Mathematical Models for the Problem of Recovery of the Heat Source Distribution Function

• Date: 2021
• Main Authors: Верлань, Анатолій, Федорчук, Володимир
• Format: Article
• Language: Ukrainian
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2021
• Online Access: http://mcm-math.kpnu.edu.ua/article/view/251137
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

The article is devoted to the problem of obtaining mathematical models in integral form for thermal objects from the initial equation of thermal conductivity, given in differential form. The case of the inverse incorrect problem for the thermal conductivity equation is considered. When solving both direct and inverse dynamics problems using computational methods, it is important to choose the form of mathematical description of the model. Even models derived from the original models as a result of equivalent transformations in the numerical implementation give non-equivalent solutions. Therefore, it is recommended to use integral mathematical models that have high computational stability to solve inverse dynamics problems. In the integral formulation, such incorrect inverse problems are successfully solved using regularization methods. The article considers two variants of the inverse problem. In the first case, the inverse problem is considered as Dirichlet's task, and in the second case, as Neumann's problem is considered. In both cases, the inverse problems presented in differential form, by equivalent transformations are given in the form of integral equations of the first kind. For the obtained integral models it is shown that the solutions of the equations are unique. The advantage of the obtained integrated models is their relative simplicity and a wide range of developed methods of their numerical implementation based on the use of different quadrature formulas. In addition, the kernels of the obtained integral equations can be physically interpreted as impulse transition characteristics of the heat-conducting medium. This makes it possible to identify them by the transient characteristics of the heat-conducting medium, which can be obtained experimentally.