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One of the powerful asymptotic methods of the theory of differential equations is the well-known averaging method associated with the famous researchers Krylov and Bogolyubov. This method is well developed for ordinary differential and integral equations as well as for many classes of partial differential equations, but not studied sufficiently for hyperbolic systems. The method was justified for semilinear hyperbolic systems in the articles by Mitropolsky, Khoma, and some other authors. Moreover, some authors proposed and justified an algorithm for constructing complete asymptotics of solutions to these systems; the solution of the averaged problem is the main term of the asymptotics. We study the Cauchy problem in a multidimensional space-time layer for a hyperbolic system of first-order quasilinear differential equations with rapidly time-oscillating terms. Such terms on the right-hand side may be large and proportional to the square root of the high-frequency oscillations; the large terms have zero mean in the fast variable (the product of frequency and time). The peculiarity of the problem is the fact that the summands of the equations do not depend explicitly on spatial variables. We construct some limit (averaged) problem as the oscillation frequency tends to infinity and justify passage to the averaging method. This proves the unique solvability of the original (perturbed) problem and substantiates the asymptotic proximity of solutions to the original (perturbed) and averaged problems uniformly through the layer. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a one-dimensional hyperbolic equation whose lower-order coefficient and right-hand side oscillate in time with a high frequency and the amplitude of the lower-order coefficient is small. Under study is the reconstruction of the cofactors of these rapidly oscillating functions independent of the space variable from a partial asymptotics of a solution at some point of the space. The classical theory of inverse problems examines the numerous problems of determining unknown sources, and coefficients without rapid oscillations for various evolutionary equations, where the exact solution to the direct problem appears in the additional overdetermination condition. Equations with rapidly oscillating data are often encountered in modeling the physical, chemical, and other processes that occur in media subjected to high-frequency electromagnetic, acoustic, vibrational, and others fields, which demonstrates the topicality of perturbation theory problems on the reconstruction of unknown functions in high-frequency equations. We give some nonclassical algorithm for solving such problems that lies at the junction of asymptotic methods and inverse problems. In this case the overdetermination condition involves a partial asymptotics of solution of a certain length rather than the exact solution. [ABSTRACT FROM AUTHOR]

A multipoint boundary value problem for a nonlinear normal system of ordinary differential equations with a rapidly time-oscillating right-hand side is considered on a positive time semiaxis. For this problem, which depends on a large parameter (high oscillation frequency), a limiting (averaged) multipoint boundary value problem is constructed and a limiting transition in the Hölder space of bounded vector functions defined on the considered semiaxis is justified. Thus, for normal systems of differential equations in the case of a multipoint boundary value problem, the Krylov–Bogolyubov averaging method on the semiaxis is justified. [ABSTRACT FROM AUTHOR]

We consider a multipoint boundary value problem for a nonlinear normal system of ordinary differential equations with a rapidly oscillating (in time) right-hand side. Some summands on the right-hand side can be of a large amplitude proportional to the square root of the oscillation frequency. The Krylov–Bogolyubov averaging method is justified for this problem depending on a large parameter (high frequency oscillations). The passage to the limit is realized in this problem that is called perturbed in the Hölder space of vector functions on the time interval under consideration and we construct the (averaged) multipoint boundary value problem (i.e., we prove that the solutions to the perturbed and averaged problems are asymptotically close). The approach of this paper relies on the classical implicit function theorem in a Banach space and was firstly used by Simonenko for abstract parabolic equations in the case of the Cauchy problem and the problem of time-periodic solutions. The Krylov–Bogolyubov averaging method is most important, widely used, and developed rather fully for various classes of equations. The numerous articles on the systems of ordinary differential equations mainly study the Cauchy problem on a segment and the problems of periodic almost periodic and general solutions bounded on the entire time axis. However, boundary-value problems, especially multipoint boundary value problems, are still represented insufficiently in the literature. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the one-dimensional hyperbolic equation with an unknown right-hand side, which rapidly oscillates in time. This right-hand side is the product of two functions, with one being a function of a time variable and a space variable and the other one being a function of a time variable and a so-called fast time variable. We formulate and solve four asymptotic problems of exact reconstruction of the source from certain information about the partial asymptotics of the solution. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a one-dimensional hyperbolic equation. The lowest order coefficient and the right-hand side oscillate in time at a high frequency, and the amplitude of the lowest order coefficient is small. The way of reconstructing these oscillating functions from partial solution asymptotics given at a certain point of the domain is studied. [ABSTRACT FROM AUTHOR]

For an abstract parabolic equation with initial condition and multidimensional parabolic initial-boundary value problem with absolute terms rapidly oscillating in time, inverse problems of finding these absolute terms from some information about the partial asymptotics of the solutions of the original problems are stated and solved. In this case, the absolute terms are the products of two factors, one of which is described by a rapidly oscillating function (i.e., depends on fast time), while the second term may depend on ordinary time but does not depend on fast time. The following three cases are considered: where, in this pair, only one of the factors is known and where only the mean value of the rapidly oscillating factor is known. [ABSTRACT FROM AUTHOR]

The paper is devoted to the development of the theory of inverse problems for evolution equations with summands rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints (overdetermination conditions) are imposed only on several first terms of the asymptotics of the solution rather that on the whole solution. This approach is realized in the case of a multidimensional hyperbolic equation with unknown absolute term. [ABSTRACT FROM AUTHOR]

We construct and justify asymptotics of linearly independent system of solutions to an algebraic-differential system with large rapidly oscillating coefficients. [ABSTRACT FROM AUTHOR]

The problem of time-periodic solutions is considered for a system of partial differential equations with the Stokes operator in the main part with coefficients rapidly oscillating by time in the case of multiple degeneration of the averaged stationary operator. The results concerning the existence and uniqueness of these solutions are proved and their asymptotic expansions are constructed and justified. [ABSTRACT FROM AUTHOR]

Four problems are solved in which a high-frequency source in the one-dimensional heat equation with homogeneous initial-boundary conditions is recovered from partial asymptotics of its solution. It is shown that the source can be completely recovered from an incomplete (two-term) asymptotic representation of the solution. The formulation of each source recovery problem is preceded by constructing and substantiating asymptotics of the solution to the original initial-boundary value problem. [ABSTRACT FROM AUTHOR]

We justify the Krylov-Bogolyubov averaging method for linear and quasilinear parabolic second order initial-boundary value problems with rapidly time-oscillating principal terms. [ABSTRACT FROM AUTHOR]

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 a linear system of partial differential equations with the Stokes operator in the principal part and coefficients that rapidly oscillate in time. In the case where the limiting (averaged) stationary part of the system degenerates, we use the boundary layer method for constructing a complete formal asymptotic expansion of a time-periodic solution. [ABSTRACT FROM AUTHOR]

Asymptotic methods are used to investigate a second-order linear parabolic problem with lower order coefficients rapidly oscillating in time. The coefficient multiplying the principal stationary operator is assumed to be singular, i.e., it has a simple zero eigenvalue. Under certain additional conditions, the problem is proved to have a unique time-periodic solution and its complete asymptotic expansion is constructed and justified. [ABSTRACT FROM AUTHOR]

For a linear second-order parabolic equation with high-frequency coefficients, the complete asymptotics of time-periodic solutions is constructed and justified. The limit averaged stationary problem is assumed degenerate. [ABSTRACT FROM AUTHOR]

A second-order linear parabolic problem with high-frequency terms is considered. The elliptic operator of the corresponding limiting (averaged) problem is assumed to be degenerate. A complete formal asymptotic expansion of a time-periodic solution of the perturbed problem is constructed. [ABSTRACT FROM AUTHOR]

We construct completely justified asymptotics for solutions to some classes of linear ordinary differential equations with slow and fast coefficients in presence of resonance conditions. [ABSTRACT FROM AUTHOR]

For a linear normal system of ordinary differential equations with rapidly oscillating coefficients in a critical case, the existence of a unique periodic solution is proved, its complete asymptotic expansion is constructed and justified, and Lyapunov stability and instability conditions are found. The asymptotic series constructed is shown to converge absolutely and uniformly to the solution. [ABSTRACT FROM AUTHOR]

We consider the second order ordinary differential equations whose coefficients of the unknowns contain smooth and rapidly oscillating summands proportional to the positive powers of the oscillation frequency. The complete asymptotic expansions are constructed and justified of solutions to the Cauchy problem. [ABSTRACT FROM AUTHOR]

Two classes of systems of linear ordinary differential equations containing high-frequency terms proportional to certain positive powers of the oscillation frequency are considered. Complete asymptotic expansions of fundamental systems of solutions are constructed and substantiated. [ABSTRACT FROM AUTHOR]

A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential equations with rapidly oscillating coefficients, some of which may be proportional to ω n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with rapidly oscillating terms proportional to powers ω d. For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem. [ABSTRACT FROM AUTHOR]

This paper considers systems of ordinary differential equations of the 1st order whose terms oscillate with a high frequency ω, among which there exist those proportional to $$ \sqrt \omega $$, the large high-frequency summands. For the problem on periodic solutions, the author justifies the averaging method and establishes a posteriori bounds for the error of partial sums of the complete asymptotic expansion for the solution. [ABSTRACT FROM AUTHOR]

Systems of first-order semilinear partial differential equations with terms that oscillate at a frequency ω ≫ 1 in a single variable and are proportional to $$ \sqrt \omega $$ are considered. The Krylov-Bogolyubov-Mitropol’skii averaging method is substantiated for such equations. Based on the two-scale expansion method, an algorithm for constructing complete asymptotics of solutions is proposed and justified. [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]

The paper deals with averaging problems for parabolic equations. We prove that exponential dichotomy is preserved without any assumption concerning the almost-periodicity of the coefficients. [ABSTRACT FROM AUTHOR]

Studies on the asymptotic integration of the problem on heat distribution in a thin rod with rapidly varying sources of heat. Demonstration of the uniform asymptotic approximation of the temperature distribution in the initial problem in an arbitrary subdomain of the plane rod and in arbitrary time interval; Identification of the temperature in the one-dimensional rod, which is a function of the longitudinal coordinate and which is independent of the transversal coordinate of the plane rod; Sample problem with elaborated solution in support of the study.

The present paper is a continuation of [1]; therefore, we preserve the notation and continue the numbering of sections, formulas, and theorems. [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.

We obtain a criterion for invertibility in Hölder spaces of linear parabolic operators of arbitrary order with any number of spatial variables and almost-periodic coefficients. [ABSTRACT FROM AUTHOR]

An algorithm for constructing the higher approximations of the averaging method for quasilinear parabolic equations with fast oscillating coefficients is suggested and justified. [ABSTRACT FROM AUTHOR]

We construct the complete asymptotics of a periodic solution of a linear normal system of differential equations with high-frequency coefficients. We study the Lyapunov stability and instability of that solution. More specifically, we consider the critical case in which the matrix coefficient of the formally averaged stationary system has one eigenvector and one generalized (in the Vishik-Lyusternik sense) associated vector. [ABSTRACT FROM AUTHOR]