Application of the numerically obtained fundamental solutions in the field point-source method

Abstract

The work objective is to obtain an integral equation by which, using the known fundamental solution to the other equation, it is possible to find a fundamental solution to the linear elliptic equation. The concept of a numerical fundamental solution (NFS) is introduced. The so obtained numerical fundamental solutions (NFS) can be used for solving boundary value problems for N-dimensional elliptic equations by the field point source method (PSM). The research result is the development of the effective numerical method for solving boundary value problems using the NFS. This allows expanding the range of solvable problems using PSM, making PSM a universal numerical method for solving boundary value problems for linear elliptic equations. It admits solutions to various types of boundary value problems. Especially effective is the use of the proposed method for solving three-dimensional Dirichlet problems for equations with spherically symmetric fundamental solutions. The Schrodinger equation for a one-dimensional quantum oscillator is solved by the proposed method as a test problem. It is shown that it is possible to find the eigenvalues and eigenfunctions of the quantum oscillator using numerically obtained fundamental solutions to the Schrodinger equation. The oscillator eigenfunctions obtained by the proposed method are in good agreement with the known analytical solutions to quantum problems. Then, as another test example, a two-dimensional boundary value problem for the Helmholtz equation is solved. In this case, it is necessary to obtain a numerical fundamental solution to the Helmholtz equation first. Dependences of the numerical solution error on the nodes number in the problem solution domain are calculated. Upon the results obtained, the following conclusion is made. The results of solving test problems confirm the efficiency of the proposed numerical method.