Задача инициализации сингулярно возмущенных интегро-дифференциальных и интегральных уравнений Вольтерра второго рода

Abstract

Initialization problems were encountered for the first time in meteorology. The problems of assimilating the observation data in the mathematical models of atmospheric motion emerged in this field. At small Rossby numbers, the solution of atmospheric models depends on two time scales: the “slow” time and the “fast” time t/ε, where ε is the Rossby number. The rapidly oscillating terms in the solution are irrelevant for weather forecasting on large time intervals. Therefore, there is a need to carry out a special procedure of initialization, which would suppress rapidly oscillating or rapidly growing waves of scale expansions. In V.M. Ipatov’s paper on the initialization problems for general atmosphere circulation models, solvability of the initialization problem for a two- layer quasi geostrophic model of general atmospheric circulation was proven. A semi-explicit spectral-difference scheme was constructed. In this article, the initialization problem is considered for singularly perturbed integro-differential and integral systems of Volterra equations of the second kind from the standpoint of S.А. Lomov’s regularization method. The cases of both a purely imaginary spectrum of the limit operator (which characterizes the presence of fast oscillations in the solution) and a spectrum with a positive real part (which corresponds to an exponentially growing term) are considered. The procedure of cancelling these terms or reducing their influence by choosing the initial conditions and separating a class of functions (the right-hand sides of the systems) is shown.