This paper considers one-dimensional heat transfer in a media with temperature-dependent thermal conductivity. To model the transient behavior of the system, we solve numerically the one-dimensional unsteady heat conduction equation with certain initial and boundary conditions. Contrary to the traditional approach, when the equation is first discretized in space and then in time, we first discretize the equation in time, whereby a sequence of nonlinear two-point boundary value problems is obtained. To carry out the time-discretization, we use the implicit Euler scheme. The second spatial derivative of the temperature is a nonlinear function of the temperature and the temperature gradient. We derive expressions for the partial derivatives of this nonlinear function. They are needed for the implementation of the Newton method. Then, we apply the finite difference method and solve the obtained nonlinear systems by Newton method. The approach is tested on real physical data for the dependence of the thermal conductivity on temperature in semiconductors. A MATLAB code is presented.
Keyword: boundary value problem
THEORETICAL FOUNDATIONS OF SECURITY
Authors’ original problem-solution-approach concerning aviation security management in civil aviation applying parallel calculation processes’ method and neural computers’ usage is considered in this paper. Problem statement by setting secure environment simulation tasks for grid models, and neural networks’ usage is presented. The research subject area of this paper is airport services in civil aviation, considered from the point of view of aviation security, defined as the state of aviation security against unlawful interference into the aviation field. The key issue in this subject area is aviation safety provision at an acceptable level. In this case, airport security level management becomes one of the main objectives of aviation security. Aviation security management is the organizational regulations in modern systems that can no longer correspond to changing and increasingly complex requirements determined by factors of external and internal environment, associated with a set of potential threats to airport activity. Optimal control requires the most accurate identification of management parameters and their quantitative assessment. The authors examine the possibility of applying mathematical methods for processes and procedures’ security management modeling in their latest works. Parallel computing methods and network neurocomputing for modeling control processes of airport security are examined in this paper. It is shown that the methods’ practical application is most effective in the decision support system, where the decision maker plays a leading role. Decision support system on the aviation safety management should include risk assessment subsystem of adverse events.
THEORETICAL FOUNDATIONS AND SPECIFICITY OF MATHEMATICAL MODELLING
LAPLACIAN PRESERVING TRANSFORMATION OF SURFACES AND APPLICATION TO BOUNDARY VALUE PROBLEMS FOR LAPLACE’S AND POISSON’S EQUATIONS
This paper shows that the constrained similarity transformation of surfaces under boundary constraints is a Laplacian preserving transformation. First, a general proof is presented and then the result is verified for mesh-functions through particular examples. The fact that the constrained similarity transformation, subject to boundary constraints, is a Laplacian preserving transformation is used to construct a method for solving boundary value problems for the Laplace’s and the Poisson’s equations. Given any solution to these equations we apply the constrained similarity transformation to get the particular solution that satisfies the given boundary conditions.