Abstract

Under some conditions, an asymptotic solution containing boundary functions of two types was constructed by V.F. Butuzov and N.N. Nefedov for initial value problems for differential equations involving the second power of a small parameter multiplying the derivative with a right-hand side consisting of a singular matrix \(A(t)\) times the unknown function (as a linear part of the equation) plus the same small parameter multiplying a nonlinear function. In the present paper, an algorithm for constructing asymptotics using the orthogonal projectors onto \(\text{ker}A(t)\) and \(\text{ker}A(t)'\) (the prime denotes transposition) is given. This approach can be useful for understanding the algorithm underlying the construction of asymptotics. It allows us to present formulas for finding asymptotic terms of any order in an explicit form.

中文翻译：

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临界情况下一类奇摄动问题渐近解的Butuzov-Nefedov算法的投影机方法

摘要

在某些条件下，由VF Butuzov和NN Nefedov构造了包含两种类型的边界函数的渐近解，以解决微分方程的初值问题，该问题涉及小参数的二次幂将导数乘以由奇异矩阵组成的右侧\（A（t）\）乘以未知函数（作为方程的线性部分）加上相同的小参数乘以非线性函数。在本文中，使用正交投影仪在\（\ text {ker} A（t）\）和\（\ text {ker} A（t）'\）上构造渐近线的算法（素数表示换位）给出。这种方法对于理解渐近结构基础的算法可能很有用。它使我们能够以显式形式给出用于查找任何阶的渐近项的公式。