Reliability and Quality of Complex Systems

Abstract

Background. Mathematical modeling of processes of various physical and chemical nature is the most important stage in solving problems of design and operation of various physical and chemical systems. The most important requirement for mathematical models is adequacy (i.e. consistency with physical laws) and the possibility of setting the required accuracy (given a sufficient number of experimental data). To construct mathematical models that meet the above requirements, the author proposed a method of mathematical prototyping of energy processes within the framework of mechanics, electrodynamics and modern non-equilibrium thermodynamics – a unified approach to modeling processes of various physical and chemical nature. To obtain the above method for the equations of the dynamics of physical and chemical processes, it is necessary to set the state functions for the properties of substances and processes, including the dissipative matrix, with an accuracy of up to the experimentally studied coefficients. The dissipative matrix must be positive definite (or non-degenerate non-negative definite in the case of inertia in the system). Therefore, the state functions of a dissipative matrix must a priori satisfy the condition of positive definiteness. This work is devoted to the construction of state functions of a dissipative matrix that satisfy the condition of positive definiteness (or non-degeneracy and non-negative definiteness). Matherials and methods. The synthesis of equations of the dynamics of physical and chemical processes is carried out on the basis of the method of mathematical prototyping of energy processes. The state functions for the dissipative matrix are constructed using the methods of uniform approximation of functions and methods of symbolic regression. Results. The assignment of the state functions of the dissipative matrix included in the equations of the method of mathematical prototyping of energy processes, guaranteeing positive definiteness (or non-negative definiteness with non-degeneracy) of the dissipative matrix guarantees the correctness of the assignment of the mentioned functions. Conclusions. The methods proposed in this paper for specifying state functions for a dissipative matrix included in the equations of the method of mathematical prototyping of energy processes make it possible to form a class of correct state functions for dissipative matrices.

Keywords

• method of mathematical prototyping of energy processes
• dissipative matrix
• approximation
• symbolic regression