Newton–Kantorovich Based Numerical Methods for Nonlinear Integral Equations

Mohammed Faleh  Aswad; Bahman  Ghaznafar

Authors

Mohammed Faleh Aswad Mathematics Department, College of pure Science University of Lorestan, Iran
Bahman Ghaznafar Mathematics Department, College of pure Science University of Lorestan, Iran

Keywords:

Newton–Kantorovich Method, Nonlinear Volterra Integral Equations, Adaptive Trapezoidal Rule, Numerical Integration, Iterative Methods, Linearization Technique, Convergence Stability, MATLAB Implementation, Quadrature Methods, Computational Accuracy

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.

References

K. Sekine, A. Takayasu, and S. Oishi, “An algorithm of identifying parameters satisfying a sufficient condition of Plum’s Newton-Kantorovich like existence theorem for nonlinear operator equations,” Nonlinear Theory and Its Applications, IEICE, vol. 5, no. 1, pp. 64–79, 2014, doi: 10.1587/nolta.5.64.

Q. Jin, “Inexact Newton–Landweber iteration for solving nonlinear inverse problems in Banach spaces,” Inverse Probl., vol. 28, no. 6, p. 65002, Apr. 2012, doi: 10.1088/0266-5611/28/6/065002.

K. E. Atkinson, “A Survey of Numerical Methods for Solving Nonlinear Integral Equations,” Journal of Integral Equations and Applications, vol. 4, no. 1, Mar. 1992, doi: 10.1216/jiea/1181075664.

H. Wang and L. Fang, “ONE FIXED POINT THEOREM FOR MIXED MONOTONE OPERATORS AND APPLICATIONS TO NONLINEAR INTEGRAL EQUATIONS,” Journal of Integral Equations and Applications, vol. 36, no. 3, Oct. 2024, doi: 10.1216/jie.2024.36.371.

S. Heikkilä, “On differential equations in ordered Banach spaces with applications to differential systems and random equations,” Differential and Integral Equations, vol. 3, no. 3, Jan. 1990, doi: 10.57262/die/1371571152.

A. M. Linkov, “Complex Variable Integral Equations,” in Boundary Integral Equations in Elasticity Theory, Springer Netherlands, 2002, pp. 87–111. doi: 10.1007/978-94-015-9914-6_7.

I. Jaradat, “Simulating time delays and space–time memory interactions: An analytical approach,” Partial Differential Equations in Applied Mathematics, vol. 11, p. 100881, Sep. 2024, doi: 10.1016/j.padiff.2024.100881.

P. Liu, M. Lei, J. Yue, and R. Niu, “Improved Meshless Finite Integration Method for Solving Time Fractional Diffusion Equations,” Int. J. Comput. Methods, vol. 21, no. 10, Apr. 2023, doi: 10.1142/s0219876223410025.

W. Shen and C. Li, “Kantorovich-type convergence criterion for inexact Newton methods,” Applied Numerical Mathematics, vol. 59, no. 7, pp. 1599–1611, Jul. 2009, doi: 10.1016/j.apnum.2008.11.002.

D. Spencer and C. Jerman, “Finding solutions,” in Risk-Led Safety: Evidence-Driven Management, Second Edition, CRC Press, 2019, pp. 71–87. doi: 10.1201/9780367823429-7.

J. M. F. Castillo and W. Okrasinski, “A New Proof of Existence of Solutions for a Class of Nonlinear Volterra Equations,” Journal of Integral Equations and Applications, vol. 6, no. 2, Jun. 1994, doi: 10.1216/jiea/1181075803.

R. Vinodhini, “Modeling Infectious Disease Spread: Differential Equations in Population Dynamics and Epidemiology,” Communications on Applied Nonlinear Analysis, vol. 32, no. 7s, pp. 196–209, Jan. 2025, doi: 10.52783/cana.v32.3377.

I. K. Argyros, “The Newton Kantorovich (NK) Method,” in Convergence and Applications of Newton-type Iterations, Springer New York, 2008, pp. 1–92. doi: 10.1007/978-0-387-72743-1_2.

M. Saffarzadeh, M. Heydari, and G. B. Loghmani, “Convergence analysis of an iterative numerical algorithm for solving nonlinear stochastic Itô-Volterra integral equations with m-dimensional Brownian motion,” Applied Numerical Mathematics, vol. 146, pp. 182–198, Dec. 2019, doi: 10.1016/j.apnum.2019.07.010.

R. Gunasridharan, “Numerical Integration Approximation of Integral over a given Interval,” SSRN Electronic Journal, 2017, doi: 10.2139/ssrn.3081757.