一类高次非线性奇摄动问题的匹配解法

The Matching Solution to a Class of High Power Nonlinear Singularly Perturbed Problems

在适当的条件下,利用奇摄动理论中的收缩变换以及匹配技巧,讨论了一类高次非线性奇摄动两点边值问题.相应于边界条件,构造了原问题的外部解和内部解.按ε的增幂展开非线性项、比较同次幂的系数.再利用特异极限,由匹配原问题的内、外部解得出了相应于参数k=0、k=1、k<0、k>1和0<k<1等情形下的原问题解的渐近展开式,推广了相应结论.这个问题的探讨是奇摄动理论在研究微分方程上的一个应用,并为估计一些相关类型的非线性问题的解提供了一种简捷有效的方法.

By using the contracted transformation and the matching techniques theory of singularly perturbations under appropriate conditions, this paper discusses a class of high power nonlinear singularly perturbed two - point boundary value problems and constructs the exterior and interior solutions of the original problem according to the boundary conditions. By expanding the nonlinear term according to the increasing powers for e, comparing the coefficient with the same powers of ε, and using special limit, we obtain the interior and exterior matching solutions of original problem, and asymptotic expansions of solution for the problem relating to the parameter in k=0,k=1,k〈0,k〉1, and 0〈k〈1, and popularize the corresponding results. This discussion is an application of the singularly perturbation theory to research differential equations and gives a simple and valid method to estimate the solution of some class of corresponding nonlinear problems.