Modeling Geometric Varieties with Given Differential Characteristics and Its Application

Abstract

The article describes an approach to modeling geometric manifolds with specified differential properties, which is based on the use of geometric interpolants of a multidimensional space. A geometric interpolant is understood as a geometric object of a multidimensional space passing through predetermined points in advance, the coordinates of which correspond to the initial experimental and statistical information. Principles for determining geometric interpolants and an example of an analytical description of a 3-parameter geometric interpolant belonging to a 4-dimensional space in the form of a geometric scheme and a computational algorithm based on a sequence of point equations are given. The main direction of practical use of geometric interpolants is geometric modeling of multifactor processes and phenomena, but they can also be an effective tool for multivariate approximation. Based on this, the article presents a general approach to modeling geometric manifolds with given differential properties and its application in the form of a numerical solution of differential equations by approximating it using geometric interpolants of a multidimensional space. To implement an approach to modeling geometric manifolds with specified differential properties, it is proposed to use a computational algorithm consisting of 10 points. The advantages of using geometric interpolants for the numerical solution of differential equations are highlighted.