Ovsky O. Instrumental system of analysis of problems of mathematical physics on the basis of analytical - numerical methods

The dissertation is devoted to the development of an instrumental system that allows to automate the analysis of boundary value problems of the theory of elasticity by means of the analytical method of initial functions. Nowadays, analytical (algebraic and symbolic) calculations, which have greater clarity of results than numerical ones, are gaining more and more recognition. Analytical calculations are implemented in computer mathematics (SCM) systems. Therefore, SCM is widely used in science, technology and, especially, in the educational process. Their main advantages are ease of use, accuracy of the obtained solutions, high efficiency of processing algebraic operations. However, they have a common drawback: partial support for transformations of transcendental and logarithmic expressions. Before application in specialized fields of science and technology SCM must be software improved. The obtained final and some intermediate results of calculations should be investigated for compliance with the content of the problem, accuracy, etc. Especially for the case of boundary value problems of the theory of elasticity, where SCM was rarely used for research. In the dissertation work the instrumental system of solving problems of mathematical physics by analytical-numerical methods is considered. The solution of mathematical physics problems for static linear theory of elasticity on a computer is automated. An analytical-numerical method of initial functions for solving two-dimensional and three-dimensional problems is proposed. Algorithms have been developed to derive analytical solutions of differential equations of the theory of elasticity on a computer. Algorithms for converting differentiation and integration operations into symbolic operations of multiplication and division have been created. The main results of the work were used to analyze the problems of mathematical physics that arise in the theory of elasticity. A new approach to finding the elastic characteristics of an elastic two-layer foundation has been developed. It is based on a synthesis of analytical and numerical methods. It should be noted that the tool system due to the language of the interpretive type SCM is designed so that if at some stage of its work results are obtained that do not satisfy the user, he will be able to make the necessary changes online to refine and correct the results. The instrumental system is aimed at giving the researcher the opportunity to modify the mathematical model in the analysis process, to interfere in the calculation process. Its main advantage is the ability to make changes to the program in the input language of the system.