Asymptotics Solutions of a Singularly Perturbed Integro-Differential Fractional Order Derivative Equation with Rapidly Oscillating In-homogeneity

Abstract

The main objective of the present article is to identify the influence of an exponentially oscillating heterogeneity and an integral operator on the structure of the asymptotic of the solution of the initial value problem for a linear singularly perturbed integro-differential equation with a fractional derivative and a rapidly oscillating heterogeneity. To construct an asymptotic solution to the problem, the algorithm of the regularization method used. The case of absence of resonance is considered, i.e. the case when the frequency of exponentially oscillating heterogeneity does not coincide with the spectrum of the limit operator of the differential part of the equation in the considered time interval. It is shown that both the rapidly oscillating heterogeneity and the kernel of the integral operator have a significant effect on the leading term of the asymptotic of the solution of the original problem.