The Variational Iterations Method for the Three-dimensional Equations Analysis of Mathematical Physics and the Solution Visualization with its Help

Abstract

The aim of the work is to use the variational iterations method to study the three-dimensional equations of mathematical physics and visualize the solutions obtained on its basis and the 3DsMAX software package. An analytical solution of the three-dimensional Poisson equations is obtained for the first time. The method is based on the Fourier idea of variables separation with the subsequent application of the Bubnov-Galerkin method for reducing partial differential equations to ordinary differential equations, which in the Western scientific literature has become known as the generalized Kantorovich method, and in the Eastern European literature has known as the variational iterations method. This solution is compared with the numerical solution of the three-dimensional Poisson equation by the finite differences method of the second accuracy order and the finite element method for two finite element types: tetrahedra and cubic elements, which is a generalized Kantorovich method, based on the solution of the three-dimensional stationary differential heat equation. As the method study, a set of numerical methods was used. For the results reliability, the convergence of the solutions by the partition step is checked. The results comparison with a change in the geometric parameters of the heat equation is made and a conclusion is drawn on the solutions reliability obtained. Solutions visualization using the 3Ds max program is also implemented.