Regularized Asymptotic Solutions of Singularly Perturbed Integral Equations with Two Independent Variables

Abstract

Lomov’s regularization method is generalized to singularly perturbed integral equations with one-fold and multiple integral operators. We consider the case in which the kernel of the one-fold integral only depends on the time variable and is independent of the spatial variable. In this case, in contrast to Imanaliev’s works, we construct a regularized asymptotic solution of any order (with respect to the parameter). We also study the initialization problem, i.e., the problem of choosing a class of initial data of the problem for which it is possible to pass to the limit in its solution (as the small parameter tends to zero) to some limit operation mode on the whole prescribed set of independent variables, including the boundary layer region.