微分方程导出的多项式系统的求解
Solution of polynomial systems derived from differential equations
AI总结：
本文采用数值延拓方法求解非线性两点边值问题的有限差分离散化，通过同伦变形逐步细化网格，并利用滤波器限制追踪路径数量以应对解随网格点增长的问题。
摘要：
非线性两点边值问题出现在众多应用领域。从理论角度研究了各种情况下解的存在性和数量。这些结果通常依赖于非线性的增长条件。然而，通常无法预测边值问题可能有多少解，甚至无法确定解的存在性。近年来，发展了数值延拓方法，允许对多项式方程系统的所有复解进行数值逼近。本文中，数值延拓方法被用于数值计算非线性两点边值问题的有限差分离散化的解。这里采用的方法是通过同伦变形逐步细化离散化。这样，从粗网格上的解获得更细网格上的额外新解。复多项式系统设置带来的复杂问题是解的数量随离散化网格点数量增长。为应对此问题，考虑使用滤波器来限制每个阶段要追踪的路径数量。
英文摘要：
Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions of the nonlinearity. However, in general, one cannot forecast how many solutions a boundary value problem may possess or even determine the existence of a solution. In recent years numerical continuation methods have been developed which permit the numerical approximation of all complex solutions of systems of polynomial equations. In this paper, numerical continuation methods are adapted to numerically calculate the solutions of finite difference discretizations of nonlinear two-point boundary value problems. The approach taken here is to perform a homotopy deformation to successively refine discretizations. In this way additional new solutions on finer meshes are obtained from solutions on coarser meshes. The complicating issue which the complex polynomial system setting introduces is that the number of solutions grows with the number of mesh points of the discretization. To counter this, the use of filters to limit the number of paths to be followed at each stage is considered.
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