Untitled

Generalization of the Regularization Method to Singularly Perturbed Integro-Differential Systems of Equations with Rapidly Oscillating Inhomogeneity

“…It is easy to see that these identities coincide with the identities (9). Thus, the conditions (9) are necessary and sufficient for the solvability of the system (8) in the space U. The theorem is proved.…”

“…Let conditions (1), (2a) hold and the right-hand side H(t, τ) = H 0 (t) + ∑ n+1 j=1 H j (t)e τ j of the system (8) belongs to the space U. Then, for the solvability of system (8) in U it is necessary and sufficient that the identities…”

“…Let conditions (1), (2a) be satisfied, the right-hand side H(t, τ) of the system (8) belongs to the space U and satisfies the orthogonality condition (9). Then system (8) under additional conditions (13a) is uniquely solvable in U.…”

“…, and hence, we define the solution (12) of the system (8) in the space in a unique way. The theorem is proved.…”

“…The presence of such forces creates serious problems for the numerical integration of the corresponding differential equations. Therefore, asymptotic methods are usually applied to such equations, the most famous of which are the Feshchenko-Shkil-Nikolenko splitting method [1,2] and the Lomov's regularization method [3][4][5][6][7][8][9][10][11][12][13][14]. The splitting method is especially effective when applied to equations with a rapidly oscillating inhomogeneity, and in the case of an inhomogeneity containing both rapidly and slow components, the Lomov's regularization method turned out to be the most effective.…”