A Generalized Iterative Framework for Solving Nonlinear Integral Equations

Abstract

Nonlinear integral equations play a fundamental role in modeling complex phenomena across mathematics, physics, engineering, and applied sciences. However, obtaining accurate analytical solutions remains challenging, particularly for equations involving nonlinear operators and diverse integration domains. This study develops a generalized relaxation-based iterative framework for the numerical solution of nonlinear Fredholm and Volterra integral equations of the second kind. The proposed approach extends the classical successive approximation method by incorporating an adjustable relaxation parameter into the iterative process, providing additional flexibility in constructing successive approximations. A unified operator formulation is established to accommodate both Fredholm equations with fixed integration limits and Volterra equations with variable upper limits within a consistent mathematical framework. Theoretical convergence is investigated under bounded-kernel, Lipschitz-continuity, and contraction assumptions, establishing sufficient conditions for convergence to a unique solution. The numerical performance of the framework is assessed through three benchmark problems with known exact solutions, encompassing quadratic and cubic nonlinearities. After ten iterations, the maximum absolute errors for the quadratic Fredholm, quadratic Volterra, and cubic Fredholm problems are (1.9189\times10^{-4}), (5.9045\times10^{-11}), and (2.4028\times10^{-6}), respectively. The corresponding mean absolute errors are (9.5943\times10^{-5}), (2.6195\times10^{-11}), and (1.2014\times10^{-6}). These findings demonstrate high numerical accuracy across the investigated benchmark problems, with particularly small approximation errors observed for the quadratic Volterra equation. The numerical results are consistent with the theoretical convergence analysis and demonstrate the applicability of the unified formulation to different nonlinear integral equation structures. The proposed framework offers a mathematically justified and straightforward computational approach for approximating solutions to nonlinear Fredholm and Volterra integral equations under suitable contraction conditions.