V.F. Butuzov

Singularly perturbed problems with boundary and internal layers

This paper is an expanded version of the talk given by the authors at the International Conference “Differential Equations and Topology” dedicated to the centenary of the birth of L.S. Pontryagin. We present a brief survey and describe new ideas and methods of analysis in the asymptotic theory of solutions with internal layers, which is one of the topical fields of singular perturbation theory.

Asymptotic Methods in the Theory of Ordinary Differential Equations

The main aim of investigations undertaken during the early stages of research into differential equations was the derivation of exact solutions. It was subsequently found, however, that the effective representation of the exact solution in terms of elementary functions is possible only for a limited number of special classes of differential equations. Therefore, the question of methods for the construction of approximate solutions of differential equations was recognized to be the main area of research. Work in this area proceeded along two directions: a) the development of numerical methods of solution and b) the development of the so-called asymptotic methods of solution. The aim of this review is to describe asymptotic methods at their present level of development.

On the special properties of the boundary layer in singularly perturbed problems with multiple root of the degenerate equation

Using the boundary-value problem for the singularly perturbed second-order differential equation as an example, we show that the multiplicity of the root of the degenerate equation significantly affects the asymptotics of the solution, especially in the boundary layer.

A Singularly Perturbed Boundary Value Problem for a Second-Order Equation in the Case of Variation of Stability

A boundary value problem for a second-order nonlinear singularly perturbed differential equation is considered for the case in which there is variation of stability caused by the intersection of roots of the degenerate equation. By the method of differential inequalities, we prove the existence of a solution such that the limit solution is nonsmooth.

Periodic Solutions with a Boundary Layer of Reaction–Diffusion Equations with Singularly Perturbed Neumann Boundary Conditions

We consider singularly perturbed reaction–diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x, t, e) with boundary layers and derive conditions for their asymptotic stability. The boundary layer part of u(x, t, e) is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order e. Another peculiarity of our problem is that — in contrast to the case of Dirichlet boundary conditions — it may have several asymptotically stable time-periodic solutions, where these solutions differ only in the description of the boundary layers. Our approach is based on the construction of sufficiently precise lower and upper solutions.

Region of attraction of a periodic solution to a singularly perturbed parabolic problem

We consider the singularly perturbed parabolic differential equation under the assumption that f is T-periodic in t and that the degenerate equation f(u, x, t, 0) = 0 has two intersecting roots. In a previous paper [V.F. Butuzov, N.N. Nefedov, L. Recke, and K.R. Schneider, On a class of periodic solutions of a singularly perturbed parabolic problem, J. Math. Anal. Appl. 348 (2008), pp. 508--515] we presented conditions under which there exists an asymptotically stable T-periodic solution u p (x, t, ϵ) satisfying no-flux boundary conditions. In this note we characterize a set of initial functions belonging to the region of attraction of u p (x, t, ϵ).

Asymptotics, Stability, and Region of Attraction of Periodic Solution to a Singularly Perturbed Parabolic Problem with Double Root of a Degenerate Equation

For a singularly perturbed parabolic problem with Dirichlet boundary conditions, the asymptotic decomposition of a solution periodic in time and with boundary layers near the ends of the segment where the degenerate equation has a double root is constructed and substantiated. The construction algorithm for the asymptotics and the behavior of the solution in the boundary layers turn out to differ significantly as compared to the case of a simple root of a degenerate equation. The stability of the periodic solution and its region of attraction are also studied.

On the dependence of the structure of boundary layers on the boundary conditions in a singularly perturbed boundary-value problem with multiple root of the related degenerate equation

We consider the two-point boundary-value problem for a singularly perturbed secondorder differential equation for the case in which the related degenerate equation has a double root. It is shown that the structure of boundary layers essentially depends on the degree of proximity of the given boundary values of the solution to the root of the degenerate equation; this phenomenon is determined by the multiplicity of the root.

Singularly perturbed boundary value problem with multizonal interior transitional layer

The paper discusses a two-point boundary value problem for a singularly perturbed ordinary second-order differential equation in the case when the degenerate equation has three nonintersecting roots from which one root is twofold and two roots are onefold. It is proved that the problem has a solution with transition from the twofold root of the degenerate equation to the onefold root in the neighborhood of a point of the interval for sufficiently small parameter values. An asymptotic expansion of this solution is constructed. It is distinguished from the known expansion when all the roots of the degenerate equation are onefold; in particular, the transitional layer is multizonal.

Corner boundary layer in nonlinear elliptic problems containing first-order derivatives

AbstractIn a rectangular domain, the first boundary-value problem is considered for the following singularly perturbed elliptic equation: