基于对称理论的近似约化与同伦近似约化

For nonlinear systems in various disciplines or fields, approximate analytical solutions are generally more practical and more applicable than exact analytical solutions. Approximate reduction method (mainly including the approximate symmetry method and approximate direct reduction method) and approximate homotopy reduction method (mainly including the approximate homotopy symmetry method and approximate homotopy direct reduction method) are effective methods for constructing approximate analytical solutions. In this topic, these two kinds of methods are used to study approximate analytical solution for various nonlinear systems: 1. improve the approximate symmetry method and approximate direct reduction method so as to propose ameliorated approximate reduction methods; 2. improve the approximate homotopy symmetry method and approximate homotopy direct reduction method so as to propose ameliorated homotopy approximate reduction methods; 3. devising new approximate methods by combining perturbation method and various methods for exact solutions of differential equations; 4. compare the above methods and explore the practical significance of approximate analytical solutions.

对于各学科领域中的非线性系统，其近似解析解通常比精确解析解更具有实际意义与应用价值。近似约化法（主要包括近似对称法和近似直接约化法）与同伦近似约化法（主要包括同伦近似对称法和同伦近似直接约化法）是构造近似解析解的有效方法。本课题主要利用这两类方法研究各种非线性系统的近似解析解：1、改进近似对称法和近似直接约化法，提出各种改进的近似约化方法；2、改进同伦近似对称法和同伦近似直接约化法，提出各种改进的同伦近似约化方法；3、结合摄动法和微分方程的各种精确求解方法设计新的近似求解方法；4、比较上述所有方法，探讨近似解析解的实际意义。

本项目主要利用近似相似约化方法以及同伦近似相似约化方法构造扰动非线性方程的近似解析解。1、对于具有一定特征扰动项的一类扰动非线性方程，近似对称方法以及同伦近似对称方法可以求出整类扰动方程的统一形式的无穷多约化方程，以及前几阶约化方程的统一形式的雅可比椭圆函数解或者双曲正切函数解，从而得出截断级数解；2、级数解的收敛性受扰动项影响，可以通过一些常数调整，或者通过扰动参数与辅助参数进行更有效的调整；3、近似直接约化方法以及同伦近似直接约化方法可以分别求出与近似对称方法以及同伦近似对称方法同样的结果。项目的科学意义：首先，扰动非线性方程的可解性可以通过无穷多约化方程关于雅可比椭圆函数解或者双曲正切函数解的存在性进行评价；其次，级数解收敛性的改进有助于提高截断级数解的精确度。