Numerical method for finding the mathematical expectation of the solution of the equation of the polymer destruction model, taking into account the influence of random factors

Abstract

The article is devoted to the development and analysis of a mathematical model of the polymer destruction process under the action of shear stresses and temperature. The process model is an ordinary Riccati differential equation containing a random process and includes elementary reactions occurring in the polymer matrix: destruction and recombination of macromolecules. The equation cannot be written in the form using the Ito or Stratonovich integrals. It is assumed that the random coefficient is specified by the characteristic functional. The problem is set of finding the mathematical expectation of the solution of the model under consideration. Since the problem cannot be solved by exact analytical methods, a numerical solution method has been developed. A systems approach has been used to solve the problem. To date, we do not know any methods for solving such a class of problems, so the following approach has been used. By changing the variable, the Riccati equation is reduced to a second-order linear differential equation with a random coefficient. For the resulting equation, an auxiliary equation is found containing ordinary and variational derivatives, from the solution of which the mathematical expectation of the solution of this equation is easily found. Methods for solving differential equations with ordinary and variational derivatives were developed by V. G. Zadorozhny. Since it is impossible to obtain an analytical solution to this problem, a numerical method for solving an equation with ordinary and variational derivatives was developed, which is an analogue of difference methods for solving partial differential equations, with the most fundamental being the method of approximating the variational derivative on a grid. Parametric identification of the model was carried out based on the data of a full-scale experiment using a genetic algorithm. Analysis of the modeling results showed good agreement between the experimental and calculated values of the radical concentration. The modeling results are presented in graphical form. The proposed method is implemented as an application program on a computer.