【作者】 于欢欢；

【导师】 张金良；

【作者基本信息】 河南科技大学 ， 应用数学， 2011， 硕士

【摘要】 20世纪中后期,非线性科学迅速发展成为科学技术研究的前沿领域。在非线性科学的研究中,非线性方程的求解一直是研究的难点、热点。孤子方程、积分方程的求解是非线性科学的研究中的两个重要领域。经过数学家和物理学家不懈努力,已经找到一些孤子方程求解的有效方法,如反散射方法、Hirota双线性法、齐次平衡法、F-展开法、指数展开法等等,但是,由于孤子方程的多样性及复杂性,导致只有有限的孤子方程才能够得到精确解,因此寻找有效的近似方法就成为孤子方程、积分方程的一个重要研究方向。变分迭代算法是何吉欢在Inokuti-Sekine-Mura方法基础上提出来的。变分迭代法自提出以后,得到了科技工作者的普遍关注,已经应用于非线性问题的研究中。首先,本文借助于变分迭代法求得了具强非线性、耦合孤子方程(组)的近似解,并对近似解进行了分析,讨论了强非线性项系数对近似解的影响;其次,利用变分迭代法,对FKPP方程进行近似求解;最后,研究了三个二维积分方程,求得了方程的近似解;借助Matlab对所得的近似解及误差进行近似模拟。更多还原

【Abstract】 In the late 20th century, the nonlinear science is rapidly developing and become the frontier of science and technology. In the research of the nonlinear science, solving nonlinear equation is one of the most difficult and hottest topics. Solving soliton equations and integral equations are two of the most important fields of the nonlinear science. Many efficient methods for exploring the exact solutions of the soliton equations are presented by the mathematicians and physicists, such as inverse scattering method, Hirota bilinear methods, the homogeneous balance method, F-expansion method, exponential expansion method, etc. However, due to the variety and complexity of soliton equations, the only special soliton equations can be solved. Thus, looking for an effective approximate method of solving soliton equations and integral equations has become an important problem.Based on the Inokuti-Sekine-Mura’s method, the variational iteration method is presented by Prof. He Ji-huan. Since the variational iteration method is introduced, more and more scientists have paid attention. As the rapid development of variational iteration method, it has been widely applied to the nonlinear problems. In this thesis, the approximate solutions of the soliton equation with high-order nonlinear terms, two coupled soliton equations are respectively presented with the aids of the variational iteration method and the approximate solutions are analyzed, the influence of coefficients of high-order nonlinear terms on the approximate solutions is discussed. And then the approximate solutions of FKPP equation are derived by variational iteration method. In the last, three two-dimensional integral equations are analyzed and the approximate solutions are obtained. The approximate solutions are simulated by Matlab.更多还原