Параболическая модель в форме пространства состояний динамики накоплений

Abstract

An important place in the theory of partial differential equations and its applications is occupied by the heat equation, a representative of the class of the so-called parabolic equations. It is known that to check the correctness of a mathematical model based on a parabolic equation, the existence of its solution is very important since a mathematical model is not always adequate to a specific phenomenon and the existence of a solution to a corresponding mathematical problem does not follow from the existence of a solution to a real applied problem. Therefore, methods for solving partial differential equations, both analytical and numerical, are always relevant. Nowadays, a computational experiment has become a powerful tool for theoretical research. It is carried out over a mathematical model of the object under study, but at the same time, other parameters are calculated using one of the parameters of the model and conclusions are drawn about the properties of the object or phenomenon under study. The problem of passive parametric identification of systems with distributed parameters for resource accumulation dynamics of many households using a stochastic distributed model in the form of a state space with regard to the white noise of the dynamics model of the object under study and the white noise of the model of a linear-type measuring system is considered in the paper. The use of the finite difference method allowed us to reduce the solution of partial differential equations of a parabolic type to the solution of a system of linear finite difference and algebraic equations represented by models in the form of a state space. It was also proposed to use a filtering algorithm based on the Kalman scheme for reliable estimation of the object behavior. Calculations were carried out using the Matlab mathematical system based on statistical data for five years, taken from the site “Agency for Strategic Planning and Reforms of the Republic of Kazakhstan Bureau of National Statistics”. Estimation of the coefficients of the equations for the household resource accumulation in the form of a state space using this technique is sufficiently universal and can be applied in other fields of science and technology.