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Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method
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Chaos, Solitons and Fractals 36 (2008) 309–313 www.elsevier.com/locate/chaos
Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method M. Javidi *, A. Golbabai Department of Mathematics, Faculty of science, Razi University, Kermanshah 67149, Iran Accepted 14 June 2006
Abstract In this paper, exact and numerical solutions are obtained for the generalized Zakharov equation (GZE) by the well known variational iteration method (VIM). This method is based on Lagrange multipliers for identification of optimal values of parameters in a functional. Using this method creates a sequence which tends to the exact solution of the problem. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction At the classical level, a set of coupled nonlinear wave equations describing the interaction between high-frequency Langmuir waves and low-frequency ion-acoustic waves were firstly derived by Zakharov [1]. Since then, this system has been the subject of a large number of studies. In one dimension, the Zakharov equations (ZE) may be written as iEt þ Exx ¼ FE; ð1Þ F tt F xx ¼ ðjEj2 Þxx ; where E is the envelope of the high-frequency electric field, F is the plasma density measured from its equilibrium value. The system can be derived from a hydrodynamic description of the plasma [2,3]. However, some important effects such as transit-time damping and ion nonlinearities, which are also implied by the fact that the values used for the ion damping have been anomalously large from the point of view of linear ion-acoustic wave dynamics, have been ignored in the ZE. This is equivalent to say that, the ZE is a simplified model of strong Langmuir turbulence. Thus we have to generalize the ZE by taking more elements into account. Starting from the dynamical plasma equations with the help of relaxed Zakharov simplification assumptions, and through taking use of the time-averaged two-time-scale two-fluid plasma description, the ZE are generalized to contain the self-generated magnetic field [4]. The generalized Zakharov equations (GZE) are a set of coupled equations and may be written as [5] ( iEt þ Exx 2bjEj2 E þ 2FE ¼ 0; ð2Þ F tt F xx þ ðjEj2 Þxx ¼ 0;
*
Corresponding author. Tel./fax: +98 831 4274569. E-mail address: [email protected] (M. Javidi).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.088
310
M. Javidi, A. Golbabai / Chaos, Solitons and Fractals 36 (2008) 309–313
where E is the envelope of the high-frequency electric field, and F is the plasma density measured from its equilibrium value. This system is reduced to the classical Zakharov equations of plasma physics whenever b = 0. Due to the fact that the GZE is a realistic model in plasma, it makes sense to study the solitary wave solutions of the GZE. Recently various powerful mathematical methods such as homotopy perturbation method [6], variational iteration method [7– 14], Adomian decomposition method [15] and others [16–18,24–26] have been proposed to obtain exact and approximate analytic solutions for nonlinear problems. In this paper, we use variational iteration method for solving the generalized Zakharov equation, which was first proposed by Ji-Huan He in 1998 [19,20] and systematically illustrated in 1999 [8].
2. Variational iteration method In this section, we use the following non-homogeneous system of partial differential equations L1 Eðx; tÞ þ N 1 ðEðx; tÞ; F ðx; tÞÞ ¼ f ðx; tÞ; L2 Eðx; tÞ þ N 2 ðEðx; tÞ; F ðx; tÞÞ ¼ gðx; tÞ;
ð3Þ
to illustrate the basic idea of the variational iteration method [7–14]. In the above system of equations L1 and L2 are linear differential operators with respect to t and N1 and N2 are nonlinear operators and f(x, t), g(x, t) are some given functions. According to the variational iteration method, we can construct a correct functional as follows [7–14]: ( Rt e n ðx; sÞ; Fe n ðx; sÞÞ f ðx; sÞ ds; Enþ1 ðx; tÞ ¼ En ðx; tÞ þ 0 k1 ðsÞ½L1 En ðx; sÞ þ N 1 ð E ð4Þ Rt e n ðx; sÞ; Fe n ðx; sÞÞ gðx; sÞ ds; F nþ1 ðx; tÞ ¼ F n ðx; tÞ þ k2 ðsÞ½L2 F n ðx; sÞ þ N 2 ð E 0
where k1 and k2 are general Lagrange multipliers, which can be identified optimally via variational theory [7–14,21,22]. The second terms on the right-hand side in (4) are called the correction and the subscript n denotes the nth order e n and Fe n are considered as a restricted varapproximation. Under a suitable restricted variational assumptions (i.e. E e n ¼ 0 and d Fe n ¼ 0), then the iation), we may assume that the above correctional functional are stationary (i.e. d E Lagrange multipliers are identified. Now we may start the procedures with the given initial approximation and using the above iteration formulas to obtain the approximate solutions.
3. Application of the variational iteration method In this section the application of the variational iteration method is discussed for solving problem (2). According to the variational iteration method we consider the correction functional in t-direction in the following form (see [7–14]): ! Z t en oEn o2 E 2e e e e Enþ1 ðx; tÞ ¼ En ðx; tÞ þ k1 ðsÞ i ðx; sÞ 2bj Eðx; sÞj Eðx; sÞ þ 2 Eðx; sÞ F ðx; sÞ ds; ðx; sÞ þ os ox2 0 ! ð5Þ Z t o2 F n o2 Fe n o2 e 2 k2 ðsÞ ðx; sÞ ðx; sÞ þ 2 j Eðx; sÞj ds; F nþ1 ðx; tÞ ¼ F n ðx; tÞ þ os2 ox2 ox 0 where k1 and k2 are the general Lagrange multiplier which its optimal value is found by using variational theory. The e n and Fe n are the restricted values i.e. values of E0 and F0 are initial approximations and must be chosen suitably and E e n ¼ 0 and d Fe n ¼ 0 [5]. Now we have dE ! Z t en oEn o2 E 2e e e e dEnþ1 ðx; tÞ ¼ dEn ðx; tÞ þ d ðx; sÞ þ k1 ðsÞ i ðx; sÞ 2bj Eðx; sÞj Eðx; sÞ þ 2 Eðx; sÞ F ðx; sÞ ds; os ox2 0 ! ð6Þ Z t o2 F n o2 Fe n o2 e 2 k2 ðsÞ ðx; sÞ ðx; sÞ þ 2 j Eðx; sÞj ds; dF nþ1 ðx; tÞ ¼ dF n ðx; tÞ þ d os2 ox2 ox 0 to find the optimal value of ki
oEn ðx; sÞ k1 ðsÞ i ds; os 0 2 Z t o F n ðx; sÞ k2 ðsÞ ds dF nþ1 ðx; tÞ ¼ dF n ðx; tÞ þ d os2 0 dEnþ1 ðx; tÞ ¼ dEn ðx; tÞ þ d
Z
t
ð7Þ
M. Javidi, A. Golbabai / Chaos, Solitons and Fractals 36 (2008) 309–313
311
and this yields to the following stationary conditions: k01 ðsÞ ¼ 0;
k002 ðsÞ ¼ 0;
1 þ ik1 ðsÞ ¼ 0js¼t ;
1 k02 ðsÞ ¼ 0js¼t ;
ð8Þ
k2 ðsÞ ¼ 0js¼t :
Therefore the Lagrange multipliers is defined as the following form: k1 ðsÞ ¼ i;
k2 ðsÞ ¼ s t:
ð9Þ
Substituting Eq. (9) into the correctional functional Eq. (5) we get the following iteration formula: Z t oEn o2 En ðx; sÞ þ 2 ðx; sÞ 2bjEðx; sÞj2 Eðx; sÞ þ 2Eðx; sÞF ðx; sÞ ds; i Enþ1 ðx; tÞ ¼ En ðx; tÞ þ i os ox 0 2 Z t o Fn o2 F n ðx; sÞ o2 e 2 F nþ1 ðx; tÞ ¼ F n ðx; tÞ þ ds: ðs tÞ ðx; sÞ þ j Eðx; sÞj ox2 os2 ox2 0
ð10Þ
We start with initial approximations E0(x, t) = E(x, 0), and F0(x, t) = F(x, 0) and by the above formulas, we can obtain the E1(x, t) and F1(x, t) as follows: E0 ¼ r tanhðpxÞ expðikxÞ; F 0 ¼ s þ r2 tanh2 ðpxÞ=ð4k 2 þ 1Þ; E1 ¼ fr tanhðpxÞ cosðkxÞ ð2r tanhðpxÞð1 tanh2 ðpxÞÞp2 sinðkxÞ þ 2rð1 tanh2 ðpxÞÞp cosðkxÞk r tanhðpxÞk 2 sinðkxÞ 2br tanhðpxÞ sinðkxÞðr2 tanh2 ðpxÞ cos2 ðkxÞ þ r2 tanh2 ðpxÞ sin2 ðkxÞÞ þ 2r tanhðpxÞ sinðkxÞðs þ 1=ð4k 2 þ 1Þr2 tanh2 ðpxÞÞÞtg þ ifr tanhðpxÞ sinðkxÞ ð2r tanhðpxÞð1 tanh2 ðpxÞÞp2 cosðkxÞ þ 2rð1 tanh2 ðpxÞÞp sinðkxÞk þ rk 2 tanhðpxÞ cosðkxÞ
ð11Þ
þ 2br tanhðpxÞ cosðkxÞðr2 tanh2 ðpxÞ cos2 ðkxÞ þ r2 tanh2 ðpxÞ sin2 ðkxÞÞ 2r tanhðpxÞ sinðkxÞðs þ 1=ð4k 2 þ 1Þr2 tanh2 ðpxÞÞÞtg; F 1 ¼ s þ 1=ð4k 2 þ 1Þr2 tanh2 ðpxÞ 1=2t2 ð2=ð4k 2 þ 1Þ r2 ð1 tanh2 ðpxÞÞ2 p2 4=ð4k 2 þ 1Þr2 p2 tanh2 ðpxÞð1 tanh2 ðpxÞÞ þ 2r2 ð1 tanh2 ðpxÞÞ2 p2 cos2 ðkxÞ 4r2 p2 tanh2 ðpxÞ cos2 ðkxÞð1 tanh2 ðpxÞÞ þ 2r2 ð1 tanh2 ðpxÞÞ2 p2 sin2 ðkxÞ 4r2 tanhðpxÞ2 sin2 ðkxÞð1 tanh2 ðpxÞÞp2 Þ .. .
Table 1 The numerical results for E2 and F2 in comparison with the exact solution of E and F, when p = 0.05, k = 1, s = 0.033, b = 1 xintj
0.1
0.2
0.3
0.4
0.5
jE E2jij 0.1 0.2 0.3 0.4 0.5
1.4608e–6 7.9233e–6 2.2588e–5 4.5141e–5 7.5104e–5
5.1669e–6 6.0321e–6 2.3554e–5 5.7301e–5 1.0651e–4
2.2610e–5 5.8359e–6 1.4271e–5 4.7565e–5 1.0487e–4
6.5098e–5 1.8824e–5 6.2559e–6 2.7326e–5 8.1291e–5
1.4358e–4 5.6289e–5 1.0996e–5 8.1121e–6 4.7195e–5
jF F2jij 0.1 0.2 0.3 0.4 0.5
8.0050e–7 4.2449e–7 2.2588e–5 2.5925e–7 2.2192e–7
1.5395e–6 7.8721e–7 5.6225e–7 4.5684e–7 3.8258e–7
2.2200e–6 1.0906e–6 7.5269e–7 5.9470e–7 4.8400e–7
2.8472e–6 1.3390e–6 8.8704e–7 6.7601e–7 5.2919e–7
3.4286e–6 1.5388e–6 9.7055e–7 7.0526e–7 5.2219e–7
312
M. Javidi, A. Golbabai / Chaos, Solitons and Fractals 36 (2008) 309–313
Table 2 The numerical results for E2 and F2 in comparison with the exact solution of E and F, when p = 0.1, k = 1, s = 0, b = 10 xintj
0.1
0.2
0.3
0.4
0.5
jE E2jij 0.1 0.2 0.3 0.4 0.5
1.5477e–7 9.7112e–6 3.1873e–5 6.6155e–5 1.1183e–4
1.4845e–6 1.3204e–6 2.6737e–5 7.7094e–5 1.5124e–4
2.4450e–5 4.6353e–6 4.9143e–6 5.2629e–5 1.3726e–4
8.9053e–5 1.2268e–5 1.3044e–5 1.3147e–5 8.9821e–5
2.1445e–4 7.1940e–5 6.7569e–6 2.0792e–5 2.9368e–5
jF F2jij 0.1 0.2 0.3 0.4 0.5
1.6866e–6 9.0176e–7 6.7054e–7 5.6412e–7 4.8981e–7
3.2381e–6 1.6685e–6 1.2062e–6 9.9364e–7 8.4539e–7
4.6618e–6 2.3074e–6 1.6141e–6 1.2955e–6 1.0736e–6
5.9701e–6 2.8307e–6 1.9062e–6 1.4816e–6 1.1861e–6
7.1802e–6 3.2554e–6 2.0996e–6 1.5689e–6 1.1996e–6
Thus the exact value of E(x, t) and F(x, t) in a closed form are Eðx; tÞ ¼ lim En ðx; tÞ ¼ r tanhðpx xtÞ exp½iðkx XtÞÞ; n!1
F ðx; tÞ ¼ lim F n ðx; tÞ ¼ s þ r2 tanh2 ðpx xtÞ=ð4k 2 þ 1Þ;
ð12Þ
n!1
where x = 2kp, X = 2s + k2 + 2p2. This result can be verified through substitution. It is the same exact solution in [23]. Continuing this procedure we may get E2(x, t), F2(x, t) and other high order approximations. The numerical results of this example is given in Tables 1 and 2. In the numerical calculation, E2(x, t) and F2(x, t) are approximations of E(x, t) and F(x, t) respectively which are calculated by the recurrence formula (10) for the values of t = 0.1(0.1)0.5 and x = 0.1(0.1)0.5. By the variational iteration method, one can get a better result by calculating more terms of the sequences {En} and {Fn}. 4. Conclusions In this paper, variational iteration method was employed successfully for solving the generalized Zakharov equation. Using this method the problem may be solved without any discretization of variables. Therefore, it is not affected by computation roundoff errors and one is not faced to the necessity of using large computer memory and time. This method provides a solution of the problem in a closed form while the mesh point techniques only provide the approximation at mesh points. This method is also useful for finding an accurate approximation of the exact solution. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Zakharov VE. Collapse of Langmuir waves. Zh Eksp Teor Fiz 1972;62:1745–51. Golman Mv. Langmuir wave solitons and spatial collapse in plasma physics. Physica D 1986;18:67–76. Nicolson DR. Introduction to plasma theory. New York: Wiley; 1983. Li LH. Langmuir turbulence equations with the self-generated magnetic field. Phys Fluids B 1993;5:350–6. Malomed B, Anderson D, Lisak M, Quiroga-Teixeiro ML. Dynamics of solitary waves in the Zakharov model equations. Phys Rev E 1977;55:962–8. He JH. Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons & Fractals 2005;26(3):695–700. He JH, Wu XH. Construction of solitary solution and compacton-like solution by variational iteration method. Chaos, Solitons & Fractals 2006;29(1):108–13. He JH. Variational iteration method – a kind of nonlinear analytical technique: some examples. Int J Nonlinear Mech 1999;34(4):699–708. He JH. Variational iteration method for autonomous ordinary differential systems. Appl Math Comput 2000;114(2–3):115–23. Odibat ZM, Momani S. Application of variational iteration method to nonlinear differential equations of fractional order. Int J Nonlinear Sci Numer Simul 2006;7(1):27–36. Bildik N, Konuralp A. The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int J Nonlinear Sci Numer Simul 2006;7(1):65–70.
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[12] Momani S, Abuasad S. Application of He’s variational iteration method to Helmholtz equation. Chaos, Solitons & Fractals 2006;27(5):1119–23. [13] Soliman AA. Numerical simulation of the generalized regularized long wave equation by He’s variational iteration method. Math Comput Simul 2005;70(2):119–24. [14] Abdou MA, Soliman AA. Variational iteration method for solving Burger’s and coupled Burger’s equations. J Comput Appl Math 2005;181(2):245–51. [15] Kaya D, El-Sayed SM. A numerical simulation and explicit solutions of the generalized Burger-Fisher equation. Appl Math Comput 2004;152:403–13. [16] He JH. Non-perturbative methods for strongly nonlinear problems, Berlin: dissertation. de-Verlag im Internet GmbH, 2006. [17] He JH. Some asymptotic methods for strongly nonlinear equations. Int J Modern Phys B 2006;20(10):1141–99. [18] Abulwafa EM, Abdou MA, Mahmoud AA. The solution of nonlinear coagulation problem with mass loss. Chaos, Solitons & Fractals 2006;29(2):313–30. [19] He JH. Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput Meth Appl Mech Eng 1998;167(1–2):69–73. [20] He JH. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput Meth Appl Mech Eng 1998;167(1–2):57–68. [21] He JH. Variational theory for linear magneto-electro-elasticity. Int J Nonlinear Sci Numer Simul 2001;2(4):309–16. [22] Inokuti M, Sekine H, Mura T. General use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Nasser S, editor. Variational methods in the mechanics of solids. Permagon Press; 1978. p. 56–162. [23] Wang M, Li X. Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations. Phys Lett A 2005;343:48–54. [24] Marinca V. An approximate solution for one-dimensional weakly nonlinear oscillations. Int J Nonlinear Sci Numer Simul 2002;3(2):107–20. [25] Draganescu GE, Capalnasan V. Nonlinear relaxation phenomena in polycrystalline solids. Int J Nonlinear Sci Numer Simul 2003;4(3):219–25. [26] Soliman AA. A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations. Chaos, Solitons & Fractals 2006;29(2):294–302.
Chaos, Solitons and Fractals 36 (2008) 309–313 www.elsevier.com/locate/chaos
Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method M. Javidi *, A. Golbabai Department of Mathematics, Faculty of science, Razi University, Kermanshah 67149, Iran Accepted 14 June 2006
Abstract In this paper, exact and numerical solutions are obtained for the generalized Zakharov equation (GZE) by the well known variational iteration method (VIM). This method is based on Lagrange multipliers for identification of optimal values of parameters in a functional. Using this method creates a sequence which tends to the exact solution of the problem. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction At the classical level, a set of coupled nonlinear wave equations describing the interaction between high-frequency Langmuir waves and low-frequency ion-acoustic waves were firstly derived by Zakharov [1]. Since then, this system has been the subject of a large number of studies. In one dimension, the Zakharov equations (ZE) may be written as iEt þ Exx ¼ FE; ð1Þ F tt F xx ¼ ðjEj2 Þxx ; where E is the envelope of the high-frequency electric field, F is the plasma density measured from its equilibrium value. The system can be derived from a hydrodynamic description of the plasma [2,3]. However, some important effects such as transit-time damping and ion nonlinearities, which are also implied by the fact that the values used for the ion damping have been anomalously large from the point of view of linear ion-acoustic wave dynamics, have been ignored in the ZE. This is equivalent to say that, the ZE is a simplified model of strong Langmuir turbulence. Thus we have to generalize the ZE by taking more elements into account. Starting from the dynamical plasma equations with the help of relaxed Zakharov simplification assumptions, and through taking use of the time-averaged two-time-scale two-fluid plasma description, the ZE are generalized to contain the self-generated magnetic field [4]. The generalized Zakharov equations (GZE) are a set of coupled equations and may be written as [5] ( iEt þ Exx 2bjEj2 E þ 2FE ¼ 0; ð2Þ F tt F xx þ ðjEj2 Þxx ¼ 0;
*
Corresponding author. Tel./fax: +98 831 4274569. E-mail address: [email protected] (M. Javidi).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.088
310
M. Javidi, A. Golbabai / Chaos, Solitons and Fractals 36 (2008) 309–313
where E is the envelope of the high-frequency electric field, and F is the plasma density measured from its equilibrium value. This system is reduced to the classical Zakharov equations of plasma physics whenever b = 0. Due to the fact that the GZE is a realistic model in plasma, it makes sense to study the solitary wave solutions of the GZE. Recently various powerful mathematical methods such as homotopy perturbation method [6], variational iteration method [7– 14], Adomian decomposition method [15] and others [16–18,24–26] have been proposed to obtain exact and approximate analytic solutions for nonlinear problems. In this paper, we use variational iteration method for solving the generalized Zakharov equation, which was first proposed by Ji-Huan He in 1998 [19,20] and systematically illustrated in 1999 [8].
2. Variational iteration method In this section, we use the following non-homogeneous system of partial differential equations L1 Eðx; tÞ þ N 1 ðEðx; tÞ; F ðx; tÞÞ ¼ f ðx; tÞ; L2 Eðx; tÞ þ N 2 ðEðx; tÞ; F ðx; tÞÞ ¼ gðx; tÞ;
ð3Þ
to illustrate the basic idea of the variational iteration method [7–14]. In the above system of equations L1 and L2 are linear differential operators with respect to t and N1 and N2 are nonlinear operators and f(x, t), g(x, t) are some given functions. According to the variational iteration method, we can construct a correct functional as follows [7–14]: ( Rt e n ðx; sÞ; Fe n ðx; sÞÞ f ðx; sÞ ds; Enþ1 ðx; tÞ ¼ En ðx; tÞ þ 0 k1 ðsÞ½L1 En ðx; sÞ þ N 1 ð E ð4Þ Rt e n ðx; sÞ; Fe n ðx; sÞÞ gðx; sÞ ds; F nþ1 ðx; tÞ ¼ F n ðx; tÞ þ k2 ðsÞ½L2 F n ðx; sÞ þ N 2 ð E 0
where k1 and k2 are general Lagrange multipliers, which can be identified optimally via variational theory [7–14,21,22]. The second terms on the right-hand side in (4) are called the correction and the subscript n denotes the nth order e n and Fe n are considered as a restricted varapproximation. Under a suitable restricted variational assumptions (i.e. E e n ¼ 0 and d Fe n ¼ 0), then the iation), we may assume that the above correctional functional are stationary (i.e. d E Lagrange multipliers are identified. Now we may start the procedures with the given initial approximation and using the above iteration formulas to obtain the approximate solutions.
3. Application of the variational iteration method In this section the application of the variational iteration method is discussed for solving problem (2). According to the variational iteration method we consider the correction functional in t-direction in the following form (see [7–14]): ! Z t en oEn o2 E 2e e e e Enþ1 ðx; tÞ ¼ En ðx; tÞ þ k1 ðsÞ i ðx; sÞ 2bj Eðx; sÞj Eðx; sÞ þ 2 Eðx; sÞ F ðx; sÞ ds; ðx; sÞ þ os ox2 0 ! ð5Þ Z t o2 F n o2 Fe n o2 e 2 k2 ðsÞ ðx; sÞ ðx; sÞ þ 2 j Eðx; sÞj ds; F nþ1 ðx; tÞ ¼ F n ðx; tÞ þ os2 ox2 ox 0 where k1 and k2 are the general Lagrange multiplier which its optimal value is found by using variational theory. The e n and Fe n are the restricted values i.e. values of E0 and F0 are initial approximations and must be chosen suitably and E e n ¼ 0 and d Fe n ¼ 0 [5]. Now we have dE ! Z t en oEn o2 E 2e e e e dEnþ1 ðx; tÞ ¼ dEn ðx; tÞ þ d ðx; sÞ þ k1 ðsÞ i ðx; sÞ 2bj Eðx; sÞj Eðx; sÞ þ 2 Eðx; sÞ F ðx; sÞ ds; os ox2 0 ! ð6Þ Z t o2 F n o2 Fe n o2 e 2 k2 ðsÞ ðx; sÞ ðx; sÞ þ 2 j Eðx; sÞj ds; dF nþ1 ðx; tÞ ¼ dF n ðx; tÞ þ d os2 ox2 ox 0 to find the optimal value of ki
oEn ðx; sÞ k1 ðsÞ i ds; os 0 2 Z t o F n ðx; sÞ k2 ðsÞ ds dF nþ1 ðx; tÞ ¼ dF n ðx; tÞ þ d os2 0 dEnþ1 ðx; tÞ ¼ dEn ðx; tÞ þ d
Z
t
ð7Þ
M. Javidi, A. Golbabai / Chaos, Solitons and Fractals 36 (2008) 309–313
311
and this yields to the following stationary conditions: k01 ðsÞ ¼ 0;
k002 ðsÞ ¼ 0;
1 þ ik1 ðsÞ ¼ 0js¼t ;
1 k02 ðsÞ ¼ 0js¼t ;
ð8Þ
k2 ðsÞ ¼ 0js¼t :
Therefore the Lagrange multipliers is defined as the following form: k1 ðsÞ ¼ i;
k2 ðsÞ ¼ s t:
ð9Þ
Substituting Eq. (9) into the correctional functional Eq. (5) we get the following iteration formula: Z t oEn o2 En ðx; sÞ þ 2 ðx; sÞ 2bjEðx; sÞj2 Eðx; sÞ þ 2Eðx; sÞF ðx; sÞ ds; i Enþ1 ðx; tÞ ¼ En ðx; tÞ þ i os ox 0 2 Z t o Fn o2 F n ðx; sÞ o2 e 2 F nþ1 ðx; tÞ ¼ F n ðx; tÞ þ ds: ðs tÞ ðx; sÞ þ j Eðx; sÞj ox2 os2 ox2 0
ð10Þ
We start with initial approximations E0(x, t) = E(x, 0), and F0(x, t) = F(x, 0) and by the above formulas, we can obtain the E1(x, t) and F1(x, t) as follows: E0 ¼ r tanhðpxÞ expðikxÞ; F 0 ¼ s þ r2 tanh2 ðpxÞ=ð4k 2 þ 1Þ; E1 ¼ fr tanhðpxÞ cosðkxÞ ð2r tanhðpxÞð1 tanh2 ðpxÞÞp2 sinðkxÞ þ 2rð1 tanh2 ðpxÞÞp cosðkxÞk r tanhðpxÞk 2 sinðkxÞ 2br tanhðpxÞ sinðkxÞðr2 tanh2 ðpxÞ cos2 ðkxÞ þ r2 tanh2 ðpxÞ sin2 ðkxÞÞ þ 2r tanhðpxÞ sinðkxÞðs þ 1=ð4k 2 þ 1Þr2 tanh2 ðpxÞÞÞtg þ ifr tanhðpxÞ sinðkxÞ ð2r tanhðpxÞð1 tanh2 ðpxÞÞp2 cosðkxÞ þ 2rð1 tanh2 ðpxÞÞp sinðkxÞk þ rk 2 tanhðpxÞ cosðkxÞ
ð11Þ
þ 2br tanhðpxÞ cosðkxÞðr2 tanh2 ðpxÞ cos2 ðkxÞ þ r2 tanh2 ðpxÞ sin2 ðkxÞÞ 2r tanhðpxÞ sinðkxÞðs þ 1=ð4k 2 þ 1Þr2 tanh2 ðpxÞÞÞtg; F 1 ¼ s þ 1=ð4k 2 þ 1Þr2 tanh2 ðpxÞ 1=2t2 ð2=ð4k 2 þ 1Þ r2 ð1 tanh2 ðpxÞÞ2 p2 4=ð4k 2 þ 1Þr2 p2 tanh2 ðpxÞð1 tanh2 ðpxÞÞ þ 2r2 ð1 tanh2 ðpxÞÞ2 p2 cos2 ðkxÞ 4r2 p2 tanh2 ðpxÞ cos2 ðkxÞð1 tanh2 ðpxÞÞ þ 2r2 ð1 tanh2 ðpxÞÞ2 p2 sin2 ðkxÞ 4r2 tanhðpxÞ2 sin2 ðkxÞð1 tanh2 ðpxÞÞp2 Þ .. .
Table 1 The numerical results for E2 and F2 in comparison with the exact solution of E and F, when p = 0.05, k = 1, s = 0.033, b = 1 xintj
0.1
0.2
0.3
0.4
0.5
jE E2jij 0.1 0.2 0.3 0.4 0.5
1.4608e–6 7.9233e–6 2.2588e–5 4.5141e–5 7.5104e–5
5.1669e–6 6.0321e–6 2.3554e–5 5.7301e–5 1.0651e–4
2.2610e–5 5.8359e–6 1.4271e–5 4.7565e–5 1.0487e–4
6.5098e–5 1.8824e–5 6.2559e–6 2.7326e–5 8.1291e–5
1.4358e–4 5.6289e–5 1.0996e–5 8.1121e–6 4.7195e–5
jF F2jij 0.1 0.2 0.3 0.4 0.5
8.0050e–7 4.2449e–7 2.2588e–5 2.5925e–7 2.2192e–7
1.5395e–6 7.8721e–7 5.6225e–7 4.5684e–7 3.8258e–7
2.2200e–6 1.0906e–6 7.5269e–7 5.9470e–7 4.8400e–7
2.8472e–6 1.3390e–6 8.8704e–7 6.7601e–7 5.2919e–7
3.4286e–6 1.5388e–6 9.7055e–7 7.0526e–7 5.2219e–7
312
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Table 2 The numerical results for E2 and F2 in comparison with the exact solution of E and F, when p = 0.1, k = 1, s = 0, b = 10 xintj
0.1
0.2
0.3
0.4
0.5
jE E2jij 0.1 0.2 0.3 0.4 0.5
1.5477e–7 9.7112e–6 3.1873e–5 6.6155e–5 1.1183e–4
1.4845e–6 1.3204e–6 2.6737e–5 7.7094e–5 1.5124e–4
2.4450e–5 4.6353e–6 4.9143e–6 5.2629e–5 1.3726e–4
8.9053e–5 1.2268e–5 1.3044e–5 1.3147e–5 8.9821e–5
2.1445e–4 7.1940e–5 6.7569e–6 2.0792e–5 2.9368e–5
jF F2jij 0.1 0.2 0.3 0.4 0.5
1.6866e–6 9.0176e–7 6.7054e–7 5.6412e–7 4.8981e–7
3.2381e–6 1.6685e–6 1.2062e–6 9.9364e–7 8.4539e–7
4.6618e–6 2.3074e–6 1.6141e–6 1.2955e–6 1.0736e–6
5.9701e–6 2.8307e–6 1.9062e–6 1.4816e–6 1.1861e–6
7.1802e–6 3.2554e–6 2.0996e–6 1.5689e–6 1.1996e–6
Thus the exact value of E(x, t) and F(x, t) in a closed form are Eðx; tÞ ¼ lim En ðx; tÞ ¼ r tanhðpx xtÞ exp½iðkx XtÞÞ; n!1
F ðx; tÞ ¼ lim F n ðx; tÞ ¼ s þ r2 tanh2 ðpx xtÞ=ð4k 2 þ 1Þ;
ð12Þ
n!1
where x = 2kp, X = 2s + k2 + 2p2. This result can be verified through substitution. It is the same exact solution in [23]. Continuing this procedure we may get E2(x, t), F2(x, t) and other high order approximations. The numerical results of this example is given in Tables 1 and 2. In the numerical calculation, E2(x, t) and F2(x, t) are approximations of E(x, t) and F(x, t) respectively which are calculated by the recurrence formula (10) for the values of t = 0.1(0.1)0.5 and x = 0.1(0.1)0.5. By the variational iteration method, one can get a better result by calculating more terms of the sequences {En} and {Fn}. 4. Conclusions In this paper, variational iteration method was employed successfully for solving the generalized Zakharov equation. Using this method the problem may be solved without any discretization of variables. Therefore, it is not affected by computation roundoff errors and one is not faced to the necessity of using large computer memory and time. This method provides a solution of the problem in a closed form while the mesh point techniques only provide the approximation at mesh points. This method is also useful for finding an accurate approximation of the exact solution. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Zakharov VE. Collapse of Langmuir waves. Zh Eksp Teor Fiz 1972;62:1745–51. Golman Mv. Langmuir wave solitons and spatial collapse in plasma physics. Physica D 1986;18:67–76. Nicolson DR. Introduction to plasma theory. New York: Wiley; 1983. Li LH. Langmuir turbulence equations with the self-generated magnetic field. Phys Fluids B 1993;5:350–6. Malomed B, Anderson D, Lisak M, Quiroga-Teixeiro ML. Dynamics of solitary waves in the Zakharov model equations. Phys Rev E 1977;55:962–8. He JH. Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons & Fractals 2005;26(3):695–700. He JH, Wu XH. Construction of solitary solution and compacton-like solution by variational iteration method. Chaos, Solitons & Fractals 2006;29(1):108–13. He JH. Variational iteration method – a kind of nonlinear analytical technique: some examples. Int J Nonlinear Mech 1999;34(4):699–708. He JH. Variational iteration method for autonomous ordinary differential systems. Appl Math Comput 2000;114(2–3):115–23. Odibat ZM, Momani S. Application of variational iteration method to nonlinear differential equations of fractional order. Int J Nonlinear Sci Numer Simul 2006;7(1):27–36. Bildik N, Konuralp A. The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int J Nonlinear Sci Numer Simul 2006;7(1):65–70.
M. Javidi, A. Golbabai / Chaos, Solitons and Fractals 36 (2008) 309–313
313
[12] Momani S, Abuasad S. Application of He’s variational iteration method to Helmholtz equation. Chaos, Solitons & Fractals 2006;27(5):1119–23. [13] Soliman AA. Numerical simulation of the generalized regularized long wave equation by He’s variational iteration method. Math Comput Simul 2005;70(2):119–24. [14] Abdou MA, Soliman AA. Variational iteration method for solving Burger’s and coupled Burger’s equations. J Comput Appl Math 2005;181(2):245–51. [15] Kaya D, El-Sayed SM. A numerical simulation and explicit solutions of the generalized Burger-Fisher equation. Appl Math Comput 2004;152:403–13. [16] He JH. Non-perturbative methods for strongly nonlinear problems, Berlin: dissertation. de-Verlag im Internet GmbH, 2006. [17] He JH. Some asymptotic methods for strongly nonlinear equations. Int J Modern Phys B 2006;20(10):1141–99. [18] Abulwafa EM, Abdou MA, Mahmoud AA. The solution of nonlinear coagulation problem with mass loss. Chaos, Solitons & Fractals 2006;29(2):313–30. [19] He JH. Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput Meth Appl Mech Eng 1998;167(1–2):69–73. [20] He JH. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput Meth Appl Mech Eng 1998;167(1–2):57–68. [21] He JH. Variational theory for linear magneto-electro-elasticity. Int J Nonlinear Sci Numer Simul 2001;2(4):309–16. [22] Inokuti M, Sekine H, Mura T. General use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Nasser S, editor. Variational methods in the mechanics of solids. Permagon Press; 1978. p. 56–162. [23] Wang M, Li X. Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations. Phys Lett A 2005;343:48–54. [24] Marinca V. An approximate solution for one-dimensional weakly nonlinear oscillations. Int J Nonlinear Sci Numer Simul 2002;3(2):107–20. [25] Draganescu GE, Capalnasan V. Nonlinear relaxation phenomena in polycrystalline solids. Int J Nonlinear Sci Numer Simul 2003;4(3):219–25. [26] Soliman AA. A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations. Chaos, Solitons & Fractals 2006;29(2):294–302.