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The averaging method is justified for normal systems of differential equations with rapidly oscillating summands proportional to the square root of the oscillation frequency in the case of the boundary-value problem on a finite interval and for the problem of bounded solutions on the positive semiaxis with boundary condition at its left endpoint. [ABSTRACT FROM AUTHOR]

We consider the problem on the periodic solutions of a system of ordinary differential equations of arbitrary order n containing terms oscillating at a frequency ω ≫ 1 with coefficients of the order of ω n/2. For this problem, we construct the averaged (limit) problem and justify the averaging method as well as another efficient algorithm for constructing the complete asymptotics of the solution. [ABSTRACT FROM AUTHOR]

We develop the averaging method theory for parabolic problems with rapidly oscillating summands some of which are large, i.e., proportional to the square root of the frequency of oscillations. In this case the corresponding averaged problems do not coincide in general with those obtained by the traditional averaging, i.e., by formally averaging the summands of the initial problem (since the principal term of the asymptotic expansion of a solution to the latter problem is not in general a solution to the so-obtained problem). In this article we consider the question of time periodic solutions to the first boundary value problem for a semilinear parabolic equation of an arbitrary order 2 k whose nonlinear terms, including the large, depend on the derivatives of the unknown up to the order k-1. We construct the averaged problem and the formal asymptotic expansion of a solution. When the large summands depend on the unknown rather than its derivatives we justify the averaging method and the complete asymptotic expansion of a solution. [ABSTRACT FROM AUTHOR]

This article presents information on asymptotic integration of differential equations with oscillatory terms of large amplitudes. The main theorems of the classical theory of the Bogolyubov averaging method deal with systems of ordinary differential equations in the so-called standard form, that is, representable in the form. In numerous subsequent studies, many results of the classical theory were generalized to various new classes of equations. In the theory of ordinary differential equations, a large class including equations in standard form and well-known systems with rapidly rotating phase is formed by systems of equations with rapid and slow variables.

The article presents a discourse on higher-order approximations of the averaging method for Parabolic Initial-Boundary Value Problems with Rapidly Oscillating Coefficients. For abstract parabolic equations with initial conditions and for semi-linear parabolic initial boundary value problems whose coefficients rapidly change in the course of time, it was shown in a previous article that the solution of the averaged problem is an asymptotic approximation (in certain norms) to the solution of the original (perturbed) problem.