Zaluzhna H. Mathematical modeling of unsteady heat transfer in a heterogeneous environment using functions interlination

The object of research is the process of heat propagation in two-dimensional bodies. The purpose of study is research and development of the finite elements method for solution of non-stationary heat conduction problem with two space variables in the areas of complex shapes using functions spline interlination. The methods of research are common methods of functional analysis, computational mathematics, theory of approximation of functions of several variables using the interlination apparatus; the numerical implementation is based on the method of finite elements approximation, built on the basis of the spline interlination of two spatial variables, the solution of the nonstationary problem of the temperature distribution in the areas of complex shape. Theoretical and practical results consist of the following. The developed in the work efficient computational schemes of interlination finite elements method for solving non-stationary heat conduction problems for flat areas of complex shape allow to solve the Cauchy problem for a smaller number of differential equations when the same accuracy as in the classical finite element method. Scientific novelty consists in the following. In the thesis the new method developed on the base of functions spline interlination, which is a finite elements method implementation of reduction to the system of integral-differential equations, which is applied in solving the boundary value problems for non-stationary heat equation with two space variables in the case of areas of complex geometric shapes; this method is called interlination finite elements method (IFEM) for non-stationary heat conduction problems. To test the properties of IFEM a method developed for constructing of exact solutions of non-stationary heat conduction problems for areas of complex shape. When numerical implementation for solving of non-stationary heat and mass transfer problems using IMKE proposed to use a special numbering of elements nodes, which allows in the system of ordinary differential equations to preserve the structure of a block-tridiagonal matrices. To test the properties of IFEM a method developed for constructing of exact solutions of non-stationary heat conduction problems for areas of complex shape. When numerical implementation for solving of non-stationary heat and mass transfer problems using IMKE proposed to use a special numbering of elements nodes, which allows in the system of ordinary differential equations to preserve the structure of a block-tridiagonal matrices. The results of the thesis have been partly used within the state budget theme, which is part of a plan of research work of the Department of Higher and Applied Mathematics of the Ukrainian Engineering and Pedagogical Academy, namely, "Research and development of a new method of exploration and development of mineral deposits on the basis of functions interlination" (State budget topic DR 0109U008661, 2012-2014), and introduced into the educational process of the Energy Department of the Ukrainian Engineering and Pedagogical Academy (Act of 04/22/2015). The fields of application are mechanical engineering, power engineering.