THEORETICAL STUDIES OF APPLIED PROBLEMS OF MATHEMATICAL MODELING FOR THERMOPHYSICAL SYSTEMS - LEVKIN, DMYTRO; SYNYAVINA, YULIYA; LEVKIN, ARTUR; BUTENKO, TATYANA (Herald of Khmelnytskyi National University. Technical sciences, 2025)
In this article, a computational mathematical model of thermal action on a multilayer material is constructed. It is shown that this thermophysical system is based on a nonlocal boundary value problem of a system of nonlinear, multidimensional, inhomogeneous differential heat conduction equations with partial derivatives, boundary and thermal action conditions. It is noted that without considering the material under study as a homogeneous rectilinear body, without the use of specialized methods and estimates on the solution function from the theory of existence and uniqueness of the solution of boundary value problems, it is generally impossible to guarantee the correctness of this boundary value problem. Using the parametric method, it is proved that the condition of uniform boundedness on the Fourier symbol of the main differential equation of a homogeneous boundary value problem is a necessary and sufficient condition for the computational mathematical model over the space of generalized functions constructed in this article. This guarantees the stability of the solution of the boundary value problem to minor changes in the initial data in the function space. From the point of view of mathematical modeling, the article solves the problem of nonlinear dynamic programming. Taking into account the characteristics of technical devices of thermal action, the boundary values of the control parameters of the system are set. Despite the fact that the research of this article is related to the section of mathematical modeling and optimization of systems with distributed parameters, it also has a multidisciplinary focus. After all, during the finite-difference approximation of a system of differential equations for the implementation of boundary value problems, this problem is reduced to nonlinear dynamic programming, which increases the accuracy of calculating the objective function and values of technical parameters of load sources by using more approximate calculation methods.