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.
Highlights
IntroductionTo solve the three-dimensional equations of mathematical physics (Poisson equation), a method is used that is a generalization of the Fourier variables separation method and is known in the literature as the variational iterations method
The history of the issueIn the work, to solve the three-dimensional equations of mathematical physics (Poisson equation), a method is used that is a generalization of the Fourier variables separation method and is known in the literature as the variational iterations method
For the heat equation (1), taking into account the boundary conditions (2), we compare the solutions that were obtained by the numerical method and the analytical method
Summary
To solve the three-dimensional equations of mathematical physics (Poisson equation), a method is used that is a generalization of the Fourier variables separation method and is known in the literature as the variational iterations method. This method is based on the idea of reducing partial differential equations to ordinary differential equations for elliptic equations - this is the Kantorovich method with the subsequent refinement of the function with respect to the desired variables - the variational iterations method. It allows you to get an analytically accurate solution at every step of the iterative procedure This method was first proposed and used in 1933 by T.E. Shunk [1] for calculating the bending of cylindrical panels. The method justification for the class of equations described by positive definite operators is given in [4]
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