kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2024-21-3-165-175kaz29-1379Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESЧИСЛЕННЫЙ МЕТОД РЕШЕНИЯ КРАЕВОЙ ЗАДАЧИ ДЛЯ ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯNUMERICAL METHOD FOR SOLVING THE BOUNDARY VALUE PROBLEM FOR THE PARABOLIC EQUATIONhttps://orcid.org/0000-0002-0680-4099ИскаковаН. Б.IskakovaN. B.

к.ф.-м.н., ведущий научный сотрудник 

050010, г. Алматы;050040, г. Алматы

Candidate of Phys.-Math.Sc., Leading Researcher 

050010, Almaty;050040, Almaty

 

narkesh77@gmail.comhttps://orcid.org/0000-0001-8861-4100КадирбаеваЖ. М.KadirbayevaZh. М.

к.ф.-м.н., ведущий научный сотрудник 

050010, г. Алматы;050000, г. Алматы;050000, г. Алматы

Candidate of Phys.-Math.Sc., Leading Researcher 

050010, Almaty;050000, Almaty;050000, Almaty

zhkadirbayeva@gmail.comhttps://orcid.org/0000-0002-3820-5373БакироваЭ. А.BakirovaE. А.

к.ф.-м.н., ведущий научный сотрудник 

050010, г. Алматы;050000, г. Алматы;

Candidate of Phys.-Math.Sc., Leading Researcher 

050010, Almaty;050000, Almaty;

bakirova1974@mail.ruhttps://orcid.org/0009-0006-0975-1662ҚуанышС. Қ.KuanyshS. K.

докторант, преподаватель

050040, г. Алматы

Doctoral student, Teacher 

050040, Almaty

saule270898@gmail.comИнститут математики и математического моделирования; Казахский национальный университет имени аль-ФарабиКазахстанInstitute of Mathematics and Mathematical Modeling; Al-Farabi Kazakh National UniversityKazakhstanИнститут математики и математического моделирования; Казахский национальный женский педагогический университет; Международный университет информационных технологийКазахстанInstitute of Mathematics and Mathematical Modeling; Kazakh National Women's Teacher Training University; International Information Technology UniversityKazakhstanИнститут математики и математического моделирования; Казахский национальный женский педагогический университет;КазахстанInstitute of Mathematics and Mathematical Modeling; Kazakh National Women's Teacher Training University;KazakhstanКазахский национальный университет имени аль-ФарабиКазахстанAl-Farabi Kazakh National UniversityKazakhstan202401102024213165175Copyright © Искакова Н.Б., Кадирбаева Ж.М., Бакирова Э.А., Қуаныш С.Қ., 20242024Искакова Н.Б., Кадирбаева Ж.М., Бакирова Э.А., Қуаныш С.Қ.Iskakova N.B., Kadirbayeva Z.М., Bakirova E.А., Kuanysh S.K.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/1379

В замкнутой области рассматривается линейная краевая задача для параболического уравнения. На основе метода ломаных краевая задача для параболического уравнения заменяется двухточечной краевой задачей для системы линейных обыкновенных дифференциальных уравнений путем дискретизации неизвестной функции u(t,x) по переменной x. Полученная двухточечная краевая задача исследуется методом параметризации профессора Джумабаева. На основе данного метода строится алгоритм нахождения численного решения для двухточечной краевой задачи для системы линейных обыкновенных дифференциальных уравнений. Построенный алгоритм реализуется путем применения известных численных методов. Конструктивность и эффективность метода параметризации позволяет также построить численное решение рассматриваемой линейной краевой задачи для параболического уравнения. Для проверки и иллюстрации предложенного алгоритма приводится один численный пример.

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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The authors declare that there are no conflicts of interest present.