Comparison of numerical implementations of the quadrature method for solving an integral equation

Integral equations are encountered in various sections of physics — in acoustics, filtration theory, in electrophysics, in mechanics of solid deformable bodies. Fredholm integral equations can be used for mathematical description of processes of radiation and scattering of sound waves. Often finding an exact analytical solution to integral equations seems difficult, in connection with which numerical methods are used to solve such equations. This work is devoted to the analysis of algorithms for the numerical solution of the Fredholm integral equation of the second kind. The paper presents the results of the implementation of some algorithms for the numerical quadrature method for solving integral equations. The quadrature method reduces the solution of the integral equation to the solution of a system of linear algebraic equations, which can be implemented in various ways. The choice of the quadrature formula is assumed to be arbitrary; in the study under consideration, the trapezoid formula was used as a quadrature formula. When reducing an integral equation to a system of linear algebraic equations, there is also arbitrariness in the choice of the method for solving linear equations. To solve the system of linear equations, the Gauss method, the method of simple iterations and the iterative Jacobi method were used. The study of the properties of the integral equation shows that the kernel of the integral operator satisfies the properties under which the integral operator is a contracting operator and, therefore, the integral equation has a unique solution. Comparison of numerical calculations for various methods of solving a system of linear equations demonstrates that the Jacobi method for solving a system of linear algebraic equations converges to an exact solution significantly faster than the method of simple iterations. The results obtained can be used in developing algorithms for solving integral equations.