Scientific Journal of KubSAU

The review of resolvability of the beginning-boundary problem describing dispersion of an impurity in turbulent atmosphere, correctness of mathematical models, describing impurity dispersion in atmosphere and Koshi’s problem, the first right problem, the third right problem is given in the article

The article by continues the cycle of their studies related to the formulation and development of methods of constructing non-negative solutions of inverse problems of balance models (in this case, the model of world trade). Method of constructing nonnegative solutions of the studied inverse problems is developed. This technique is based on the following scheme of the solution. Initially we convinced of a correct formulation of the direct problem, then of the solvability of the inverse. Further, by specified tabular solutions of the direct problem, a system of algebraic equations containing the unknown, the estimated parameters of the studied model is built. Then the inverse problem reduces to solving the following quadratic programming, the solution of which is determined in MS Excel. The theoretical material is accompanied by solution of specific example, using statistical data of the Karachay-Cherkess Republic that shows how actually in practice it is possible to solve the inverse problem, i.e. to organize a process of balanced trade of the Karachay-Cherkess Republic with each of the subjects of Noth – Caucasion Federal District. Found the non-negative elements of a matrix, by which we can judge what proportion of national income, y, the subject has to spend on the purchase of goods in the Karachay-Cherkess Republic, to trade between this pair was balanced. So, the inverse problem posed in relation to the trading countries, it is possible to put and solve the following method and to trade between the subjects of one country

The article continues the cycle of their studies associated with the formulation and development of methods of construction of nonnegative solutions of inverse problems for dynamic systems. In this article the authors formulated and investigated inverse problems for dynamic systems: model of Samuelsson– Hicks. The technique of constructing non-negative solutions of the studied inverse problems. This method is based on the following scheme of the solution. First, we have to identify the formulation of the direct problem, then the formulation of the inverse. This work investigates how correct the mathematical models describing the dynamic economic system are. Further, in the specified tabular solutions of the direct problem, we have built a system of algebraic equations containing the unknown estimated parameters of the studied model. Then posed inverse problem is reduced to solution of a problem of quadratic programming, the solutions of which are defined in MS Excel. The theoretical material is accompanied by the specific example

The article continues the cycle of their studies associated with the formulation and development of methods of construction of nonnegative solutions of inverse problems for dynamic systems. In practice, we have developed and tested mathematical models of dynamic systems. The basis of these models was based on the apparatus of linear algebra, mathematical analysis, mathematical programming, differential equations, optimization methods, optimal control theory, probability theory, stochastic processes, operations research, game theory, statistical analysis. The inverse problem in various models of mathematical Economics was considered rare. These tasks were sufficiently well investigated in the study of physical processes. As shown by the analysis of the theoretical and applied studies of economic processes they represent considerable interest for practice. Therefore, the article considered the inverse problem of the mathematical model, as shown already introduced the results of other mathematical models, are of considerable interest in applied and theoretical research. In this article the authors formulated and investigated the inverse problem for dynamical systems zero-order and the model of Keynes. For their solution, the authors propose to build a system of algebraic equations, then, using methods of quadratic programming, to find the best average of mean square estimation of the model parameter, which are defined in MS Excel

In practice, there were developed and tested some mathematical models of balance relationships (balance model), economic growth, expanding economy, labour market, theories of consumption, production, competitive equilibrium models of the economy in conditions of imperfect competition and others. The basis of these models were based on linear algebra, mathematical analysis, mathematical programming, differential equations, optimization methods, optimal control theory, probability theory, stochastic processes, operations research, game theory, statistical analysis. The inverse problem in various models of mathematical Economics was considered quite rare. These tasks were sufficiently investigated in the study of physical processes. As shown by the analysis of the theoretical and applied studies of economic processes, they represent considerable interest for practice. Therefore, the considered in the study inverse problems of the mathematical model, as it is shown by the already introduced results of other mathematical models, are of considerable interest in applied and theoretical research. In this article, the authors have formulated and investigated an inverse problem for a model of economic growth. For its solution the authors propose to build a system of algebraic equations, using a reproduction model of national income; then, using methods of quadratic programming, to find the best average quadratic estimates of the model parameter

In practice, we often encounter the problem of determining a system state based on results of various measurements. Measurements are usually accompanied by random errors; therefore, we should not talk about the definition of the system state but its estimation through stochastic processing of measurement results. In the monograph by E. A. Semenchina and M. Z. Laipanova [1] it was investigated for one-step filtering of the measurement errors of the vector of demand in balance model of Leontiev, as well as multistage optimal filtering of measurement errors of the vector of demand. In this article, we have delivered and investigated the inverse problem for the optimal one-step and multi-step filtering of the measurement errors of the vector of demand. For its solution, the authors propose the method of conditional optimization and using given and known disturbance to determine (estimate) the matrix elements for one-step filtering of measurement errors and for multi-stage filtration: for given variables and known disturbance to determine the elements of the matrix. The solution of the inverse problem is reduced to the solution of constrained optimization problems, which is easily determined using in MS Excel. The results of the research have been outlined in this article, they are of considerable interest in applied researches. The article also formulated and the proposed method of solution of inverse in a dynamic Leontiev model