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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., Author(s): E. V. Korablina [sup.1], V. B. Levenshtam [sup.1] [sup.2] Author Affiliations: (1) https://ror.org/01tv9ph92, grid.182798.d, 0000 0001 2172 8170, Southern Federal University, , Milchakov Street, 8A, 344058, Rostov-on-Don, Russia (2) [...]

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

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.