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## Application of the modified variational iteration method in the fourth-order Cahn-Hilliard equation BBM-Burgers equation

Zhong Ming, Tian Shou-Fu, Shi Yi-Qing
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• #### 摘要

变分迭代法是一种基于变分原理, 具有高数值精度的数值格式, 目前已广泛应用于各类强非线性孤立波方程的数值求解中. 本文利用修正的变分迭代法对两类非线性方程进行研究. 该格式是对原数值方法的一种改进, 即在变分项前引入了参数h. 通过定义误差函数的离散二范数并在定义域内绘出h-曲线, 从而确定出使误差达到最小的h, 再返回原迭代过程进行求解. 同时, 参数的引入也扩大了原数值解的收敛域, 在迭代次数一定的情况下达到了数值最优. 在数值实验中, 将上述结果应用于四阶的Cahn-Hilliard方程和Benjamin-Bona-Mahoney-Burgers方程. 对于四阶的Cahn-Hilliard方程, 普通的变分迭代法绝对误差在$10^{-1}$左右, 经过修正后, 绝对误差降为$10^{-4}$, 而且修正后的方法扩大了原数值解的收敛域. 对于Benjamin-Bona-Mahony-Burgers方程, 利用带有辅助参数的变分迭代法将数值解的精度提高到$10^{-3}$, 对真解的逼近效果优于原始的变分迭代法. 此数值方法也为其他强非线性孤立波微分方程的数值求解提供了方法和参考.

#### Abstract

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.

#### 作者及机构信息

###### 通信作者: 田守富, sftian@cumt.edu.cn
• 基金项目: 国家自然科学基金(批准号: 11975306)和江苏省高校自然科学研究项目(批准号: BK20181351)资助的课题

#### Authors and contacts

###### Corresponding author: Tian Shou-Fu, sftian@cumt.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11975306) and the Natural Science Foundation of Higher Education Institutions of Jiangsu Province, China (Grant No. BK20181351)

#### 施引文献

• 图 1  原始算法的误差

Fig. 1.  Error of the original algorithm.

图 2  h-曲线

Fig. 2.  h-curve

图 3  带有参数的算法的误差

Fig. 3.  Error of algorithm with parameter

图 4  精确解

Fig. 4.  Exact solution

图 5  数值解

Fig. 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

Fig. 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

Fig. 7.  Error of the original algorithm.

图 8  h-曲线

Fig. 8.  h-curve

图 9  带有参数的算法的误差

Fig. 9.  Error of algorithm with parameters

图 10  精确解

Fig. 10.  Exact solution

图 11  数值解

Fig. 11.  Numerical solution

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##### 出版历程
• 收稿日期:  2020-12-17
• 修回日期:  2021-05-24
• 上网日期:  2021-09-17
• 刊出日期:  2021-10-05

## 修正的变分迭代法在四阶Cahn-Hilliard方程和BBM-Burgers方程中的应用

• 中国矿业大学数学学院, 徐州　221100
• ###### 通信作者: 田守富, sftian@cumt.edu.cn
基金项目: 国家自然科学基金(批准号: 11975306)和江苏省高校自然科学研究项目(批准号: BK20181351)资助的课题

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