Singular perturbed integral equations with rapidly oscillation coefficients

B. T. Kalimbetov ; V. F. Safonov ; O. D. Tuichiev

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 2068-2083

Résumé

The article considers a singularly perturbed integral equation with a slowly varying kernel and a rapidly oscillating coefficient. The main idea with which the construction of asymptotic solutions of such problems is carried out is the transition (by differentiating the original system with respect to the independent variable) to an equivalent integro-differential equation and the subsequent application of the S.A. Lomov's regularization method. In this paper, we have implemented the case of a singular perturbed integral equation containing (along with a slowly varying kernel and a slowly varying inhomogeneity) a rapidly varying coefficient of an unknown function. Previously, such integral equations were not considered from the standpoint of the regularization method. The presence of a rapidly oscillating coefficient significantly complicates the structure of the solution space for the corresponding iterative problems, which contain (in contrast to problems with slowly varying coefficients) nonlinear exponents of regu-larizing functions. Therefore, the study of the solvability of iterative problems must be carried out in the presence of both nonresonant and resonant spectral relations. All these issues are reflected in this work.