Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity

Burkhan T Kalimbetov,Olim D Tuychiev

doi:10.1515/math-2021-0021

Abstract

Abstract In this paper, the regularization method of S. A. Lomov is generalized to integro-differential equations with rapidly oscillating coefficients and with a rapidly oscillating right-hand side. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution of this problem. The case of coincidence of the frequencies of a rapidly oscillating coefficient and a rapidly oscillating inhomogeneity is considered. In this case, only the identical resonance is observed in the problem. Other cases of the relationship between frequencies can lead to so-called non-identical resonances, the study of which is nontrivial and requires the development of a new approach. It is supposed to study these cases in our further work.

Highlights

Perturbed integro-differential equations have been the subject of research for many decades, starting with the work of A
Lomov [6–8], integro-differential equations were considered under the conditions of the spectrum of the matrix of the first variation lying in the open left half-plane, which significantly narrowed the scope of the above work in problems with purely imaginary points of the spectrum
We show that the class Mε = U∣τ=ψ(x)∕ε is asymptotically invariant with respect to the operator J

Summary

Introduction

Perturbed integro-differential equations have been the subject of research for many decades, starting with the work of A. A class Mε is said to be asymptotically invariant (with ε → +0) with respect to an operator P0 if the following conditions are fulfilled: (1) Mε ⊂ D(P0) for each fixed ε > 0; (2) the image P0μ(x, t, ε) of any element μ(x, t, ε) ∈ Mε decomposes in a power series. N=0 convergent asymptotically for ε → +0 (uniformly in (x, t) ∈ [x0, X] × [0, T]) From this definition, it can be seen that the class Mε depends on the space U , in which the operator P0 is defined. The operator Jis formally defined, its utility is obvious; since in practice, it is usual to construct the N th approximation of the asymptotic solution of problem (2), which impose only N th partial sums of the series (6), which have not a formal, but a true meaning.

Iterative problems and their solvability in the space U

Construction of the solution of the first iteration problem

Conclusion