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The variational iteration method, based on the variational principle, is a numerical format with high numerical accuracy and convenience, has been widely applied in the numerical solution of various strong-nonlinear solitary wave equations. In this paper, the modified variational iteration method is used to improve the original numerical method, that is, the parameter h is introduced before the variational item. By defining the discrete two norm of the error function and drawing the h-curve in the domain of definition, the h that minimizes the error is determined and then returned to the original iteration process for solution. In this paper, We choose the uniform grid points to caculate the error. At the same time, the introduction of parameters also expands the convergence domain of the original numerical solution and achieves the numerical optimization under the condition of a certain number of iterations, which shows better than the general method. In the numerical experiment, the above results are applied to the fourth-order Cahn-Hilliard equation and the Benjamin-Bona-Mahony-Burgers equation. The Cahn-Hilliard equation was first found to describe the phenomeno in interface dynamics. It has vital application in physical. For the fourth-order Cahn-Hilliard equation, the error order of the ordinary variational iteration method is about
$10^{-1}$ , and the absolute error is reduced to$10^{-4}$ after the modification. Moreover, the modified method expands the convergence domain of the original numerical solution. And the Benjamin-Bona-Mahony-Burgers equation can be degenerated to the Benjamin-Bona-Mahony and the Burgers equation under the appropriate parameter selection. For the Benjamin-Bona-Mahony-Burgers equation, if using the normal method, we can find that the numerical solution will not converge. But the accuracy of the numerical solution is decreased to$10^{-3}$ by using the variational iteration method with auxiliary parameters, which is superior to the original variational iteration method in the approximation effect of the true solution. This numerical method also provides a scheme and reference for the numerical solution of other strong-nonlinear solitary wave differential equations. This scheme provieds a continuous solution in the time and space domain, which differs from the finite difference method, finite volume scheme and so on. That means we can use this method independently without using any other scheme to match our approarch, this is also the advantage of the modified variational iteration method.-
Keywords:
- modified variational iteration method /
- nonlinear solitary wave equation /
- fourth order Cahn-Hilliard equation /
- Benjamin-Bona-Mahony-Burgers equation
[1] Kartashov Y V, Astrakharchik G E, Malomed B A, Torner L 2019 Nat. Rev. Phys. 1 185
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 原始算法的误差
Figure 1. Error of the original algorithm.
图 2 h-曲线
Figure 2. h-curve
图 3 带有参数的算法的误差
Figure 3. Error of algorithm with parameter
图 4 精确解
Figure 4. Exact solution
图 5 数值解
Figure 5. Numerical solution
图 6
$ x=4$ 时原方法和修正后方法的数值解以及真解的图像Figure 6. The numerical solution of the original method and the corrected method, as well as the image of the exact solution when
$ x=4$ 
图 7 原始算法的误差
Figure 7. Error of the original algorithm.
图 8 h-曲线
Figure 8. h-curve
图 9 带有参数的算法的误差
Figure 9. Error of algorithm with parameters
图 10 精确解
Figure 10. Exact solution
图 11 数值解
Figure 11. Numerical solution
表 1 不同点处真解与数值解对比
Table 1. Comparison of true and numerical solutions at different points
$ (x, t)$ 原始数值解$ u_1$ 修正数值解$ u_2$ 精确解u (1, 0.2) –1.855 × 10–1 –1.855 × 10–1 –1.855 × 10–1 (1, 0.4) –2.176 × 10–1 –1.857 × 10–1 –1.864 × 10–1 (1, 0.6) –2.970 × 10–1 –1.860 × 10–1 –1.873 × 10–1 (1, 0.8) –4.234 × 10–1 –1.862 × 10–1 –1.881 × 10–1 (1, 1.0) –6.297 × 10–1 –1.863 × 10–1 –1.889 × 10–1 (1, 1.2) –8.914 × 10–1 –1.866 × 10–1 –1.893 × 10–1 (1, 1.4) –1.212 –1.868 × 10–1 –1.903 × 10–1 (1, 1.6) –1.590 –1.870 × 10–1 –1.910 × 10–1 (1, 1.8) –2.033 –1.872 × 10–1 –1.916 × 10–1 (1, 2.0) –2.429 –1.874 × 10–1 –1.922 × 10–1 
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[1] Kartashov Y V, Astrakharchik G E, Malomed B A, Torner L 2019 Nat. Rev. Phys. 1 185
Google Scholar
[2] Kengne E, Liu W M, Malomed B A 2020 Phys. Rep. 1 899
[3] 张大军 2020 物理学报 69 010202
Google Scholar
Zhang D J 2020 Acta Phys. Sin. 69 010202
Google Scholar
[4] 宋彩芹, 朱佐农 2020 物理学报 69 010204
Google Scholar
Song C Q, Zhu Z N 2020 Acta Phys. Sin. 69 010204
Google Scholar
[5] Wang P, Zheng Y L, Chen X F, Huang C M, Kartashov Y V, Torner L, Konotop, V V, Ye F W 2020 Nature 577 42
Google Scholar
[6] Fu Q D, Wang P, Huang C M, Kartashov Y V, Torner L, Konotop V V, Ye F W 2020 Nat. Photonics 14 663
Google Scholar
[7] Tian S F 2016 Proc. Math. Phys. Eng. Sci. 472 20160588
Google Scholar
[8] Tian S F 2017 J. Differ. Equ. 262 506
Google Scholar
[9] Ma W X 2020 Appl. Math. Lett. 102 106161
Google Scholar
[10] Zhang G Q, Yan Z Y 2020 Physica D 402 132170
Google Scholar
[11] Wang X B, Han B 2020 J. Math. Anal. Appl. 487 123968
Google Scholar
[12] Li B Q, Ma Y L 2020 Appl. Math. Comput. 386 125469
Google Scholar
[13] Su J J, Gao Y T, Ding C C 2019 Appl. Math. Lett. 88 201
Google Scholar
[14] Zhang X, Wang R, Zhang Y Q, Kartashov Y V, Li F, Zhong H, Guan H, Gao K, Li F, Zhang Y P, Xiao M 2020 Nat. Commun. 11 1902
Google Scholar
[15] Jin K, Li Y, Li F, Belic M R, Zhang Y P, Zhang Y Q 2020 Adv. Photonics 2 046002
Google Scholar
[16] Zeng L, Zeng J 2019 Adv. Photonics. 1 046004
Google Scholar
[17] Zeng L, Zeng J 2020 Commun. Phys. 3 26
Google Scholar
[18] Yang J K 2008 J. Comput. Phys. 227 6862
Google Scholar
[19] Bao W Z, Yin J 2019 Res. Math. Sci. 6 1
Google Scholar
[20] Antoine X, Bao W Z, Besse C 2013 Comput. Phys. Commun. 184 2621
Google Scholar
[21] Cockburn B, Shu C W 1998 SIAM J. Math. Anal. 35 2440
Google Scholar
[22] Jiang G S, Shu C W 1996 J. Comput. Phys. 126 202
Google Scholar
[23] He J H 1999 Int. J. Non. Linear Mech. 34 699
Google Scholar
[24] He J H 2007 J. Comput. Appl. Math. 207 3
Google Scholar
[25] He J H 2007 Comput. Math. Appl. 54 881
Google Scholar
[26] Hesameddini E, Latifizadeh H 2009 J. Nonlinear Sci. Numer. Simul. 10 1377
Google Scholar
[27] Salkuyeh D K 2008 Comput. Math. Appl. 56 2027
Google Scholar
[28] Noor M A, Mohyud-Din S T 2008 J. Nonlinear Sci. Numer. Simul. 9 141
Google Scholar
[29] Liao S J 2004 Appl. Math. Comput. 147 499
Google Scholar
[30] Zayed E M, Rahman H 2009 J. Nonlinear Sci. Numer. Simul. 10 1093
Google Scholar
[31] Hosseini M M, Mohyud-Din S T, Ghaneai H, Usman M 2010 J. Comput. Appl. Math. 11 495
Google Scholar
[32] Cahn J W, Hilliard J E 1958 J. Chem. Phys. 28 258
Google Scholar
[33] Barrett J W, Blowey J F 1997 Numer. Math. 77 1
Google Scholar
[34] Wells G N, Kuhl E, Garikipati K 2006 J. Comput. Phys. 218 860
Google Scholar
[35] Kay D, Welford R 2006 J. Comput. Phys. 212 288
Google Scholar
[36] Benjamin T B, Bona J L, Mahony J J 1972 Philos. Trans. R. Soc. A 272 47
Google Scholar
[37] Karakoc S B G, Bhowmik S K 2019 Comput. Math. Appl. 77 1917
Google Scholar
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