【作者】 李彦刚；

【导师】 周宇斌；

【作者基本信息】 兰州大学 ， 计算数学， 2007， 硕士

【摘要】 求解微分方程的精确解在理论和实际中都是重要而古老的研究课题，显式解，特别是行波解可以很好地描述各种物理现象，如振动、传播波等。基于非线性方程的复杂性，至今能够求出精确解的方程很少，因此，寻求新的求解方法和拓展已有方法都是重要而有价值的工作。首先，本文对原有“Tanh-”函数法做了新的扩展，用变系数式构造解的形式，实现常系数非线性方程和变系数非线性方程求解式的统一构造，从而同一方法可获得更多方程更多形式的精确解，也便于实现计算机上统一编程处理；为获得更广泛的解，本文对辅助Riccati方程也做了相应改进，方程中构造了任意函数项，即辅助Riccati方程改进为：φ′(ξ)=φ~2(ξ)+b(ξ)，b(ξ)为ξ的任意函数，使得构造形式更灵活，文中具体给出了b(ξ)=kξ~n的解的情形，应用该方法求解KdV方程，(2+1)维KD方程等非线性演化方程，获得了丰富的精确解簇。其次，基于齐次平衡方法，通过引入参数，给出了求解非线性演化方程精确解的一种变换方法，对于特定方程通过变换，可以讨论方程的多孤子解，变换求解自相似解，进一步通过引入GM(Gardner-Morikawa)变换求解非线性演化方程的截断级数解。更多还原

【Abstract】 It is an important and old research subject for obtaining the exact solution of differential equations. The explicit solution, especially the traveling wave solution, can be used to describe many physical phenomena well, such as oscillation, propagation wave etc. Up to now the exact solutions for many important equations can not still be received since the complexity of the nonlinear equations. Thus seeking new method and extending existed method, both come to be an important and valuable work.In this paper, we generalize the Tanh-function method, exhibit the united formulae to solutions of constant coefficient equations and variable coefficient equations, by the generalized method we can get more exact solutions. Meanwhile it is convenience to be dealt with by the computer. In order to receive more solutions, we also modify the subsidiary equation to beφ’(ξ) =φ~2(ξ) + b(ξ), where b(ξ) is an arbitrary function ofξ. The arbitrariness of b(ξ) makes the solutions more flexible. Specially, the case when b(ξ) = kξ~n is discussed in this paper. And by applying the generalized method to KdV equation, (2+1) dimension KD equations and other nonlinear evolution equations, we obtain abundant exact solution families of those equations.Based on the homogenous balance method, by introducing the parameters we give a new method of solving nonlinear partial differential equation (NLPDE)—Method of parametric transformation. For some specific NLPDE we obtain the multi-soliton solutions, self-similar solutions and trun- cation series solutions associated with the GM (Gardner-Morikawa) transformation.更多还原