Abstract

The article studies the asymptotic behavior of the solutions of a singularly perturbed three boundary value problems on an interval. The object of the study is a linear inhomogeneous ordinary differential equation of the second order with a small parameter with the highest derivative of the unknown function. The singularities of the problem are that the small parameter is found at the highest derivative of the unknown function and the corresponding unperturbed first-order differential equation has higher order an irregular singular point at the left end of the segment. At the ends of the segment, boundary conditions are imposed. Three problems are considered, in one Dirichlet problem, in two Neumann problem and in the three Roben problem. Asymptotic expansions of problems are constructed by the classical method of Vishik–Lyusternik–Vasilyeva–Imanaliev boundary functions. However, this method cannot be applied directly, since the external solution has a singularity. We first remove this singularity from the external solution, then apply the method of boundary functions. The constructed asymptotic expansions are substantiated using the maximum principle, i.e. estimates for the residual functions are obtained.

中文翻译：

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有限方程具有奇异点时边值问题解的渐近性

摘要

本文研究了一个区间上奇摄动三个边值问题的解的渐近行为。研究的目的是二阶线性不均匀常微分方程，其参数很小，未知函数的导数最高。问题的奇异之处在于，在未知函数的最高导数处发现了较小的参数，并且相应的无扰动的一阶微分方程在该段的左端具有较高阶的不规则奇异点。在段的末端，施加边界条件。在一个Dirichlet问题，两个Neumann问题和三个Roben问题中考虑了三个问题。问题的渐近扩展是通过Vishik–Lyusternik–Vasilyeva–Imanaliev边界函数的经典方法构造的。但是，由于外部解决方案具有奇异性，因此无法直接应用此方法。我们首先从外部解中消除这种奇异性，然后应用边界函数的方法。构造的渐近展开采用最大原理进行证实，即获得残差函数的估计值。