Chorna O. Mathematical modeling of spatial distribution of aggregate of minerals by methods of interlineation of matrix-functions

The object of research is the process of restoring the spatial distribution of the density of the aggregate of minerals as a three-dimensional function between a system of unevenly spaced curves in vertical planes. Research methods - the main results of the work were obtained on the basis of the theory of functional analysis (for the study and development of methods for assessing mineral reserves), computational mathematics (for obtaining the form of basis polynomials (or splines) on the system of spatial noncontinuous curves), the theory of approximation of functions of several variables using the interlineation of functions (for constructing mathematical models of restoration of the distribution of the density of the investigated object according to the data of spatial curves). The basis of numerical realization is the interlinеation of the functions of three variables using the data of the system of spatial noncontinuous curves in vertical planes. Theoretical and practical results - developed mathematical models and methods for solving the problem of restoring the density distribution of the aggregate of minerals as a three-dimensional function by the methods interlineation the functions of three variables on the system of non-perturbed spatial curves allows a significant approximation to the general model of the distribution of investigated objects. The developed modifications of methods allow, based on information about the presence of an object, to restore the corresponding layer of one or another mineral. The scientific novelty of the obtained results is that for the first time the method of restoring the matrix function from three variables between a given system of lines describing a given well system is developed and investigated; from this method, as a partial case, the method of restoring the function of three variables between a system of predetermined lines describing the well system using the polynomial, global, and spline-interlineation functions of three variables follows from this method; the theoretical comparison of the methods of the interlineation of matrix functions is carried out and recommendations are made on which auxiliary functions in matrix functions operators are better to use. For the first time, theorems have been formulated and proved that the operators of the interlineation of matrix functions of three variables have traces of this matrix function at the points of each given line. For the first time, a matrix mathematical model of the spatial distribution of the density of the investigated object between a system of spatial curves in vertical planes according to the data of spatial non-perturbed curves by methods interlineation of functions of three variables was constructed. The further development of the methods for restoring the function of three variables between the system of given lines describing the system of inclined wells by the methods of polynomial, global and spline-interlineation of functions of three variables has been developed, which, unlike the previous approaches, allows us to retrieve the appropriate layer on the basis of information about the presence of an object or other minerals. The method of estimating mineral resources according to the data from the cores of inclined wells is improved on the basis of the mathematical models of their distribution offered in the dissertation, which provides opportunities for optimization of the choice of directions of mining of minerals. Dissertation work was carried out in accordance with the plan of research work of the Department of Higher and Applied Mathematics of the Ukrainian Engineering and Pedagogical Academy within the framework of research work # 13-01 "Development and research of a new method of exploration and development of mineral deposits on the basis interlineation of functions" (NDR 0109U008661). The calculated computational methods for restoring the distribution of the density of the aggregate of minerals have been introduced into the educational process at the Ukrainian Engineering and Pedagogical Academy in the preparation of students and masters studying in the specialty "Naftohazov case". The calculated computational methods for restoring the distribution of the density of the aggregate of minerals have been introduced into the educational process at the Ukrainian Engineering and Pedagogical Academy in the preparation of students and masters studying in the specialty "Naftohazov case". The computational methods developed can be used in finding the estimation of mineral resources, taking into account the results of drilling of inclined wells, and the design of mining enterprises.