Analysis of efficient discretization technique for nonlinear integral equations of Hammerstein type

Abstract

Purpose

This study focuses on investigating the numerical solution of second-kind nonlinear Volterra–Fredholm–Hammerstein integral equations (NVFHIEs) by discretization technique. The purpose of this paper is to develop an efficient and accurate method for solving NVFHIEs, which are crucial for modeling systems with memory and cumulative effects, integrating past and present influences with nonlinear interactions. They are widely applied in control theory, population dynamics and physics. These equations are essential for solving complex real-world problems.

Design/methodology/approach

Demonstrating the solution’s existence and uniqueness in the equation is accomplished by using the Picard iterative method as a key technique. Using the trapezoidal discretization method is the chosen approach for numerically approximating the solution, yielding a nonlinear system of algebraic equations. The trapezoidal method (TM) exhibits quadratic convergence to the solution, supported by the application of a discrete Grönwall inequality. A novel Grönwall inequality is introduced to demonstrate the convergence of the considered method. This approach enables a detailed analysis of the equation’s behavior and facilitates the development of a robust solution method.

Findings

The numerical results conclusively show that the proposed method is highly efficacious in solving NVFHIEs, significantly reducing computational effort. Numerical examples and comparisons underscore the method’s practicality, effectiveness and reliability, confirming its outstanding performance compared to the referenced method.

Originality/value

Unlike existing approaches that rely on a combination of methods to tackle different aspects of the complex problems, especially nonlinear integral equations, the current approach presents a significant single-method solution, providing a comprehensive approach to solving the entire problem. Furthermore, the present work introduces the first numerical approaches for the considered integral equation, which has not been previously explored in the existing literature. To the best of the authors’ knowledge, the work is the first to address this equation, providing a foundational contribution for future research and applications. This innovative strategy not only simplifies the computational process but also offers a more comprehensive understanding of the problem’s dynamics.