Herald of Kazakh-British technical university

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ON A BOUNDARY VALUE PROBLEM FOR A LOADED FRACTIONAL DIFFERENTIAL EQUATION
(2026-09-25)
Narkesh B. Iskakova
;
E. А. Bakirova
;
K. I. Usmanov
;
N.T. Orumbayeva
The paper investigates a two-point boundary value problem for a loaded differential e quation with a Caputo fractional derivative. The Dzhumabaev parametrization method is applied to study the problem. Within this approach, the initial interval is divided into a finite n umber of s ubintervals, a nd additional parameters and new functions are introduced, which makes it possible to reduce the original boundary value problem to an equivalent multipoint boundary value problem with unknown parameters. The equivalence of the original and transformed problems is established. Based on the obtained integral representations, a system of algebraic equations with respect to the introduced parameters is constructed. Sufficient conditions for the existence and uniqueness of a solution are established and related to the invertibility of the corresponding matrix. An algorithm based on the method of successive approximations is developed to determine the unknown parameters and functions. A numerical example demonstrating the application of the proposed algorithm and confirming its e ffectiveness is presented. The ob tained results de monstrate the efficiency of the pa rametrization me thod for in vestigating bo undary va lue pr oblems for lo aded fractionalorder differential e quations and c onstructing t heir a pproximate solutions.
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ON THE SOLUTION OF A BOUNDARY VALUE PROBLEM FOR AN IMPULSIVE HYPERBOLIC EQUATION WITH A PIECEWISE-CONSTANT ARGUMENT
(2026-06-27)
Altynai Molybaikyzy
;
Symbat Seisenbekovna Kabdrakhova
;
Narkesh B. Iskakova
;
B. B. Мinglibayeva
This article investigates a boundary value problem for an impulsive hyperbolic equation with a piecewise constant argument. Impulsive hyperbolic equations with a piecewise constant argument arise as mathematical models of physical processes in neural networks, continuous dynamical systems, hybrid systems, and other fields. Issues related to the existence of boundary value problems and the construction of their solutions for such equations remain among the most relevant and challenging problems at present. To obtain solvability conditions for the considered problem, the Dzhumabaev parametrization method is employed, and an iterative algorithm for constructing an approximate solution is developed. For each step of the iterative process, integral formulas are derived and expressed in terms of the matrix Q(x), which describes the relationship between the functional parameters. If this matrix is invertible, the existence and uniqueness of the solution are proved both for the parametric and the original problems. The proposed method is not limited to proving theoretical solvability; it also provides a concrete constructive procedure for obtaining the solution. This is of significant importance for subsequent numerical implementations and for the analysis of solution stability. Moreover, the proposed approach can be applied to other types of problems with a piecewise constant argument, including systems with impulsive conditions, neural networks with memory effects, and nonlinear hybrid models.
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ON THE APPROXIMATION OF A BOUNDARY VALUE PROBLEM FOR DELAY INTEGRO-DIFFERENTIAL EQUATIONS
(2025-07-06)
E. А. Bakirova
;
Narkesh B. Iskakova
;
Askarbek E. Imanchiyev
;
Sayakhat G. Karakenova
A linear two-point boundary value problem for a system of integro-differential equations with a constant delay argument is investigated on a finite interval. By dividing the interval by parts, the integral term of the integrodifferential equation with constant delay argument is replaced by the quadrature formula. With this replacement, the linear two-point boundary value problem for a system of integro-differential equations with a constant delay argument is approximated by the linear boundary value problem for a system of loaded differential equations with a constant delay argument. Definitions of correct solvability of boundary value problem for system of integro-differential equations with delay argument and constructed boundary value problem for system of loaded differential equations with constant delay argument are introduced. Conditions of correct solvability of linear boundary value problem for system of integro-differential equations with delay argument and linear boundary value problem for system of loaded differential equations with delay argument are established. Relationship between correct solvabilities of linear two-point boundary value problem for system of integro-differential equations with constant delay argument and approximating linear two-point boundary value problem for system of loaded differential equations with constant delay argument is shown.

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