МОДЕЛЮВАННЯ ЛІНІЙНИМИ РІЗНИЦЕВИМИ РІВНЯННЯМИ ПЕРШОГО ПОРЯДКУ

Abstract

The article considers economic and mathematical models and studies the socio-economic processes that develop over time, as well as mathematical models that describe them. These are dynamic models. All variables in dynamic models generally depend on the time that acts as an independent variable. In economic research, there are often problems in which variables acquire discrete numerical values. For example, at the end of the month, quarter, year, etc., production results are optimized; accrual of interest on the bank deposit at the end of the month, six months, at the end of the year. In addition, because computers operate only with numbers, so when using computer technology, all continuous processes are reduced to discrete. In this case, from differential equations that describe certain economic processes, we move to difference equations. There are dynamic models with continuous and discrete time, ie continuous and discrete models. Therefore, depending on the type of dynamics of the system under study, dynamic models can be divided into discrete and continuous. In discrete dynamic models, difference equations or systems of difference equations are used; differential equations or systems of differential equations are used in continuous dynamic models. In addition, in some cases there may be systems with mixed dynamics, then differential-equation equations are used to describe them. Difference equations and systems of equations are used successfully in modeling dynamic processes (in economics, banking, etc.). It is when the change of process occurs abruptly, or discretely, that it is convenient and expedient to apply difference equations and systems of equations. The theory of dynamical systems with discrete time, which arose as a result of building mathematical models of real economic and physical processes at the junction of the theory of difference equations and discrete random processes, is currently experiencing a period of rapid development and widespread use in various spheres of human life. In this paper, we investigate the following equations, as well as show their application to solve economic problems. In particular, discrete models described by first-order difference equations are considered. Considerable attention is paid to the analysis of specific models that are meaningful and widely used in economic theory, banking, etc.