О численном решении уравнения Фредгольма I-го рода

Abstract

In solving various applied problems of mathematical physics, integral equations are increasingly used. This arouses interest in methods for solving such equations. This article discusses an approximate method for solving the Fredholm integral equation of the first kind with a Fredholm kernel. The method is based on the approximation of the solution of an integral equation by a system of point potentials. The method of point potentials is successfully used to solve a number of problems in mathematical physics. This is due to its algorithmicity and ease of use for a wide class of areas. These advantages remain for the method proposed in the article. An approximate solution to the integral equation is sought in the form of a linear combination of point potentials. To determine the coefficients of this linear combination, a variational problem is constructed. The convergence of the method is proved. For the numerical implementation, a stable algorithm based on the regularization of the initial variational problem is proposed. The problem of finding an approximate solution is reduced to a system of linear algebraic equations. Using the proposed method, the problem of flowing around an infinitely thin plate with a potential flow of an ideal fluid, which reduces to the Fredholm integral equation of the first kind with a logarithmic kernel, is solved. The results of numerical calculations are presented.