几个非线性偏微分方程的近似解

发布时间：2018-06-21 04:48

本文选题：Drinfel’d-Sokolov-Wilson方程 + Boussinesq方程　；参考：《江苏大学》2016年硕士论文

【摘要】：在自然科学和社会科学领域中,很多现象都能够借助非线性方程来表述。因此,对这些问题的探讨就等价于对非线性方程的研究。近年来,由于计算机符号运算软件的蓬勃发展,使得求解线性方程变得比过去更加简单。但是,对非线性方程的求解还是比较困难。因此,对非线性方程近似解的研究已经变成热点问题。近年来,很多求非线性方程近似解的途径已经被提出。其中,同伦摄动法和同伦分析法是两种比较重要的方法。本文主要利用同伦分析法研究常系数非线性偏微分方程,利用同伦摄动法和傅里叶变换求解变系数非线性偏微分方程,从而得到其近似解。以Drinfel’d-Sokolov-Wilson方程为例,当取三角函数形式和指数函数形式的基函数时,我们得到了两种不同形式的近似解。同样地,当取指数函数形式的基函数时,我们得到了广义Boussinesq方程的指数形式的近似解。通过调节近似解中辅助参数η来进行误差分析,从而选取恰当的辅助参数η以确保解的收敛性,最终保证了近似解的有效性。这表明了运用同伦分析法求解这一类常系数非线性偏微分方程是一个好的方法。然而,应用同伦分析法求解变系数非线性偏微分方程时碰到了困难。在研究变系数的Boussinesq方程和变系数的KdV-Burgers方程时,本文运用同伦摄动法求得了方程不同形式的二阶近似解。
[Abstract]:In the field of natural science and social science, many phenomena can be expressed by nonlinear equations. Therefore, the discussion of these problems is equivalent to the study of nonlinear equations. In recent years, due to the rapid development of computer symbolic computing software, solving linear equations becomes more simple than before. However, it is difficult to solve nonlinear equations. Therefore, the study of approximate solutions of nonlinear equations has become a hot issue. In recent years, many approaches to approximate solutions of nonlinear equations have been proposed. Homotopy perturbation and homotopy analysis are two important methods. In this paper, the homotopy analysis method is used to study the nonlinear partial differential equations with constant coefficients, and the homotopy perturbation method and Fourier transform are used to solve the nonlinear partial differential equations with variable coefficients, and the approximate solutions are obtained. Taking Drinfeld-Sokolov-Wilson equation as an example, we obtain two kinds of approximate solutions in the form of trigonometric function and exponential function. Similarly, when we take the basis function in the form of exponential function, we obtain the approximate solution of the exponential form of the generalized Boussinesq equation. By adjusting the auxiliary parameter 畏 in the approximate solution, the error analysis is carried out, and the appropriate auxiliary parameter 畏 is selected to ensure the convergence of the solution, and finally the validity of the approximate solution is guaranteed. This shows that homotopy analysis is a good method for solving this class of nonlinear partial differential equations with constant coefficients. However, it is difficult to solve nonlinear partial differential equations with variable coefficients by homotopy analysis. When we study the Boussinesq equation with variable coefficients and the KdV-Burgers equation with variable coefficients, we obtain the second order approximate solutions of the equation in different forms by using the homotopy perturbation method.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175.29

【相似文献】