danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"dan-481Research ArticleТЕХНИЧЕСКИЕ НАУКИTECHNICAL SCIENCESВЫСОКОТОЧНЫЕ ПОЛИНОМИАЛЬНЫЕ РЕШЕНИЯ КЛАССИЧЕСКОЙ ЗАДАЧИ СТЕФАНАHIGH-ACCURACY POLYNOMIAL SOLUTIONS OF THE CLASSICAL STEFAN PROBLEMКотВ. А.KotValery A.

канд. техн. наук, ст. науч. сотрудник

ул. П. Бровки, 15, 220072

Ph. D. (Engineering), Senior researcher

15, P. Brovka Str., 220072

valery.kot@hmti.ac.byИнститут тепло- и массообмена имени А. В. Лыкова Национальной академии наук Беларуси, МинскA. V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus, Minsk201721012018616112122Copyright © Кот В.А., 20182018Кот 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/481

Задача Стефана, под которой понимают класс математических моделей, описывающих в основном тепловые и диффузионные процессы с фазовыми превращениями, занимает чрезвычайно важное место во многих физических процессах и технических приложениях. Решение задачи Стефана состоит в вычислении температурного (концентрационного) профиля с определением закона перемещения межфазной границы. Представлены высокоточные полиномиальные решения задачи Стефана для полуограниченного пространства с граничными условиями Дирихле, Неймана, а также общего вида. Начальная температура принималась равной температуре фазового превращения. На основе интегрального метода граничных характеристик, основанного на многократном интегрировании уравнения теплопроводности, получены последовательности из тождественных равенств для разных граничных условий. Далее построены полиномиальные решения. На тестовых примерах продемонстрирована высокая эффективность предложенного подхода. При полиномах второй и третьей степени полученные решения значительно превзошли по точности аппроксимации известные. При полиномах четвертой и пятой степени точность расчета межфазной границы на несколько порядков превзошла точность численных методов. Полученные решения можно условно считать точными, поскольку ошибки расчета межфазной границы и температурного профиля составляют ничтожно малые величины. 

The Stefan problem is of extreme importance in investigating many physical processes and technologies. Solving the Stefan problem reduces to calculating a temperature (concentration) profile when an interphase boundary is to be determined. High-accuracy polynomial solutions of the Stefan problem for a semi-infinite medium with Dirichlet/Neumann boundary conditions and general conditions are presented. An initial medium temperature is assumed to be equal to a phase change temperature. With the use of the integral method of boundary characteristics, based on multiple integration of the heat conduction equation, sequences of identical equalities with different boundary conditions are obtained and, as a result, polynomial solutions are constructed. The high efficiency of the approach proposed is illustrated with various examples. The solutions based on the 2nd and 3rd degree polynomials are more exact in comparison to the known solutions. The accuracy of calculating the position of the interphase boundary by means of 4th and 5th degree polynomials is several orders of magnitude higher than that of numerical methods. The solutions obtained can be considered as conditionally exact because of negligibly small errors in determining the interphase boundary and the temperature profile.

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