@article{Mirzakulova_Dauylbaev_Akhmet_Dzhetpisbaeva_2018, title={The Cauchy problem for singularly perturbed higher-order integro-differential equations}, volume={97}, url={https://bm.kaznu.kz/index.php/kaznu/article/view/481}, DOI={10.26577/jmmcs-2018-1-481}, abstractNote={<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>}, number={1}, journal={Journal of Mathematics, Mechanics and Computer Science}, author={Mirzakulova, A. E. and Dauylbaev, M. K. and Akhmet, M. U. and Dzhetpisbaeva, A. K.}, year={2018}, month={Aug.}, pages={14–24} }