The Cauchy problem for singularly perturbed higher-order integro-differential equations

Abstract

The article is devoted to research the Cauchy problem for singularly perturbed higher-order linear
integro-differential equation with a small parameter at the highest derivatives, provided that the
roots of additional characteristic equation have negative signs. The aim of this paper is to bring
asymptotic estimation of the solution of a singularly perturbed Cauchy problem and the asymptotic
convergence of the solution of a singularly perturbed initial value problem to the solution of
an unperturbed initial value problem. In this paper the fundamental system of solutions, initial
functions of a singularly perturbed homogeneous differential equation are constructed and their
asymptotic estimates are obtained. By using the initial functions, we obtain an explicit analytical
formula of the solution. The theorem about asymptotic estimate of a solution of the initial value
problem is proved. The unperturbed Cauchy problem is constructed. We find the solution of the
unperturbed Cauchy problem. An estimate difference of the solution of a singularly perturbed
and unperturbed initial value problems. The asymptotic convergence of solution of a singularly
perturbed initial value problem to the solution of the unperturbed initial value problem is proved