A new Block by Block scheme via quadrature rule of Lobatto-Gaussian for nonlinear Volterra integral equations

Abstract

In this note, a multistage technique called Block by Block technique is proposed to solve nonlinear Volterra integral equations by combining quadrature rule of Lobatto-Gaussian. This procedure calculates several values of unknown functions at once and  it is  the most appropriate method which shows high accuracy for entire points of intervals, especially at the end points of large intervals. Also, the convergence of the presented method via the Gronwall inequality is proven and the rate of convergence is at least $O(h^8)$. Some numerical experiments report the ability and accuracy of the proposed method.