Improved polynomial approximations for the solution of nonlinear integral equations

In this paper, the solutions of nonlinear integral equations, including Volterra, Fredholm, Volterra–Fredholm of first and second kinds, are approximated as a linear combination of some basic functions. The unknown parameters of an approximate solution are obtained based on minimization of the residual function. In addition, the existence and convergence of these approximate solutions are investigated. In order to use Newton’s method for minimization of the residual function, a suitable initial point will be introduced. Moreover, to confirm the efficiency and accuracy of the proposed method, some numerical examples are presented. It is shown that there are considerable improvements in our results compared with the results of the existing methods. All numerical computations have been performed on a personal computer using Maple 12.