kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2025-22-2-279-289kaz29-2008Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESЧИСЛЕННОЕ РЕШЕНИЕ УРАВНЕНИЯ КОНВЕКЦИИ-ДИФФУЗИИ ДРОБНОГО ПОРЯДКА ДЛЯ ПРОГНОЗА ЗАГРЯЗНЕНИЯ ВОЗДУХАNUMERICAL SOLUTION OF A FRACTIONAL CONVECTION-DIFFUSION EQUATION FOR AIR POLLUTION PREDICTIONhttps://orcid.org/0009-0004-7847-1926БакишевА. К.BakishevA. K.

докторант 

г. Усть-Каменогорск 

PhD student 

Ust-Kamenogorsk

b.aibek86@mail.ruhttps://orcid.org/0000-0002-9890-0589МадияровМ. Н.MadiyarovM. N.

ассоц. профессор, к.т.н. 

г. Усть-Каменогорск 

 Cand. Tech. Sci., Associate Professor 

Ust-Kamenogorsk

madiyarov_mur@mail.ruhttps://orcid.org/0000-0002-1078-0480АлимбековаН. Б.AlimbekovaN. B.

ассоц. профессор, PhD 

г. Усть-Каменогорск 

PhD, Associate Professor 

Ust-Kamenogorsk

 

nalimbekova@vku.edu.kzhttps://orcid.org/0000-0003-4396-9914БайгереевД. Р.BaigereyevD. R.

ассоц. профессор, PhD 

г. Усть-Каменогорск 

PhD, Associate Professor

Ust-Kamenogorsk

dbaigereyev@gmail.comhttps://orcid.org/0000-0002-4812-4104БайшемировЖ. Б.BaishemirovZ. D.

ассоц. профессор, PhD 

г. Алматы 

PhD, Associate Professor

Almaty

zbai.kz@gmail.comВосточно-Казахстанский университет имени Сарсена АманжоловаКазахстанSarsen Amanzholov East Kazakhstan UniversityKazakhstanКазахстанско-Британский технический университетКазахстанKazakh-British Technical UniversityKazakhstan202506072025222279289Copyright © Бакишев А.К., Мадияров М.Н., Алимбекова Н.Б., Байгереев Д.Р., Байшемиров Ж.Б., 20252025Бакишев А.К., Мадияров М.Н., Алимбекова Н.Б., Байгереев Д.Р., Байшемиров Ж.Б.Bakishev A.K., Madiyarov M.N., Alimbekova N.B., Baigereyev D.R., Baishemirov Z.D.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/2008

В работе представлен численный метод решения уравнения конвекции-диффузии с производными дробного порядка в смысле Капуто для моделирования загрязнения воздуха в городской среде. Разработанная конечно-элементная схема учитывает эффекты памяти, обеспечивая более точное описание переноса загрязняющих веществ по сравнению с классическими моделями. Теоретически доказаны устойчивость и сходимость метода, что подтверждено численными экспериментами. Модель эффективно определяет зоны накопления загрязнений и позволяет прогнозировать качество воздуха при различных метеоусловиях. Результаты исследования имеют практическое значение для совершенствования систем экологического мониторинга и планирования мер по снижению загрязненности воздуха.

This paper presents a numerical method for solving the convection-diffusion equation with a fractional-order Caputo derivative to model air pollution in urban environments. The developed finite element scheme accounts for memory effects, offering a more accurate representation of pollutant transport compared to classical models. Stability and convergence of the method are theoretically proven and supported by numerical experiments. The model effectively identifies pollutant accumulation zones and can forecast air quality under various weather conditions. The results have practical value for improving environmental monitoring systems and planning measures to reduce pollution levels.

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