kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2025-22-2-242-259kaz29-2005Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESОПТИМИЗАЦИОННЫЙ АЛГОРИТМ ДЛЯ ЧИСЛЕННОЙ РЕАЛИЗАЦИИ ДРОБНОЙ ПРОИЗВОДНОЙ ГРЮНВАЛЬДА-ЛЕТНИКОВА НА ОСНОВЕ ПРИНЦИПА МЕМОИЗАЦИИ ДЛЯ ОБЫКНОВЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙOPTIMIZATION ALGORITHM FOR NUMERICAL IMPLEMENTATION OF THE FRACTIONAL GRUNWALD-LETNIKOV DERIVATIVE BASED ON THE MEMORIZATION PRINCIPLE FOR ORDINARY DIFFERENTIAL EQUATIONShttps://orcid.org/0000-0002-1937-8615ИсаховА. А.IssakhovA. A.

 профессор 

 г. Алматы 

Professor 

Almaty 

alibek.issakhov@gmail.comhttps://orcid.org/0000-0002-5967-6959АбылкасымоваА. Б.AbylkassymovaA. B.

 ассоциированный профессор 

 г. Алматы 

Associate Professor 

Almaty 

abylkassymova.aizhan@gmail.comhttps://orcid.org/0009-0002-6295-1592ЮнС.YunS.

 бакалавр 

 г. Алматы 

Bachelor student 

Almaty 

s_yun@kbtu.kzhttps://orcid.org/0009-0007-6323-0689ЖайлыбаевР.ZhailybaevR.

бакалавр 

 г. Алматы 

Bachelor student

Almaty 

r_zhailybaev@kbtu.kzКазахстанско-Британский технический университетКазахстанKazakh-British Technical UniversityKazakhstan202506072025222242259Copyright © Исахов А.А., Абылкасымова А.Б., Юн С., Жайлыбаев Р., 20252025Исахов А.А., Абылкасымова А.Б., Юн С., Жайлыбаев Р.Issakhov A.A., Abylkassymova A.B., Yun S., Zhailybaev R.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/2005

Дробные производные благодаря своей нелокальности могут описывать сложные процессы, где исторические данные важны для будущих вычислений. Одновременно данное свойство приносит трудности при численных моделированиях. Эта статья представляет новый дискретный оператор для аппроксимации дробной производной на основе определения Грюнвальда-Летникова, «принципа короткой памяти», мемоизации и аналитических предположений. Данный оператор существенно уменьшает количество операций в процессе вычислений, при решении краевых задач, за счет сохранения вычисляемых данных и преобразования для дальнейшего использования с регулируемой точностью.

Fractional derivatives, due to their nonlocality, can describe complex processes where historical data are important for future calculations. At the same time, this property brings difficulties in numerical simulations. This paper presents a new discrete operator for approximating the fractional derivative based on the Grunwald-Letnikov definition, the "principle of short memory", memorization and analytical assumptions. This operator significantly reduces the number of operations in the process of calculations, when solving boundary value problems, due to the storage of calculated data and transformation for further use with adjustable accuracy. 

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