Newton–Kantorovich Based Numerical Methods for Nonlinear Integral Equations

Shaymaa A. Hameed; Abdullah Mhmood  Jasim; Nabila I.  Aziz

Authors

Shaymaa A. Hameed Department of Mathematics, College of Basic Education, Diyala University, Diyala, Iraq
Abdullah Mhmood Jasim Department of Mathematics, College of Education – Tuzkhurmatu, University of Tikrit, Salahaddin, Iraq
Nabila I. Aziz Department of Mathematics, College of Sciences, University of Kirkuk, Iraq

Keywords:

Newton–Kantorovich Method, Nonlinear Volterra Integral Equations, Adaptive Trapezoidal Rule, Numerical Quadrature, Iterative Linearization, Nonlinear Operator Equations

Abstract

This research addresses the development of an effective numerical method for solving nonlinear Volterra integral equations of the second kind based on the Newton–Cantorovich method for bypassing nonlinearity and converting the problem into a series of successive linear equations. This technique is combined with an adaptive trapezoidal rule to improve the accuracy of numerical integration by adjusting the step size according to the behavior of the function. This combination contributes to reducing cumulative error and enhancing convergence stability during iteration. The method was implemented using MATLAB and tested on several standard examples. The results showed high agreement with exact solutions and significant improvement compared to traditional methods such as Simpson's rule, confirming the efficiency and computational accuracy of the proposed method.

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