The Functional Voxel Method Applied To Solving a Linear First-Order Partial Differential Equation with Given Initial Conditions

Abstract. This paper considers an approach to solving the Cauchy problem for a linear first-order partial differential equation by the functional voxel (FV) method. The approach is based on the principles of differentiation and integration developed for functional voxel modeling (FVM) and yields local geometrical characteristics of the resulting function at linear approximation nodes. A classical approach to solving the Cauchy problem for a partial differential equation is presented on an example, and an FV-model is built as a reference for further comparison with the FVM results. An algorithm for solving differential equations by FVM means is described. The FVM results are visually and numerically compared with the accepted reference. Unlike numerical methods for solving such problems, which give the values of a function at approximation nodes, the FV-model contains local geometrical characteristics at the nodes (i.e., gradient components in the space increased by one dimension). This approach allows obtaining an implicit-form nodal local function as well as an explicit-form differential local function.

Keywords: functional voxel modeling, partial differential equation, Cauchy problem, local function, local geometrical characteristics.

Funding. This study was carried out within the scientific program of the National Center for Physics and Mathematics, direction No. 9 “Artificial Intelligence and Big Data in Technical, Industrial, Natural, and Social Systems.”