danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2026-70-5-406-414dan-1334Research ArticleТЕХНИЧЕСКИЕ НАУКИTECHNICAL SCIENCESПредельное гомотопическое разложение. Задача радиационного теплообменаLimit homotopy decomposition. Radiative heat transfer problemКотВ. А.KotV. A.

Кот Валерий Андреевич – канд. техн. наук, ст. науч. сотрудник

ул. П. Бровки, 15, 220072, Минск 

Kot Valery A. – Ph. D. (Engineering), Senior Researcher 

15, P. Brovka Str., 220072, Minsk 

valery.kot@hmti.ac.byИнститут тепло- и массообмена имени А. В. Лыкова Национальной академии наук БеларусиБеларусьA. V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of BelarusBelarus202613112026705406414Copyright © Кот В.А., 20262026Кот В.А.Kot V.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://doklady.belnauka.by/jour/article/view/1334

Предложен новый подход в гомотопии, который нашел свое выражение в методе предельного гомотопического разложения (LHDM). В данном методе игнорируется одно из основных свойств известных гомотопических построений, согласно которому полностью сохраняется исходное нелинейное дифференциальное уравнение при параметре p = 1. В предложенном новом подходе при p = 1 происходит соответствующее нарушение исходного вида нелинейного дифференциального уравнений с коэффициентом перед нелинейной частью дифференциального уравнения (параметр согласования), стремящимся в пределе порядка приближения (M → ∞) к единице. Такой подход позволяет находить приближенные решения краевых задач, описываемых нелинейными дифференциальными уравнениями, в более простой аналитической форме. На основе LHDM получено приближенное решение задачи радиационного теплообмена тонкого прямоугольного ребра. Показано, что метод LHDM дает более компактные и простые по форме аналитические решения по сравнению с соответствующими решениями основным вариантом оптимального гомотопического асимптотического метода (OHAM), причем с той же точностью. При этом значение параметра согласования   C M/N с каждым новым приближением асимптотически быстро стремится к своему предельному единичному значению. Это позволяет считать предложенный метод LHDM хорошей альтернативой основному варианту OHAM.

A new homotopy-based approach is proposed, embodied in the Limit Homotopy Decomposition Method (LHDM). This method departs from a fundamental property of standard homotopy constructions – namely, the requirement that the original nonlinear differential equation be fully preserved at the parameter value p = 1. In the proposed approach, the original form of the nonlinear differential equation is modified at p = 1 by introducing a coefficient (a matching parameter) multiplying the nonlinear term; this coefficient approaches unity as the order of approximation increases (M → ∞). This approach enables the derivation of approximate analytical solutions for boundary-value problems governed by nonlinear differential equations in a simpler form. Using LHDM, an approximate solution was obtained for the problem of radiative heat transfer in a thin rectangular fin. It is demonstrated that LHDM yields analytical solutions that are more compact and simpler in form than those obtained via the standard version of the Optimal Homotopy Analysis Method (OHAM), while maintaining comparable accuracy. Furthermore, the matching parameter rapidly C M/N approaches its limiting value of unity with each successive approximation. These results suggest that the proposed LHDM is a viable alternative to the standard OHAM.

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