (38), we have (R m,1 (u m 1, v m 1 )) = lim = L 1 ( lim n L 1 [u m (x, t) χ m u m 1 (x, t)] = L 1 ( lim u n(x, t)) = 0, n [u m (x, t) χ m u m 1 (x, t)]) (R m,2 (u m 1, v m 1 )) = lim = L 2 ( lim n L 2 [v m (x, t) χ m v m 1 (x, t)] = L 2 ( lim v n(x, t)) = 0, n [v m (x, t) χ m v m 1 (x, t)]) Sine 0, we arrive at (R m,1 (u m 1, v m 1 )) = 0, (R m,2 (u m 1, v m 1 )) = 0, Substituting Eq. (40) Eq. (41) in the above expressions, we have (R m,1 (u m 1, v m 1 )) = ( u m 1 + v m 1 ) = u m 1 + v m 1 = (S 1 ) t + (S 2 ) x = 0, WJMS for subsription: info@wjms.org.uk

8 10 H. Kheir & A. Jabbari & et al.: He s semi-inverse method for soliton solutions (R m,2 (u m 1, v m 1 )) = = = = ( v m 1 v m 1 v m 1 v m 1 m 1 + 2a + 2a + 2a + 2a j=0 (u j u m 1 j m 1 j=0 j=0 m=j+1 u j j=0 i=0 = (S 2 ) t + a(s 2 1) x b(s 1 ) xxx = 0. Moreover, due to initial onditions Eq. (39) Eq. (32), we get (u j u m 1 j ) b 3 u m 1 3 ) ) b u m 1 j (u j ) b u i b 3 u m u m u m 1 3 S 1 (x, 0) = u m (x, 0) = u 0 (x, 0) = u(x, 0), m=0 S 2 (x, 0) = v m (x, 0) = v 0 (x, 0) = v(x, 0). m=0 Therefore, S 1 (x, t) S 2 (x, t) satisfy Eq. (1) Eq. (32), they are the exat solutions of Eq. (1) with initial onditions Eq. (32). The auxiliary-parameter an be regarded as an iteration fator that is widely used in numerial omputations. It is well known that a properly hosen iteration fator an ensure the onvergene of iteration [24, 25]. Similarly, it is found that the onvergene of the homotopy-series like Eq. (11) is dependent upon the value of : one an ensure the onvergene of the homotopy-series solution simply by means of hoosing a proper value of. In fat, it is the auxiliary-parameter that provides us, for the first time, a simple way to ensure the onvergene of series solution. Due to this reason, it seems reasonable to rename the onvergene-ontrol parameter [25] ; more information is like to find in this referene. In general, by means of the so-alled -urve, it is straightforward to hoose an appropriate range for whih ensures the onvergene of the solution series. To influene of on the onvergene of solution, we plot the so-alled -urve of u( 2.5, 0.7) by 8th-order approximation of solution, as shown in Fig. 1. It is easy to disover that 1.2 < < 0.6 is the valid region of. It is easy to see that in order to have a good approximation, has to be hosen in 1.2 < < 0.6. This means that for these values of the series Eq. (44) Eq. (45) onverge to the exat solution Eq. (1). 7 Result disussion We delare the results for 8th order HAM approximations in Tab. 1 Tab. 2. The results obtained with = = are better than = 1. Hene, the outputs of HAM are better than the HPM. Moreover the absolute error for u is drawn in Fig. 2 The evolution results for the approximate solutions obtained by HPM HAM, the exat solutions of Eq. (1) are given in Fig. 7 (a)-(), respetively. In addition, we give the numerial results for the approximate solutions obtained by HPM HAM, the exat solutions of the Boussinesq system Eq. (1) with t = 0.7 in Fig. 4. From the numerial results of Fig. 7 Fig. 4, we an easily onlude that both methods present remarkable auray for the approximate solutions of Eq. (1). It is important to note that the auray an be further improved by onsidering more terms of the series Eq. (27). In Tab.3, the absolute errors of approximation results are given with [4, 4] HPadéM. It is shown the HPadéM aelerate the onvergene of the related series. WJMS for ontribution: submit@wjms.org.uk

9 World Journal of Modelling Simulation, Vol. 9 (2013) No. 1, pp Fig. 1. The urve of Boussinesq system for u(-2.5,0.7) obtained from the 8th order HAM. Fig. 2. Absolute error urve of Boussinesq system with 8th order HAM. Fig. 3. The evolution results for the Boussinesq system with a = = 1 b = 1: (a) Exat solution (He s semi-inverse solution), (b) HPM, () HAM. Fig. 4. The numerial results for the Boussinesq system with t = 0.7, a = = 1 b = 1: Exat solution (He s semi-inverse solution)(red), HPM(green), HAM (blue). 8 Conlusion We established variational formulations for Boussinesq system by He s semi-inverse method. We have also given the omparison of results with homotopy perturbation method, homotopy analysis method WJMS for subsription: info@wjms.org.uk

10 12 H. Kheir & A. Jabbari & et al.: He s semi-inverse method for soliton solutions Table 1. Absolute error of Boussinesq system for 8th order HAM with = 0.962, a = = 1 b = 1. x t=0.1 t=0.4 t=0.7 t= E E E E 07 u a u e E E E E E E E E E E E E E E E E 07 v a v e E E E E E E E E E E E E 03 Table 2. Absolute error of Boussinesq system for 8th order HPM with a = = 1 b = 1. x t=0.1 t=0.4 t=0.7 t= E E E E 06 u a u e E E E E E E E E E E E E E E E E 06 v a v e E E E E E E E E E E E E 03 Table 3. Absolute error of Boussinesq system for [4,4] HPadéM with a = = 1 b = 1. x t=0.1 t=0.4 t=0.7 t= E E E E 12 u a u e E E E E E E E E E E E E E E E E 12 v a v e E E E E E E E E E E E E 08 homotopy padé Method. It is obvious that the He s semi-inverse method is useful manageable remarkably simple to find various kinds of solitary solutions of Boussinesq system. Referenes [1] A. Alomari, M. Noorani, R. Nazar. Adaptation of homotopy analysis method for the numeri-analyti solution of hen system. Communiations in Nonlinear Siene Numerial Simulation, 2009, 14: [2] A. Alomari, M. Noorani, et al. Homotopy analysis method for solving frational lorenz system. Communiations in Nonlinear Siene Numerial Simulation, 2010, 15: [3] A. Jabbari, H. Kheiri. The (g /g)-expansion method for solving the ombined the double ombined sinh-oshgordan equations. Ata Universitatis Apulensis, 2010, 22: [4] A. Jabbari, H. Kheiri. New exat traviling wave solutions for the kawahara modified kawahara equations by using modified tanh-oth method. Ata Universitatis Apulensis, 2010, 23: [5] A. Jabbari, H. Kheiri. Homotopy analysis homotopy padse methods for (2+1)-dimensional boiti-leonpempinelli system. International Journal of Nonlinear Siene, 2011, 12: [6] A. Jabbari, H. Kheiri, A. Yildirim. Homotopy pade method for solving seond-order one-dimensional telegraph equation. International Journal of Numerial Methods for Heat Fluid Flow. WJMS for ontribution: submit@wjms.org.uk

11 World Journal of Modelling Simulation, Vol. 9 (2013) No. 1, pp [7] A. Jabbari, H. Yildirim. Homotopy analysis homotopy padse methods for (1 + 1) (2 + 1)-dimensional dispersive long wave equations. International Journal of Numerial Methods for Heat Fluid Flow, [8] A. Soliman. On the solution of two-dimensional oupled burgers equations by variational iteration method. Chaos, Solitons Fratals, 2009, 40: [9] A. Wazwaz. A variety of exat wave solutions with distint physial strutures for the boussinesq system. Communiations in Nonlinear Siene Numerial Simulation, 2006, 11: [10] H. Kheiri, A. Jabbari. Exat solutions for the double sinh-gordon generalized form of the double sinh-gordon equations by using (G /G)-expansion method. Turkish Journal of Physis. [11] H. Kheiri, A. Jabbari. Homotopy analysis homotopy padé methods for two-dimensional oupled burgers equations. Iranian Journal of Mathematial Siene Information, 2011, 6: [12] J. He. Homotopy perturbation tehnique. Computational Mathematis Appliation Mehanis, 1999, 178: [13] J. He. A lassial variational model for miropolar elastodynamis. International Journal of Nonlinear Siene Numerial Simulation, 2000, 1(2): [14] ) Coupled variational priniples of piezoeletriity. International Journal of Engineering Siene, 2000, 39(3): [15] J. He. Variational theory for linear magneto-eletro-elastiity. International Journal of Nonlinear Siene Numerial Simulation, 2001, 2(4): [16] J. He. Variational priniples for some nonlinear partial dikerential equations with variable oenients. Chaos Solitons Fratals, 2004, 19(4): 847. [17] J. He. Variational priniple for two-dimensional inompressible invisid flow. Physis Letters A, 2007, 371(1-2): [18] J. Wang. A list of (1 + 1) dimensional integrable equations their properties. Journal of Nonlinear Mathematial Physis, 2002, 9: [19] M. Ablowitz, P. Clarkson. Soliton, Nonlinear Solution Equations Inverse Satting. Cambridge University Press, New York, [20] M.Miurs. Bäklund Transformation. Springer, Berlin, [21] P. Olver. Appliations of Lie groups to dikerential equations. Springer, [22] R. Hirota. Exat solution of the korteweg-de vries equation for multiple ollisions of solitons. Physis Review Letter, 1971, 27: [23] S. Liao. Proposed homotopy analysis tehnique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University, [24] S. Liao. Beyond perturbation: An introdution to Homotopy analysis method. Boa Raton: Champan Hall/CRC, [25] S. Liao. Notes on the homotopy analysis method: Some difinitions theorems. Communiations in Nonlinear Siene Numerial Simulation, 2009, 14: [26] S. Liu, Z. Fu, et al. Jaobi ellipti funtion expansion method periodi wave solutions of nonlinear wave equations. Physis Letters A, 2001, 289: [27] S. Yousefi, A. Lotfi, M. Dehghan. He s variational iteration method for solving nonlinear mixed volterra-fredholm integral equations. Computers & Mathematis with Appliations, 2009, 58: [28] U. Gokta, E. Hereman. Symboli omputation of onserved densities for systems of nonlinear evolution equations. Journal of Symboli Computation, 1997, 24(5): [29] Y. Ye, L. Mo. He s variational method for the benjamin-bona- mahony equation the kawahara equation. Computers & Mathematis with Appliations, 2009, 58: [30] Z. Tao. Variational approah to the invisid ompressible fluid. Ata Appliae Mathematiae, 2004, 100(3): [31] Z. Tao. Frequeny amplitude relationship of the dunng-harmoni osillator. Topology Method in Nonlinear Analysis, 2008, 31(2): [32] Z. Tao. Solitary solutions of the boiti-leon-manna-pempinelli equation using he s variational method. Zeitshrift fr Naturforshung A, 2008, 63(10/11): [33] Z. Tao. Variational priniples for some nonlinear wave equations. Zeitshrift fr Naturforshung A, 2008, 63(5/6): [34] Z. Tao. Solving the breaking soliton equation by he s variational method. Computers & Mathematis with Appliations, 2009, 58: [35] Z. Tao. Variational approah to the benjamin ono equation. Nonlinear Analysis: Real World Appliations, 2009, 10(3): WJMS for subsription: info@wjms.org.uk