Asymptotic integration of integridifferential equations with two independent variables

In this paper, the method of regularization of S.A. Lomov is generalized to integro-differential equations of Volterra type with multiple integral operator.We consider the case when the operator of multiplication of the differential part depends only on the differentiation variable. In this case, in contrast to the works of M.I. Imanaliev, a regularized asymptotic solution of any order (with respect to a parameter) is constructed. In addition, we consider and solve the so-called initialization problem. The formulation of this problem is as follows. It is necessary to choose a class of given data (say, $\Sigma$) so that the passage to the limit of an exact solution to a certain limiting regime (when the small parameter tends to zero) holds true on the entire set of changes of independent variables, including the boundary layer zone.