TY - JOUR
AU - Mirzakulova, A. E.
AU - Dauylbaev, M. K.
AU - Akhmet, M. U.
AU - Dzhetpisbaeva, A. K.
PY - 2018/08/27
Y2 - 2024/10/12
TI - The Cauchy problem for singularly perturbed higher-order integro-differential equations
JF - Journal of Mathematics, Mechanics and Computer Science
JA - JMMCS
VL - 97
IS - 1
SE - Mathematics
DO - 10.26577/jmmcs-2018-1-481
UR - https://bm.kaznu.kz/index.php/kaznu/article/view/481
SP - 14-24
AB - <p>The article is devoted to research the Cauchy problem for singularly perturbed higher-order linear<br>integro-differential equation with a small parameter at the highest derivatives, provided that the<br>roots of additional characteristic equation have negative signs. The aim of this paper is to bring<br>asymptotic estimation of the solution of a singularly perturbed Cauchy problem and the asymptotic<br>convergence of the solution of a singularly perturbed initial value problem to the solution of<br>an unperturbed initial value problem. In this paper the fundamental system of solutions, initial<br>functions of a singularly perturbed homogeneous differential equation are constructed and their<br>asymptotic estimates are obtained. By using the initial functions, we obtain an explicit analytical<br>formula of the solution. The theorem about asymptotic estimate of a solution of the initial value<br>problem is proved. The unperturbed Cauchy problem is constructed. We find the solution of the<br>unperturbed Cauchy problem. An estimate difference of the solution of a singularly perturbed<br>and unperturbed initial value problems. The asymptotic convergence of solution of a singularly<br>perturbed initial value problem to the solution of the unperturbed initial value problem is proved</p>
ER -