In this paper, we consider a system with rapidly oscillating coefficients, which includes an integral operator with an exponentially varying kernel. The main goal of the work is to develop an algorithm for the regularization method for such systems and to identify the influence of the integral term on the asymptotic behavior of the solution of the original problem.

In this paper, we consider an initial problem for systems of differential equations of fractional order with a small parameter for the derivative. A regularization problem is produced, and an algorithm for normal and unique solubility of general iterative systems of differential equations with partial derivatives is given. In the environment of the computer mathematical system Maple, approximate solutions are calculated, and corresponding solution schedules for various values of the small parameter are constructed.

The integro-differential Cauchy problem with exponential inhomogeneity and with a spectral value that turns zero at an isolated point of the segment of the independent variable is considered. The problem belongs to the class of singularly perturbed equations with an unstable spectrum and has not been considered before in the presence of an integral operator. A particular difficulty is its investigation in the neighborhood of the zero spectral value of inhomogeneity. Here, it is not possible to apply the well-known procedure of Lomov’s regularization method, so the authors have chosen the method of constructing the asymptotic solution of the initial problem based on the use of the regularized asymptotic solution of the fundamental solution of the corresponding homogeneous equation whose construction from the positions of the regularization method has not been considered so far. In the case of an unstable spectrum, it is necessary to take into account its point features. In this case, inhomogeneity plays an essential role. It significantly affects the type of singularities in the solution of the initial problem. The fundamental solution allows us to construct asymptotics regardless of the nature of the inhomogeneity (it can be both slowly changing and rapidly changing, for example, rapidly oscillating). The approach developed in the paper is universal with respect to arbitrary inhomogeneity. The first part of the study develops an algorithm for the regularization method to construct the asymptotic (of any order on the parameter) of the fundamental solution of the corresponding homogeneous integro-differential equation. The second part is devoted to constructing the asymptotics of the solution of the original problem. The main asymptotic term is constructed in detail, and the possibility of constructing its higher terms is pointed out. In the case of a stable spectrum, we can construct regularized asymptotics without using a fundamental solution.

This article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form of a Volterra-type integral. The presence of an integral operator and its type significantly affect the development of an algorithm for asymptotic solutions, in the implementation of which it is necessary to take into account essential singularities generated by the rapidly decreasing kernel of the integral operator. It is shown in tise work that when passing the structure of essentially singular singularities changes from an integral operator of Volterra type to an operator of Fredholm type. If in the case of the Volterra operator they change with a change in the independent variable, then the singularities generated by the kernel of the integral Fredholm-type operators are constant and depend only on a small parameter. All these effects, as well as the effects introduced by the rapidly oscillating inhomogeneity, are necessary to take into account when developing an algorithm for constructing asymptotic solutions to the original problem, which is implemented in this work.