Sixth-Order Numerical Solver Based on Truncation Error for Solution of Nonlinear Fredholm Equations

A sixth order accurate numerical method is proposed for nonlinear integral equations of the Fredholm type. First the problem is discretized and the integral approximated by the classical second order trapezoid rule. Then, the leading truncation error terms of the quadrature rule are approximated. The first derivatives in the error term are approximated using five-point finite difference formula which has fourth order of accuracy, while the third derivative is approximated with a second order finite difference formula. This results to a new leading error term which is of sixth order, hence a sixth order method. Newton algorithm is then designed to approximate the resulting nonlinear system. Numerical examples are used to demonstrate the accuracy and performance of the method. The results are also compared with those from other published works in which the superiority of the proposed method is demonstrated.