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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.
Highlights
Differential characteristics of geometric manifolds have important theoretical and applied significance
The need to study them in parallel with the development of mathematical analysis led to the emergence of differential geometry as a separate branch of mathematics
2 Principles for determining geometric interpolants in a multidimensional space In General, a geometric interpolant is a geometric object of a multidimensional affine space that passes through pre-defined points whose coordinates correspond to the original experimental and statistical information [11]
Summary
Differential characteristics of geometric manifolds have important theoretical and applied significance. An evolute, that has found wide application in engineering practice, is a geometric place of points (a curve) that are the centers of curvature of the original curve Another example of using differential characteristics of curved lines is the work [1,2,3], in which the author proposed the use of functional curves for geometric modeling of physical processes in the form of a set of functionally interconnected lines of trajectory, speed and acceleration. In the works of Sophus Li [6,7], a detailed study of point groups of plane transformations is proposed, whereas differential equations are found not as the main object of research, but as an auxiliary apparatus Another approach based on modeling geometric manifolds with pre-defined properties is proposed in [8,9,10]. The main concept of the proposed approach to the numerical solution of differential equations [8,9,10] is to use geometric interpolants of a multidimensional space [11] to model geometric manifolds with specified differential characteristics
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