New method for the numerical solution of the Fredholmlinear integral equation on a large interval

The traditional numerical process to tackle a linear Fredholm integral equation on a large interval is divided into two parts, the first is discretization, and the second is the use of the iterative scheme to approach the solutions of the huge algebraic system. In this paper we propose a new method based on constructing a generalization of the iterative scheme, which is adapted to the system of linear bounded operators. Then we don't discretize the whole system, but only the diagonal part of the system. This system is built by transforming our integral equation. As discretization we consider the product integration method and the Gauss–Seidel iterative method as iterative scheme. We also prove the convergence of this new method. The numerical tests developed show its effectiveness.