Substantiating the pulse method for determining the time parameter of fire detectors with a thermoresistive sensing element

This paper substantiates the pulse method for determining the time parameter for fire detectors with a thermoresistive sensing element ‒ the time constant. The method is based on using the Joule-Lenz effect, which manifests itself when an electric current pulse passes through the thermoresistive sensing element of fire detectors. Thermal processes in such a sensing element are described by a mathematical model that belongs to the class of equations of mathematical physics. The solution to the differential equation of this class was derived using the Hankel integral transformation and is represented as a series relative to the Bessel functions. The resulting solution is used to construct a mathematical model of a thermoresistive sensing element in the form of a transfer function, which takes the form of the transfer function of the inertial link. To trigger the thermoresistive sensing element of fire detectors, a single pulse of electric current in the shape of a rectangular triangle is used. The integral Laplace transformation was applied to mathematically describe the response of a thermoresistive sensing element to the thermal effect of such a test influence. To obtain information about the time parameter of fire detectors with a thermoresistive sensing element, the ratio of its output signals is used, which are measured in the a priori defined moments. A two-parametric expression was built to determine the time parameter of fire detectors; a verbal interpretation of the pulse method to determine it was provided. The implementation of this method ensures the invariance of the time parameter of fire detectors with a thermoresistive sensing element relative to the amplitude of a single pulse of an electric current, as well as relative to the parameter that is included in its transfer coefficient.

Development of information culture of students when teaching equations of mathematical physics in the conditions of informatization of education

Problem and goal. In modern conditions, specialists in various subject areas who have an information culture and are able to solve complex professional problems using modern information and communication technologies are in demand. Currently, specialists in the field of applied mathematics are required, which plays an important role in the development of human civilization. Therefore, in the process of teaching various academic disciplines of applied mathematics at the university, including the discipline Equations of mathematical physics, attention should be paid to the development of students information culture. Methodology. When teaching students the discipline Equations of mathematical physics, it is extremely important that the teacher knows not only the content of teaching this discipline of applied mathematics, but also has practical experience in solving equations of mathematical physics by computer means. Such qualities of the teacher will allow him to successfully conduct training sessions in the conditions of informatization of teaching mentioned discipline. At the same time, it ought to be clearly understood that the use of computer technologies in teaching the discipline Equations of mathematical physics must be correct. The necessity to develop and implement in practice a variety of methodological approaches that allow students to develop an information culture in training sessions on that discipline is obvious. Results. The use of advanced pedagogical technologies in training sessions on the discipline Equations of mathematical physics, where computer technologies are used, will allow students to develop an information culture. Conclusion. Computer technologies that students use in the process of solving educational problems require them to have certain skills and abilities to identify their broad capabilities. Students are aware of the role of computer technologies in conducting applied scientific research, understand the role of computer modeling methodology and computational experiment in studying the world around them.

Dynamics of media with stationary parameters and telegraph equations

The paper proposes an approach for constructing exact solutions of differential equations of mathematical physics, in particular, the telegraph equation. The method is based on the theory of finite-dimensional dynamics of systems of evolutionary differential equations. This theory is a natural extension of the theory of dynamical systems to partial differential equations. It allows one to construct exact solutions of partial differential equations even in the case when equations do not have symmetry algebras sufficient for integration.

Modeling of heat transport phenomena using the equations of mathematical physics

The study of physical phenomena that include conservative principles is part of the research field of the equations of mathematical physics. To deepen in methods to solve the equations of mathematical physics is a contribution in understanding the modeling of applications in different areas. This research studies the physical phenomenon of heat transport with convection from the viewpoint of modeling with differential equations. The advantage of working with equations is to apply the techniques of mathematical analysis and numerical methods to obtain the temperature function. In the research, the solution of the heat transport model is computed according to the analytical method of separable variables in order to represent the temperature function as a trigonometric series. With the help of a simple numerical method, it is possible to derive a scheme of calculation of the temperature function. By performing a case study, the methods are compared, and their fit is verified by simulation.

Modeling of the physical phenomenon of heat generation by using partial differential equations

The equations of mathematical physics are a natural environment for modeling physical phenomena, an example of the above is evidenced by the heat equation in relation to its use in a variety of applications; directly related to the equations of mathematical physics are the solution methods that are used to construct the predictive models. This paper describes step by step the analytical method of separation of variables to perform a complete description of the heat conduction phenomenon in the presence of a heat generation source. The investigation by using mathematical arguments allowed to calculate the temperature function as the addition of a Fourier series and a function which represents the steady state; by performing a computational simulation, it was possible to demonstrate the accuracy of the results achieved.

CHARACTERISTIC FUNCTION OF THE SYSTEM D–EQUATIO

This paper is devoted to the problems of studying the multiperiodic solution of some evolutionary equations. The article also discusses the existence and uniqueness of a multiperiodic solution with respect to vector functions for an evolutionary reduced equation. Studies have been conducted on the characteristic function of a certain system of the evolutionary equation. Some properties of the vector function are proved. They can be used in the further study of oscillatory bounded solutions of evolutionary equations. Based on the argumentation of the theorem on the existence and uniqueness of an almost multiperiodic solution of the specified system, considered using the method of shortening the characteristic function. All estimates of the characteristic function are based on the enhanced Lipschitz condition, first introduced by academician K. P. Persidskiy. The results will also be useful in the study of periodic solutions of evolutionary equations of mathematical physics

Teaching students the equations of mathematical physics using educational electronic resources

Problem and goal. Currently, information and telecommunications technologies are widely used in the professional activities of most specialists in various subject areas. This circumstance initiates the training of students in higher education institutions, who must have not only deep subject knowledge, but also be able to master modern information and telecommunications technologies and be able to apply them in their activities. One of the fundamental disciplines that is included in the university curricula for preparing students of physical and mathematical fields of study is Equations of mathematical physics. In the process of teaching students the equations of mathematical physics, the goals are set not only to form students' solid subject knowledge, but also to acquire the skills and abilities to apply modern information technologies in the study of mathematical models based on the equations of mathematical physics. Methodology. Educational electronic resources are used in training sessions on mathematical physics equations. Such training sessions with students take place in the form of laboratory classes, where modern computer technologies are used to find solutions to equations of mathematical physics and then analyze them. Results . The implementation of didactic principles of teaching mathematical physics equations in laboratory classes using educational electronic resources allows students to achieve good results in the methods of studying mathematical physics equations. Conclusion. The use of educational electronic resources in the classroom on the equations of mathematical physics allows students, in addition to deep subject knowledge, to acquire the skills and abilities to use modern computer techno- logies to solve mathematical problems.

Mathematical modeling of filling the CCM mold with liquid metal during its supply from a rotating submersible nozzle

Experimental studies of the flow of liquid metal in CCM mold are long, complex and labor consuming process. Therefore, mathematical modeling by numerical methods is increasingly used for this purpose. The article considers a new technology for liquid metal supply into a mold. The authors present original patented design of the device, consisting of direct-flow and rotating bottom nozzles. The main results of investigations of the melt flow in the mold are considered. The objects of research were hydrodynamic and heat flows of liquid metal at new process of steel casting into a CCM mold of rectangular section. The result is spatial mathematical model describing flows and temperatures of liquid metal in the mold. To simulate the processes occurring during metal flow in the mold, special software was designed. Theoretical calculations are based on fundamental equations of hydrodynamics, equations of mathematical physics (equation of heat conduction taking into account mass transfer) and proven numerical method. The area under study was divided into elements of finite dimensions; for each element, resulting system of equations was written in difference form. The results are fields of velocities and temperatures of metal flow in the mold volume. A calculation program was compiled based on developed numerical schemes and algorithms. An example of calculation of steel casting into a mold of rectangular cross-section, and flow diagrams of liquid metal along various sections of the mold are given. Vector flows of liquid metal in different sections of the mold are clearly presented at different angles of rotation of the deep-bottom nozzle. The authors have identified the areas of intense turbulence. Metal flows of the described technological process were compared with traditional metal supply through a fixed bottom nozzle.