Arithmetic in Functional-Voxel Modeling

Abstract

The paper considers the claim that the function-voxel model allows arithmetic operations over the space of values of two different functions given by a single domain. At the same time, there are three possible approaches to solving the problem, leading to a similar result: functional approach -when the analytical representation of functions is involved in the calculations; functional voxel approach -when voxel data representing local geometric characteristics are used in the construction of local functions for further use in the calculation; voxel approach -when exclusively voxel data is used for sequential recalculation of local geometric characteristics of the model. The basic arithmetic operations on functional voxel models are considered, including such procedures as: addition, subtraction, modulo, exponentiation, taking root expressions, multiplication and division of functional voxel models. It is shown that the obtained applicability of arithmetic operations to functional voxel models leads to obtaining new complex functional voxel models.

Functional-Voxel Modeling of The Cauchy Problem

“…The solution of a simple partial differential equation of the first order does not seem to be a difficult problem today and is fully provided by analytical and computer numerical solutions. The proposed study is aimed rather at obtaining a fundamental answer: is it possible to apply the properties of a functional voxel model (PV model) used to solve a fairly wide class of problems [1,2] in solving a differential equation in principle. Therefore, the example [3] was initially chosen, which is not complicated, but is able to demonstrate quite clearly the answer to the question posed.…”