Three-Dimensional Nonlinear Fredholm Integral Equation of The Second Kind Solved by Comparing Fibonacci Collocation Method and Hermite-Galerkin Method

Abstract

In this article, we present the numerical solutions of three-dimensional nonlinear Fredholm integral equations using two different polynomial-based approaches. The first method employs Fibonacci polynomials within the collocation method, while the second method applies Hermite polynomials through the Galerkin projection technique. A comparative study between the two proposed approaches is carried out, and the obtained results are also contrasted with those available in related works. The numerical results are illustrated in a series of tables and figures, demonstrating the efficiency and accuracy of the methods.

Introduction: The three-dimensional nonlinear Fredholm integral equations (3D-NLFIEs) of the second kind are fundamental tools in modeling a wide range of problems in applied mathematics, physics, and engineer- ing. These equations naturally arise in mathematical models of electromagnetic scattering, quantum mechanics, heat transfer, and population dynamics [6, 12, 14]. In three dimensions, the nonlinear struc- ture combined with the multi-variable kernel makes analytical solutions intractable, which emphasizes the importance of developing efficient and accurate numerical methods. It is worth mentioning that several recent studies have applied second-kind Fredholm integral equations to advanced fields such as electromagnetic wave propagation in dielectric gratings. For example, the paper [15] presents a volume integral equation (VIE) formulation combined with the Galerkin method, which enables stable and ac- curate solutions for the scattering phenomena in multilayer dielectric gratings. This work highlights the significance of numerical techniques based on integral equations in addressing complex and practical problems in physics and applied mathematics, which is in line with our research focus on developing numerical approaches for solving nonlinear Fredholm integral equations [9].

Objectives: This paper consists of five sections: an introduction and definition of the 3D-NLFIEs, a statement of objectives, a review of Fibonacci and Hermite polynomials with the application of the Fibonacci collocation and Hermite–Galerkin methods, results including existence and uniqueness proofs and numerical comparisons, and finally a discussion analysing the accuracy and efficiency of the methods.

Methods: We employed the Fibonacci collocation method and the Hermite–Galerkin method to solve 3D-NLFIEs.

Results: Established the existence and uniqueness of the solution for the 3D-NLFIE and obtained numerical solutions using MATLAB by applying both methods. A comparative analysis was then conducted between the two methods, and the results were reported in previous studies [7,8]

Conclusions: The results demonstrated that the Fibonacci collocation method consistently outperformed the Hermite–Galerkin method for different values of , achieving higher accuracy and lower numerical errors across most test points. This highlights the effectiveness of the Fibonacci approach in improving convergence and solution quality compared to the Hermite–Galerkin method under the same number of