Hypersingular nonlinear boundary-value problems with a small parameter

TL;DR

This paper introduces a new class of hypersingular nonlinear boundary-value problems with a small parameter, highlighting their unique boundary layer properties and providing exact solutions for comparison and method development.

Contribution

It describes hypersingular boundary-value problems with a small parameter, revealing their unique properties and providing exact solutions, which differ from standard singular perturbation problems.

Findings

Super thin boundary layers with extremely large derivatives are observed.

Boundary layer position is determined by boundary values, not equation coefficients.

Exact solutions are obtained for specific hypersingular problems.

Abstract

For the first time, some hypersingular nonlinear boundary-value problems with a small parameter~$\varepsilon$ at the highest derivative are described. These problems essentially (qualitatively and quantitatively) differ from the usual linear and quasilinear singularly perturbed boundary-value problems and have the following unusual properties: (i) in hypersingular boundary-value problems, super thin boundary layers arise, and the derivative at the boundary layer can have very large values of the order of $e^{1/\varepsilon}$ and more (in standard problems with boundary layers, the derivative at the boundary has the order of $\varepsilon^{-1}$ or less); (ii) in hypersingular boundary-value problems, the position of the boundary layer is determined by the values of the unknown function at the boundaries (in standard problems with boundary layers, the position of the boundary layer is determined by coefficients of the given equation, and the values of the unknown function at the boundaries do not play a role here); (iii) hypersingular boundary-value problems do not admit a direct application of the method of matched asymptotic expansions (without a preliminary nonlinear point transformation of the equation under consideration). Examples of hypersingular nonlinear boundary-value problems with ODEs and PDEs are given and their exact solutions are obtained. It is important to note that the exact solutions presented in this paper can be used to compare the effectiveness of various methods of numerical integration of singularly perturbed problems with boundary layers, and also to develop new numerical and approximate analytical methods.