Two methods for numerical solving mathematical physics problems

Abstract

The work objective is to perform comparative analysis possibilities of two methods for numerical solution boundary value problems for linear equations of elliptic type - the generalized point-sources method and the direct collocation method. The first proposed method is based on the transformation of the original mathematical physics equation to a simpler inhomogeneous equation with the known fundamental solution. A feature of the second proposed method, the direct collocation method, is the irregular arrangement of the collocation nodes in the problem solving domain, which has made it possible to significantly increase the numerical solution accuracy. Both methods allow one to obtain an approximate solution of boundary value problems for almost any type of linear elliptic equations in analytical form. To confirm the effectiveness of the studied numerical methods, two-dimensional and three-dimensional test boundary value problems with known solutions were solved.The work objective is to perform comparative analysis possibilities of two methods for numerical solution boundary value problems for linear equations of elliptic type - the generalized point-sources method and the direct collocation method. The first proposed method is based on the transformation of the original mathematical physics equation to a simpler inhomogeneous equation with the known fundamental solution. A feature of the second proposed method, the direct collocation method, is the irregular arrangement of the collocation nodes in the problem solving domain, which has made it possible to significantly increase the numerical solution accuracy. Both methods allow one to obtain an approximate solution of boundary value problems for almost any type of linear elliptic equations in analytical form. To confirm the effectiveness of the studied numerical methods, two-dimensional and three-dimensional test boundary value problems with known solutions were solved.