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.

中文翻译：

---

Hammerstein型非线性积分方程高效离散化技术分析

 目的

本研究重点研究采用离散化技术的第二类非线性 Volterra-Fredholm-Hammerstein 积分方程（NVFHIE）的数值解。本文的目的是开发一种高效、准确的方法来求解 NVFHIE，这对于对具有记忆和累积效应的系统进行建模、将过去和现在的影响与非线性相互作用相结合至关重要。它们广泛应用于控制理论、群体动力学和物理学。这些方程对于解决复杂的现实问题至关重要。

设计/方法论/途径

以皮卡德迭代法为关键技术，证明了方程解的存在性和唯一性。使用梯形离散化方法是对解进行数值逼近的选择方法，从而产生代数方程的非线性系统。梯形方法 (TM) 在离散 Grönwall 不等式的应用的支持下，对解表现出二次收敛性。引入一种新的 Grönwall 不等式来证明所考虑方法的收敛性。这种方法可以对方程的行为进行详细分析，并有助于开发稳健的求解方法。

 发现

数值结果最终表明，所提出的方法在求解 NVFHIE 方面非常有效，显着减少了计算量。数值例子和比较强调了该方法的实用性、有效性和可靠性，证实了其与参考方法相比的优异性能。

 原创性/价值

与依赖多种方法组合来解决复杂问题（尤其是非线性积分方程）的不同方面的现有方法不同，当前的方法提供了重要的单一方法解决方案，提供了解决整个问题的综合方法。此外，本工作介绍了所考虑的积分方程的第一个数值方法，这在现有文献中尚未被探索过。据作者所知，这项工作是第一个解决这个方程的工作，为未来的研究和应用提供了基础性的贡献。这种创新策略不仅简化了计算过程，而且还提供了对问题动态的更全面的理解。