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Herald of Kazakh-British technical university

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1998-6688
2959-8109
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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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    NUMERICAL METHOD FOR SOLVING THE BOUNDARY VALUE PROBLEM FOR THE PARABOLIC EQUATION
    (2024-10-01)
    Narkesh B. Iskakova
    ;
    Zh. М. Каdirbayeva
    ;
    E. А. Bakirova
    ;
    Saule Kuanysh
    A linear boundary value problem for a parabolic equation is considered in a closed domain. Based on the broken line method, the boundary value problem for a parabolic equation is replaced by a two-point boundary value problem for a system of linear ordinary differential equations by discretizing the unknown function u(t,x) with respect to the variable x. The obtained two-point boundary value problem is investigated by the parameterization method of Professor Dzhumabaev. Based on this method, an algorithm for finding a numerical solution to the two-point boundary value problem for a system of linear ordinary differential equations is constructed. The constructed algorithm is realized by applying known numerical methods. The constructiveness and efficiency of the parameterization method also allows us to construct a numerical solution of the considered linear boundary value problem for the parabolic equation. One numerical example is given to verify and illustrate the proposed algorithm.
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    ON THE UNIQUE SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR DIFFERENTIAL EQUATIONS WITH PARAMETER
    (2024-03-25)
    E. А. Bakirova
    ;
    Narkesh B. Iskakova
    ;
    Svetlana Temesheva
    ;
    Zh. М. Каdirbayeva
    A linear boundary value problem for a differential equation with a parameter is investigated on a finite interval by the parameterization method. The studied boundary value problem with parameter is reduced to an equivalent multipoint boundary value problem with parameters by splitting the interval, introducing additional parameters at the points of splitting and new functions. The obtained equivalent boundary value problem contains Cauchy problems for ordinary differential equations with respect to new functions. By substituting the solution representation of the Cauchy problem into the boundary conditions and continuity conditions of the solution, a system of linear algebraic equations with respect to the introduced parameters is compiled. An algorithm for finding a solution to the boundary value problem with parameters is constructed. The formulation of the theorem on sufficient conditions of unique solvability of the boundary value problem with parameters is given. Sufficient conditions of its unique solvability are obtained in terms of the data of the original boundary value problem. An example showing the fulfillment of the conditions of the theorem is given.
    Цитирования (OpenAlex) 1Цитирования (OpenCitations) 1
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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.