Asymptotic Solution of a Singularly Perturbed Integro-Differential Equation with Exponential Inhomogeneity

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.