一类非线性奇摄动问题的匹配求解

The matching solving method for a class of nonlinear singularly perturbed problems

匹配渐近展开法基本思想是用多个展开式来表示解,其中每一个展开式在一部分区域上有效,对相邻的展开式在重叠区域内进行匹配.讨论了一类非线性奇摄动方程的求解问题,利用匹配法求出它的一阶渐近解.依据方程的特征,找出满足左边界条件的外展开式;引进伸展变换确定满足右边界条件的内展开式;对一项外展开式和一项内展开式进行匹配;得出一个一致有效的复合展开式.

The matched asymptotic expansion is a common method for researching the singularly perturbed equations. Its fundamental principle is that the solutions are given by some expansions. Every expansion is efficient in some regions. The consecutive expansions are matched in overlay region. This paper deals with concerns the solving problems for a class of the nonlinear singularly perturbed equations. Using the matching method, the first order asymptotic solutions are obtained. The outer expansions satisfying the left boundary condition are found according to the characteristic of equation. The inner expansions satisfying the right boundary condition are defined by introducing stretching transform. A term of the outer expansions and another term of the inner expansions are matched. A uniformly valid compositie expansions is obtained.