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Article · Herald of Kazakh-British technical university · March 29, 2026

ASYMPTOTIC CONVERGENCE OF THE CAUCHY PROBLEM SOLUTION FOR AN INTEGRO–DIFFERENTIAL EQUATION WITH A SMALL PARAMETER

N.U. Bukanay, A. E. Mirzakulova, A. T. Assanova

Aziza Mirzakulova
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Abstract

The Cauchy problem at point 0, where the characteristic equation built according to the homogeneous part of the linear integro-differential equation under consideration has both positive and negative roots, and the order of the derivative of the integral uo to 2, has never been examined before in the theory of equations with small parameters. It is well known that the problem’s solution may veer towards infinity when the characteristic equation’s roots are opposite. The Cauchy problem for a differential equation with a small parameter in front of two higherorder derivatives is still uncertain when the roots of the characteristic equation are opposite. This can be solved analytically by adding the integral part to the right-hand side of the differential equation and treating it as an integrodifferential equation. In this paper, an unperturbed problem is constructed for a given perturbed problem with a small parameter. In the unperturbed problem the external differential operator is one order lower than the internal differential operator. This is a non-standard case and requires special consideration. In this regard, an analytical formula for the solution of the unperturbed problem is obtained, and further analysis is carried out. Moreover, the interrelation between the perturbed and unperturbed problems was established and illustrated by an example. The theorem on the limiting transition was also formulated.

Differential Equations and Numerical MethodsDifferential Equations and Boundary ProblemsNonlinear Differential Equations AnalysisCauchy problemintegro-differential equationssmall parameter

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