Functional-Voxel Modeling of The Cauchy Problem

Abstract

The paper considers an approach to solving the Cauchy problem for an example of a partial differential equation of the first order under given boundary conditions by the functional voxel method (FVM). The proposed approach uses the accumulated experience of differentiation and integration into FV- modeling to obtain local geometric characteristics of triangular elements on the surface of the resulting function in the process of linear approximation. The analytical solution of a simple example of a partial differential equation of the first order for the Cauchy problem is analyzed. Based on the obtained analytical solution, FV-model is constructed for further comparison with the results obtained by means of FV-modeling. The algorithm for solving the example is described by means of FV-modeling. A visual and numerical comparative analysis is carried out to determine the difference between the obtained results of FV-modeling and the accepted standard. The main difference between solving such a problem by numerical methods is the results obtained. In numerical methods, the result is the value of the function at the approximation nodes, and the FV-model at the nodes contains local geometric characteristics (gradient components in a space enlarged by one), which makes it possible to obtain a nodal local function of an implicit form, as well as a differential local function of an explicit form. The proposed graphical representation of the function area on a computer provides not only visual visibility, but also compact storage compared to a traditional array of real numbers.