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Singularly perturbed linear oscillator with piecewise-constant argument
(Al-Farabi Kazakh National University, 2018-08-27)
Marat U. Akhmet
;
M. K. Dauylbayev
;
A. E. Mirzakulova
;
Н. Атахан
The Cauchy problem for singularly perturbed linear differential equation the second order with piecewise-constant argument is considered in the article. The definition of singularly perturbed linear harmonic oscillator with piecewise-constant argument is given in the paper. The system of fundamental solutions of homogeneous singularly perturbed differential equation with piecewiseconstant argument are constructed according to the nonhomogeneous singularly perturbed differential equation with piecewise-constant argument. With the help of the system of fundamental solutions, the initial functions are constructed and their asymptotic representation are obtained. By using the reduction method, the analytical formula of the solution of singularly perturbed the initial value problem with piecewise-constant argument is obtained. In addition, the unperturbed Cauchy problem is constructed according to the singularly perturbed Cauchy problem. The solution of the unperturbed Cauchy problem is obtained. When the small parameter tends to the zero, the solution of singularly perturbed the Cauchy problem with piecewise-constant argument approaches the solution of the unperturbed Cauchy problem with piecewise-constant argument. The theorem on the passage to the limit is proved.
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ASYMPTOTIC EXPANSION OF SOLUTION OF GENERAL BVP WITH INITIAL JUMPS FOR HIGHER-ORDER SINGULARLY PERTURBED INTEGRO-DIFFERENTIAL EQUATION
(2018-12-15)
M. K. Dauylbayev
;
Н. Атахан
;
A. E. Mirzakulova
In this article we constructed an asymptotic expansion of the solution undivided boundary value problem for singularly perturbed integro-differential equations with an initial jump phenomenon m – th order. We obtain the theorem about estimation of the remainder term’s asymptotic with any degree of accuracy in the smallparameter.
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ESTIMATION OF A BOUNDARY VALUE PROBLEM SOLUTION WITH INITIAL JUMP FOR LINEAR DIFFERENTIAL EQUATION
(2020-03-10)
B. Sharip
;
А.Т. Yessimova
The paper considers a boundary value problem for a singularly perturbed linear differential equation with constant third-order coefficients. In this problem, a small parameter is indicated before the highest derivatives that are part of the differential equation and the boundary condition at t = 0.The fundamental system of solutions of a homogeneous singularly perturbed differential equation is constructed on the basis of asymptotic representations obtained for the roots of the corresponding characteristic equation. This system was used to construct the Cauchy function, special functions of boundary value problems, and also the Green function. With the help of these functions, an analytical formula is obtained for solving a singularly perturbed boundary value problem and it turns out that this solution has an initial zero-order jump at t = 0. It is proved that the solution to the considered singularly perturbed boundary value problem tends to the corresponding unperturbed problem obtained from it under .
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STATIONARY DIFFUSION MODEL IN HETEROGENEOUS MEDIA WITH SMALL PARAMETERS
(2025-06-01)
O.S. Akhmetova
;
S.A. Issayev
This study presents a theoretical and applied investigation of a stationary diffusion problem in heterogeneous media incorporating small parameters and modified boundary conditions. The relevance of this work stems from the necessity to accurately represent boundary layer effects in complex physical systems with variable internal structures, such as multilayered materials, biological tissues, and natural geophysical environments. Traditional models often oversimplify such effects, resulting in distortions in the solution near domain boundaries.The primary objective of the research is to construct a rigorous mathematical framework that accounts for both the presence of small parameters and altered boundary conditions. The analysis includes deriving a priori stability estimates and constructing asymptotic expansions for the solution. The methodology relies on functional formulation in Sobolev spaces, the Galerkin method, asymptotic techniques, and weak convergence theory. Special attention is given to the existence and uniqueness of both strong and generalized solutions, as well as to their convergence properties as the small parameters tend to zero.The results confirm the solvability of the problem, provide robust energy estimates, and establish solution stability under various types of boundary modifications. It is demonstrated that changes in boundary conditions primarily affect solution behavior near the domain edges, while preserving the global structure of the field. The model is applied in various fields, including climate modeling, thermal diffusion, and hydrophysical systems. This work lays a foundation for the development of adaptive numerical schemes and for extending the analysis to non-stationary problems.

Қазақ ұлттық қыздар педагогикалық университеті / Казахский национальный женский педагогический университет

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