kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-3-106-123kaz29-3179Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESАНАЛИТИЧЕСКОЕ И ПРИБЛИЖЕННОЕ РЕШЕНИЕ ДВУМЕРНОГО УРАВНЕНИЯ ЛАПЛАСА В ЦИЛИНДРИЧЕСКИХ КООРДИНАТАХANALYTICAL AND APPROXIMATE SOLUTIONS OF THE TWO-DIMENSIONAL LAPLACE EQUATION IN CYLINDRICAL COORDINATEShttps://orcid.org/0000-0002-5596-0936СатыбалдинаА. Н.SatybaldinaA. N.

Магистр технических наук, Сениор лектор

Астана, Алматы

Master of Technical Science, Senior lecturer

Astana, Almaty

aigul1191@gmail.comhttps://orcid.org/0000-0002-4284-9446ЫдырысА. Ж.YdyrysA. Zh.

PhD по специальности Математика, Ассоциированный профессор

Алматы

PhD in Mathematics, Associate professor

Almaty

a.ydyrys@iitu.edu.kzЕвразийский Национальный Университет имени Л.Н.Гумилева; Казахстанско-Британский Технический УниверситетРоссияL.N. Gumilyov Eurasian National University; Kazakh-British Technical UniversityRussian FederationМеждународный Университет Информационных ТехнологийРоссияInternational Information Technology UniversityRussian Federation202625092026233106123Copyright © Сатыбалдина А.Н., Ыдырыс А.Ж., 20262026Сатыбалдина А.Н., Ыдырыс А.Ж.Satybaldina A.N., Ydyrys A.Z.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/3179

В данной работе исследуется стационарная задача теплопроводности в многослойной цилиндри­ческой области с неоднородными граничными условиями. Рассмотрены два подхода к решению пря­мой задачи: аналитический метод, основанный на разделении переменных и разложении в ряд Фурье, и численный метод, основанный на конечно-разностной аппроксимации в сочетании с алгоритмом последовательной верхней релаксации (SOR). Получено точное аналитическое решение уравнения Лапласа в полярных координатах для трёхслойной цилиндрической системы с условиями сопряже­ния на границах раздела слоёв и конвективным теплообменом на внешней границе. Для численного решения разработана конечно-разностная схема, а возникающая система линейных алгебраических уравнений решается итерационным методом SOR. Точность численного подхода оценена путём срав­нения с аналитическим решением и экспериментальными измерениями температуры. Полученные результаты демонстрируют отличное согласование аналитического и численного решений. Средняя абсолютная ошибка между двумя подходами составляет 0.158K, среднеквадратическая ошибка — 0.167K, а максимальная абсолютная ошибка не превышает 0.241K. Экспериментальная валидация показала, что обе модели корректно воспроизводят общее распределение температуры в многослой­ной цилиндрической системе. Средняя абсолютная ошибка относительно экспериментальных дан­ных составляет 2.872K для аналитического решения и 3.024K для решения, полученного методом SOR. Проведённый анализ подтверждает высокую точность и устойчивость разработанного конечно­разностного алгоритма и демонстрирует его применимость для моделирования процессов теплопереноса в многослойных цилиндрических конструкциях. Предложенный подход может быть использо­ван при тепловом анализе трубопроводов, геотермальных систем, теплоизоляционных конструкций и других инженерных объектов, содержащих многослойные среды.

This paper investigates a steady-state heat conduction problem in a multilayer cylindrical domain with nonhomogeneous boundary conditions. Two approaches for solving the direct problem are considered: an analytical method based on separation of variables and Fourier series expansion, and a numerical method based on the finite difference approximation combined with the Successive Over-Relaxation (SOR) algorithm. An exact analytical solution of the Laplace equation in polar coordinates is derived for a threelayer cylindrical system with continuity conditions at layer interfaces and convective heat exchange at the outer boundary. A finite d ifference sc heme is de veloped to ap proximate the go verning eq uation, wh ile the resulting system of linear algebraic equations is solved iteratively using the SOR method.The accuracy of the numerical approach is evaluated through comparison with the analytical solution and experimental temperature measurements. The obtained results demonstrate good agreement between the analytical and numerical solutions. The mean absolute error between the two approaches is 0.134K, the root mean square error is 0.141K, and the maximum absolute error does not exceed 0.209K. Experimental validation shows that both models correctly reproduce the overall temperature distribution in the multilayer cylindrical system. The mean absolute errors relative to experimental data are 2.509K for the analytical solution and 2.638K for the SOR solution. The performed analysis confirms the good a ccuracy and s tability of the developed finite difference algorithm and demonstrates its applicability for modeling heat transfer processes in multilayer cylindrical structures. The proposed approach can be employed in the thermal analysis of pipelines, geothermal systems, insulation structures, and other engineering applications involving layered media.

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