Solving nonlinear integral equations in the Urysohn form by Newton–Kantorovich–quadrature method

The Newton–Kantorovich method is a well-known method for solving nonlinear integral equations. This method attempts to solve a sequence of linear integral equations. In this article, we develop a new method, which is a combination of the Newton–Kantorovich and quadrature methods. The new method solves the nonlinear integral equations of the Urysohn form in a systematic procedure. Some numerical examples are provided, and the obtained numerical approximations are compared with the corresponding exact solutions.