Newton-Kantorovich method for solving one- and two-dimensional nonlinear Volterra integral equations of the second kind

The problems of nonlinear Volterra integral equations (VIEs) which consist of one dimensional nonlinear VIE, system of 2×2 nonlinear VIEs, system of n × n nonlinear VIEs, and two dimensional nonlinear VIE, with smooth unknown functions and continuous bounded given functions are discussed. The Newton-Kantorovich method (NKM) is used to linearize the problems. Then the Nystrom-type Gauss-Legendre quadrature formula (QF) is used to solve the linearized equations and systems. New majorant functions are found for some problems which lead to the increment of convergence interval. The new approach based on the subcollocation method is developed and motivation leads to high accurate approximate solutions. The existence and uniqueness of solution are proved and error estimation and rate of convergence are obtained. Numerical examples show that our results are coincided with the theoretical finding