kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2025-22-1-184-196kaz29-1744Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESУСЛОВИЕ УСТОЙЧИВОСТИ КОНЕЧНО-РАЗНОСТНОЙ СХЕМЫ ДЛЯ УРАВНЕНИЙ ПАРАБОЛИЧЕСКОГО И ГИПЕРБОЛИЧЕСКОГО ТИПОВ: СРАВНЕНИЕ МЕТОДА КОНЕЧНЫХ ОБЪЕМОВ ДЛЯ УРАВНЕНИЙ ДИФФУЗИИ ДРОБНОГО ПОРЯДКАSTABILITY CONDITION OF FINITE DIFFERENCE SCHEMES FOR PARABOLIC AND HYPERBOLIC EQUATIONS: A COMPARISON WITH FINITE VOLUME METHODS FOR FRACTIONAL-ORDER DIFFUSIONhttps://orcid.org/0000-0002-1937-8615ИсаховА. А.IssakhovA. A.

 PhD, профессор 

 г. Алматы 

 PhD, Professor 

 Almaty 

al.isakhov@kbtu.kzАбылкасымоваА. Б.AbylkassymovaA. B.

 PhD, ассоциированный профессор 

 г. Алматы 

 PhD, Associate Professor 

 Almaty 

https://orcid.org/0009-0007-6323-0689ЖайлыбаевР. Е.ZhailybaevR. E.

 студент 

 г. Алматы 

 undergraduate student 

 Almaty 

raim.zhailybaev@mail.ruhttps://orcid.org/0009-0002-6295-1592ЮнС. Л.YunS. L.

 студент 

 г. Алматы 

 undergraduate student 

 Almaty 

s_yun@kbtu.kzКазахстанско-Британский технический университетКазахстанKazakh-British Technical UniversityKazakhstan202525032025221184196Copyright © Исахов А.А., Абылкасымова А.Б., Жайлыбаев Р.Е., Юн С.Л., 20252025Исахов А.А., Абылкасымова А.Б., Жайлыбаев Р.Е., Юн С.Л.Issakhov A.A., Abylkassymova A.B., Zhailybaev R.E., Yun S.L.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/1744

В данной работе проведен сравнительный анализ методов конечных разностей и объемов. Данные методы широко известны для уравнений диффузии с целым порядком, но тем не менее недостаточно исследована эффективность данных методов для уравнений диффузии с дробным порядком по времени. Для аппроксимации уравнения использовано определение дробной производной Грюнвальда-Летникова. Для метода конечных разностей получена явная разностная схема и выведено условие устойчивости для разностной схемы с дробным порядком по времени, что также является обобщением для параболических и гиперболических типов уравнений, что ранее было неизвестно для схем с дробным порядком по времени. Представлена явная дискретная форма для решения уравнений субдиффузии в двумерном пространстве с дробным порядком по времени методом конечных объемов. Результаты показывают, что метод конечных разностей демонстрирует высокую точность, тогда как метод конечных объемов лучше подходит для сложных геометрических форм. Эти результаты открывают возможности для дальнейшего развития численных методов в задачах, связанных с моделированием процессов аномальной диффузии.

This paper compares the finite difference and finite volume methods for solving time-fractional diffusion equations. These methods are widely known for diffusion equations with integer order, but their effectiveness for time-fractional diffusion equations has not been sufficiently studied. The definition of the Grunwald-Letnikov fractional derivative is used to approximate the equation. An explicit difference scheme for the finite difference method is obtained and a stability condition for the fractional time order difference scheme is derived, which is also a generalisation for parabolic and hyperbolic type equations, which was previously unknown for schemes with a fractional time order. An explicit discrete form for solving subdiffusion equations in two-dimensional space with fractional time order by the finite volume method is presented. Numerical results show that the finite difference method demonstrates high accuracy, while the finite volume method is better suited for complex geometries. These findings provide insights for future developments in anomalous diffusion modeling.

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