Application of the generalized point source method for solving boundary value problems of mathematical physics

Introduction. The work objective is to develop a new universal numerical method for solving boundary value problems for linear elliptic equations. Materials and Methods. The proposed method is based on the transformation of the original mathematical physics equation to a simpler inhomogeneous equation with the known fundamental solution. From this equation, the transition to an inhomogeneous integral equation with the kernel expressed by the known fundamental solution is carried out. The obtained integral equation with boundary conditions is solved numerically. An approximate solution, the field potential being in an analytical form, is resulted. That allows not only find an approximate value of the field potential at any point in the solutions domain, but also differentiate this potential, and all without perceptible loss of accuracy. This property of the developed numerical method sets it apart from the traditional numerical methods for solving boundary value problems, such as the finite element method. Research Results. To confirm the effectiveness of the proposed numerical method, the two-dimensional and three-dimensional boundary value problems with the known solutions are solved. The dependences of the numerical solution error on the number of linear equations in the resulting system are obtained. It is shown that even at a small number of equations in the system (some hundreds) the solution accuracy is achieved at the level of hundredths of a percent. Another major illustration of the proposed method effectiveness is the solution to quantum mechanical problems for the one-dimensional and two-dimensional quantum oscillators. It is shown that the given method allows finding the energy eigenvalues and eigenfunctions with an acceptable accuracy. The developed numerical technique allows greatly extend the application domain of the traditional point source method in solving applied problems for modeling fields of different physical nature, including the eigenvalue problems. Discussion and Conclusions. The results obtained confirm that a physical field described by any linear elliptic equation can be represented as a superposition of point source fields satisfying a simpler equation, the solution of which is obtained through the method of point source of the field. Therefore, the numerical method presented in this paper can be considered as a generalized point source method.