الملخص

         Several scientific and engineering applications are usually described as integral equations.  A  new approach for solving the type of linear and nonlinear Fredholm integral equation of the second kind is proposed. Although many methods provide an analytic solution,  there are different types of integral equations are difficult to solve. Therefore, the numerical approach for solving integral equations is used. Fredholm integral equations of the second kind have been converted to unconstrained optimization problems to find their approximate solutions. This work employs particle swarm optimization combined with padé expansion to find an approximate solution of the Fredholm integral equation. This is applied by minimizing the fitness function value. The fitness function is calculated using the discrete least squares weighted function. The proposed algorithm is applied to solve linear and non-linear FIE. The results are compared to exact solutions. The stability of the proposed algorithm is also presented. The results are promising in terms of convergence , stability and accuracy of the approximate solution.