关于Duffing非线性振动方程渐近解最优性的公开问题

发布时间：2018-07-02 21:38

本文选题：渐近解 + 非线性振动方程　；参考：《大连理工大学》2015年硕士论文

【摘要】：利用重正规化群方法(RG方法),Kirkinis在[E. Kirkinis, SIAM Review 54 (2012) 374-388]文章中得到了Duffing非线性振动方程的一个渐近解(RG解).之后Kirkinis又在文章最后提出了很多公开问题,其中一个就是关于此方程所求得的渐近解(RG解)是否是最优解的问题.而在本文中,针对这个公开问题,我们将会利用同伦分析方法(HAM)给出一个肯定的答案.本文第一章主要介绍有关重正规化群方法(RG方法)和同伦分析方法(HAM)的一些基本概念、方法和定理;第二章中,为了解决公开问题,我们会利用同伦分析方法(HAM)求得Duffing非线性振动方程的渐近解；之后通过选取合适的参数值C0,Kirkinis求得的渐近解(RG解)可以被恢复成HAM渐近解；最后通过计算比较渐近解的平均剩余误差,从而证明了Kirkinis渐近解是最优的问题.第三章中,我们将应用HAM求解一个更一般振动方程,其HAM渐近解可以恢复为RG渐近解,并且会说明HAM方法是解决此类问题最优的方法.第四章中,为了进一步说明HAM的有效性,我们会用此方法解决一个含有参数7四阶边值问题[19],并且当参数γ的值越来越大时,其他的解析方法可能都会失效.最后在结论部分,我们会总结出HAM的一些结论,说明HAM方法是一种实用、有效的解析方法.
[Abstract]:In this paper, we obtain an asymptotic solution (RG solution) of duffing nonlinear vibration equation in [E. Kirkinis, Siam Review 54 (2012) 374-388] by using the method of normalized group (RG method). At the end of the paper, Kirkinis put forward many open problems, one of which is whether the asymptotic solution (RG solution) obtained by this equation is the optimal solution. In this paper, we will use homotopy analysis (ham) to give an affirmative answer to this open problem. In the first chapter, we mainly introduce some basic concepts, methods and theorems about the methods of normalized group (RG) and homotopy analysis (Ham). The asymptotic solution of duffing nonlinear vibration equation can be obtained by homotopy analysis method (Ham), and then the asymptotic solution (RG solution) obtained by selecting appropriate parameter value C _ 0 Kirkinis can be restored to the asymptotic solution of ham. Finally, it is proved that the Kirkinis asymptotic solution is an optimal problem by calculating the mean residual error of the asymptotic solution. In the third chapter, we use ham to solve a more general vibration equation, and the Ham asymptotic solution can be restored to RG asymptotic solution, and it is shown that Ham method is the best method for solving this kind of problem. In chapter 4, in order to further illustrate the effectiveness of ham, we use this method to solve a fourth order boundary value problem with parameter 7, and when the value of parameter 纬 becomes larger and larger, other analytical methods may fail. Finally, in the conclusion part, we will sum up some conclusions of Ham, which shows that Ham is a practical and effective analytical method.
【学位授予单位】：大连理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.8

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