Інтегральні моделі нестаціонарних задач теплопровідності на основі методу теплових потенціалів

• Дата: 2019
• Автори: Верлань, Анатолій Федорович, Федорчук, Володимир Анатолійович, Іванюк, Віталій Анатолійович
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Kamianets-Podilskyi National Ivan Ohiienko University 2019
• Онлайн доступ: http://mcm-tech.kpnu.edu.ua/article/view/173695
• Назва журналу: Mathematical and computer modelling. Series: Technical sciences

The article discusses the approach to the construction of integral models of non-stationary problems of heat conduction based on the application of the method of thermal potentials. The possibility of constructing integral models is considered on specific examples using different thermal potentials: a one-dimensional heat conduction problem with different formulation of a boundary value problem (conditions of the first and second kind), the two-dimensional problem of heat exchange, the problem of heat exchange with a moving boundary. It is proposed to use a combination of exact and numerical methods, which allows to take into account the advantages of various approaches. The application of the method of thermal potentials to models in the form of partial differential equations allowed us to obtain a general solution in the form of the Volterra operator, which depends on the functions that are determined from the boundary conditions. That is, the task is reduced to solving the Volterra integral equations of the second kind or their systems. A feature of the models obtained is that the cores of integral models are singular at the end point of integration. It is proposed to solve such equations using computational methods that are based on the quadrature method. To avoid features in the kernel, the offset method is used. Taking into account the properties of the core, it is proposed to apply the method of left rectangles, which will avoid the singularity. To improve the accuracy of building a solution, it is proposed to apply the adaptive algorithm for compaction of simulation step in the vicinity of a singular point. The proposed approach to solving non-stationary problems of heat conduction takes into account the advantages of exact (thermal potential method) and computational methods (quadrature method) and allows to increase the efficiency of calculations based on the parallelization of the problem.