The Legendre-Picard iteration method for the numerical solution of Fredholm integral equations of the second kind

In this study, an efficient numerical method for solving the Fredholm integral equations of the second kind is presented. The proposed method is based on the Picard iteration method, the shifted Legendre polynomials, and the shifted Legendre-Gauss integration rule. According to the orthogonal property of Legendre polynomials, the proposed method uses an iterative scheme to update the coefficients of the series of approximate solution. Also, a vector-matrix structure is introduced to increase the efficiency and reduce the computational time. The numerical results clearly indicate the feasibility and the accuracy of the proposed technique.