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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]

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]

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]

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]

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]

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]