Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No.10S Y (2025) 

 

                                                                                                                                            

                                                                                                                              3308  

 

Three-Dimensional Nonlinear Fredholm Integral Equation of The Second 

Kind Solved by Comparing Fibonacci Collocation Method and Hermite-

Galerkin Method 

 

M. H. Ahmed and S. H. Aljahdali 

Taibah University, College of Science, Department of Mathematics, P.O.Box 344, Madinah,3002, Saudi Arabia. 

 

 

Article History: 

Received: 01/01/2025 

Revised: 06/02/2025 

Accepted: 10/03/2025 

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 modelling a wide range of problems 

in applied mathematics, physics, and engineering. 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 structure 

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 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No.10S Y (2025) 

 

                                                                                                                                            

                                                                                                                              3309  

 

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 𝑁, 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 𝑁. 

Keywords: Three-dimensional nonlinear Fredholm integral equations; Fibonacci 

polynomials; Hermite polynomials; Collocation method; Galerkin method 

1. Introduction 

A general form of the three-dimensional nonlinear Fredholm integral equation of the second kind can be written as 

𝑢(𝑥1, 𝑦1, 𝑧1) − 𝜆 ∫ ∫ ∫ 𝑘(𝑥1, 𝑦1, 𝑧1, 𝑠1, 𝑡1, 𝑟1) 𝑄(𝑢(𝑠1, 𝑡1, 𝑟1))𝑑𝑠1 𝑑𝑡1 𝑑𝑟1

𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

= 𝑔(𝑥1, 𝑦1, 𝑧1), 

                              ∀ (𝑥1, 𝑦1 , 𝑧1) ∈ 𝐷                                                                                                                                   (1) 

where 𝑢 is the unknown function. In domain 𝐷 is usually taken as unit cube [0,1]3, 𝑘(𝑥1, 𝑦1, 𝑧1, 𝑠1, 𝑡1, 𝑟1) is the kernel 

function and g is a given source function In domain 𝐷 ×  𝐷 and 𝐷. Depending on the properties of the kernel (continuous, 

symmetric, degenerate, or singular), the complexity of both analysis and computation varies significantly [12]. Several 

numerical approaches have been developed for solving 3D-NLFIEs. Basseem [1] implemented the degenerate kernel 

technique to approximate solutions of three-dimensional nonlinear integral equa tions. Kazemi et al. [8] introduced a Haar 

wavelet-based procedure for solving such equations, while Maleknejad et al. [10] employed Bernstein polynomials for the 

numerical solution of two-dimensional nonlinear Fredholm integral equations, which was later extended to higher 

dimensions. These contributions demonstrate the continuous effort to establish reliable approximation schemes for three-

dimensional nonlinear integral equations. In this study, we employ and compare two polynomial-based approximation 

techniques: the Fibonacci Collocation Method and the Hermite–Galerkin Method. Fibonacci polynomials, derived from 

general izations of the Fibonacci sequence, have shown promising results in solving various types of integral and 

differential equations. Hermite polynomials, with their well-known orthogonality properties, are widely applied in 

Galerkin-type schemes for integral equations. We further compare our results with three other numerical methods available 

in the literature [1,7,8]. 

2. Objectives 

The remainder of this paper is organized as follows. Section 1 presents an introduction and a formal definition of the 

3D-NLFIEs. Section 2 outlines the content and purpose of each section of the study. In section 3 reviews the Fibonacci 

and Hermite polynomials, discusses some of their key properties, and demonstrates the application of the Fibonacci 

collocation method and the Hermite–Galerkin method to the 3D-NLFIEs. Section 4 establishes the existence and 

uniqueness of the solution for the nonlinear three-dimensional Fredholm integral equation and applies the two methods 

to several illustrative examples, followed by a comparison with results reported in previous studies. Finally, Section 5 

provides a critical analysis of the obtained results, highlighting the efficiency and accuracy of the proposed methods 

in comparison with other techniques. 

3. Methods 

3.1  Fibonacci Polynomials.  

[2–5] Leonardo of Pisa, commonly known as Fibonacci, was an Italian mathematician from the 13𝑡ℎ century. In 1202, he 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No.10S Y (2025) 

 

                                                                                                                                                                                                                                                                                                                                                                                          

                                                                                                                                    3310 

posed and solved a problem related to the growth of a rabbit population under idealized conditions. The resulting sequence 

is now known as the Fibonacci numbers: 

0,1,1,2,3,5,8, ⋯ 

This sequence gained widespread attention for its elegant structure and natural occurrence in various scientific phenomena. 

In 1883, Belgian mathematician Eugene Charles Catalan and German mathematician E. Jacobsthal studied broader classes 

of polynomials inspired by Fibonacci numbers. These are known as Fibonacci polynomials, typically denoted by 

𝐹𝑛(𝑥1).This class of polynomials plays a significant role in both theoretical and applied mathematics. Nu- merous properties 

and identities involving 𝐹𝑛(𝑥1)  have been studied due to their wide applicability. Fibonacci polynomials also appear in the 

solution of second-order ordinary differential equations:  

𝑑2𝑦1

𝑑𝑥1
2 − 𝑥1

𝑑𝑦1

𝑑𝑥1

− 𝑦1 = 0, 

whose solution for 𝑛  ∈  𝑁0is: 

𝐹𝑛(𝑥1) =
1

𝛼 − 𝛽
 (𝛼𝑛 − 𝛽𝑛), 

 where  

𝛼 =  
𝑥1 + √𝑥1

2 + 4

2
,   𝛽 =  

𝑥1 − √𝑥1
2 + 4

2
  

An explicit representation of Fibonacci polynomials is given as follows, : 

𝐹𝑛+1(𝑥1) =   ∑ (
𝑛 − 𝑘

𝑘
)

[
𝑛
2

]

𝑘=0

𝑥1
𝑛−2𝑘 , 𝑛 ≥ 0. 

The first few terms are: 

𝐹1(𝑥1) = 1, 

𝐹2(𝑥1) =  𝑥1, 

𝐹3(𝑥1) =  𝑥1
2 + 1, 

𝐹4(𝑥1) =  𝑥1
3 + 2𝑥1, 

𝐹5(𝑥1) =  𝑥1
4 + 3𝑥1

2 + 1, 

 

In what follows, we present some fundamental properties of Fibonacci polynomials: 

 

 

Properties of Fibonacci Polynomials.  

(Recurrence Relation) Fibonacci polynomials satisfy the recurrence relation: 

𝐹𝑛+2(𝑥1) = 𝑥1𝐹𝑛+1(𝑥1) + 𝐹𝑛(𝑥1), 

with: 

𝐹1(𝑥1) = 1,   𝐹2(𝑥1) =  𝑥1. 

Notably: If 𝑥1 = 1 then 𝐹𝑛(1) = 𝐹𝑛+1, if 𝑥1 = 0 then 𝐹2𝑛(0) = 0 for 𝑛 = 1,2, … 

 

(Extrema) The extrema of 𝐹𝑛(𝑥1) over [𝑎, 𝑏] are: 

𝑥1𝑘 = 𝑎 +
𝑏 − 𝑎

𝑛 + 1
𝑘,   𝑘 = 1,2, … , 𝑛 + 1. 

(Generating Function) The generating function is given by: 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No.10S Y (2025) 

 

                                                                                                                                                                                                                                                                                                                                                                                          

                                                                                                                                    3311 

∑ 𝐹𝑛(𝑥1)𝑧1
𝑛

∞

𝑛=0

=
1

1 − 𝑥1𝑧1 − 𝑧1
2. 

(Evenness and Oddness) 𝐹1(𝑥1) is even when 𝑛 is odd, and odd when 𝑛 is even. 

 

3.2 Hermite Polynomials [11,13,17]. 

Hermite polynomials, widely used in both pure and applied mathematics, have a rich historical back ground. While a form 

of these polynomials was introduced by Pierre-Simon Laplace as early as 1810, it was Charles Hermite who later provided 

a formal definition for a broader class of what are now known as generalized Hermite polynomials—an extension that 

remains relatively underexplored compared to the classical case . Hermite polynomials have gained significant importance 

in recent decades due to their applications in quantum mechanics, engineering, physics, and various other scientific 

disciplines. Their mathematical utility and orthogonality properties make them essential tools in the analysis of complex 

systems. The classical Hermite polynomials arise as solutions to the second-order linear differential equation defined on 

the interval (−∞,  ∞), given by:  

𝑑2𝑦1

𝑑𝑥1
2 − 2𝑥1

𝑑𝑦1

𝑑𝑥1

+ 2𝑛𝑦1 = 0, 

Where 𝑛 ≥ 0. The solution corresponding to a given 𝑛 is denoted by 𝐻𝑛(𝑥1) and is referred to as Hermite polynomial of 

degree 𝑛. As with other families of classical orthogonal polynomials, Hermite polynomials can be introduced and derived 

through various equivalent formulations, including differential equations, generating functions, and recurrence relations.In 

what follows, we present some fundamental properties of Hermite polynomials . 

 

Properties of the Hermite Polynomials. (Rodrigues’ Formula) Hermite polynomials 𝐻𝑛(𝑥1) can be expressed using 

Rodrigues’ formula, which involves repeated differentiation of the Gaussian function. The formula is given by:  

𝐻𝑛(𝑥1) = (−1)𝑛𝑒𝑥1
2 𝑑𝑛

𝑑𝑥1
𝑛 (𝑒−𝑥1

2
), 

For 𝑛 = 0,1,2,3, … .   

The first few Hermite polynomials in terms of 𝑥1are: 

 

𝐻0(𝑥1) = 1, 

𝐻1(𝑥1) =  2𝑥1, 

𝐻2(𝑥1) =  4𝑥1
2 − 2, 

𝐻3(𝑥1) =  8𝑥1
3 − 12𝑥1, 

𝐻4(𝑥1) =  16𝑥1
4 − 48𝑥1

2 + 12, 

𝐻5(𝑥1) =  32𝑥1
5 − 160𝑥1

3 + 120, 

𝐻6(𝑥1) =  64𝑥1
6 − 480𝑥1

4 + 720𝑥1
2 − 120, 

 

(Recurrence Relation) The fundamental recurrence relation of 𝐻𝑛(𝑥1) is: 

𝐻𝑛+2(𝑥1) = 2𝑥1𝐻𝑛+1(𝑥1) − 2𝑛𝐻𝑛(𝑥1), 

 

For 𝑛 = 0,1,2,3, …  , which can generate 𝐻𝑛(𝑥1) efficiently and easily with the initial terms: 

𝐻0(𝑥1) = 1, 𝐻1(𝑥1) =  2𝑥1. 

 

(Orthogonality) Hermite polynomials 𝐻𝑛(𝑥1) form an orthogonal set on the interval [−∞, ∞] with respect to the weight 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No.10S Y (2025) 

 

                                                                                                                                                                                                                                                                                                                                                                                          

                                                                                                                                    3312 

function 𝑒−𝑥1
2
. This means: 

⟨𝐻𝑚(𝑥1), 𝐻𝑛(𝑥1)⟩ = ∫ 𝑒−𝑥1
2
𝐻𝑚(𝑥1)𝐻𝑛(𝑥1)𝑑𝑥1

∞

−∞

= {
0 , 𝑚 ≠ 𝑛

2𝑛𝜋
1

2𝑛! ,    𝑚 = 𝑛
,   

(Generating Function) The generating function for 𝐻𝑛(𝑥1) is  

𝑒−𝑧1
2+2𝑧1𝑥1 = ∑ 𝐻𝑛(𝑥1) 

𝑧1
𝑛

𝑛!

∞

𝑛=0

. 

Oddness and Evenness It is clear that the polynomial 𝐻𝑛(𝑥1) is even or odd in 𝑥1, according to whether 𝑛 is even or odd. 

 

3.3 Fibonacci Collocation Method: Consider the which is represented the approximate solution: 

𝑢𝑁(𝑥1, 𝑦1 , 𝑧1) ≈ ∑ ∑ ∑ 𝑐𝑖𝑗𝑘𝐹𝑖(𝑥1)𝐹𝑗(𝑦1)𝐹𝑘(𝑧1)

𝑁+1

𝑘=1

𝑁+1

𝑗=1

𝑁+1

𝑖=1

.                                                                       (2)    

Where 𝐹𝑖  (𝑥1), 𝐹𝑗(𝑦1) and 𝐹𝑘(𝑧1) are Fibonacci polynomial of degree 𝑖, 𝑗, 𝑘, 𝑐𝑖𝑗𝑘  the coefficients are determined choose the 

collocation points (𝑥𝑛 , 𝑦𝑚, 𝑧𝑙) by dividing the intervals [𝑎, 𝑏], [𝑐, 𝑑] and [𝑒, 𝑓] into 𝑁 + 1, respectively: 

𝑥1𝑛 = 𝑎 + 
(𝑏 − 𝑎) 

𝑁 + 1
𝑛, 𝑦1𝑚 = 𝑐 + 

(𝑑 − 𝑐)

𝑁 + 1
 𝑚, 𝑧1𝑙 = 𝑒 +  

(𝑓 − 𝑒)

𝑁 + 1
 𝑙, 𝑛, 𝑚, 𝑙 = 1,2, . . . , 𝑁 + 1.                    (3) 

First substituting Eq.(2) into three-dimensional nonlinear Fredholm integral equations (1) at use the collocation points 

(𝑥𝑛, 𝑦𝑚 , 𝑧𝑙): 

∑ ∑ ∑ 𝑐𝑖𝑗𝑘𝐹𝑖(𝑥1)𝐹𝑗(𝑦1)𝐹𝑘(𝑧1)

𝑁+1

𝑘=1

𝑁+1

𝑗=1

𝑁+1

𝑖=1

 

− 𝜆 ∫ ∫ ∫ 𝑘(𝑥1, 𝑦1, 𝑧1, 𝑠1, 𝑡1, 𝑟1) 𝑄
𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

(∑ ∑ ∑ 𝑐𝑖𝑗𝑘𝐹𝑖(𝑥1)𝐹𝑗(𝑦1)𝐹𝑘(𝑧1)

𝑁+1

𝑘=1

𝑁+1

𝑗=1

𝑁+1

𝑖=1

. ) 𝑑𝑠1 𝑑𝑡1 𝑑𝑟1

= 𝑔(𝑥1, 𝑦1, 𝑧1).                                                                                                                                                (4) 

Then the coefficients 𝑐𝑖𝑗𝑘  are determined by solving the nonlinear system of matrix iteratively: 

𝐴  ⋅ 𝑐 = 𝐵 

Where 𝐴 is a matrix of size (𝑁 + 1)3 × (𝑁 + 1)3, B is a matrix of size (𝑁 + 1)3 × 1 and 𝑐 is a matrix of size (𝑁 + 1)3 × 1 

defined by [𝑐111 𝑐112 … 𝑐(𝑁+1)3  ]
𝑇
.  Thus we have obtained a nonlinear system of equations which contains (𝑁 + 1)2  

unknown coefficients. solving this system to obtain the values of these coefficients, we arrive at the approximate solution 

𝑢𝑁(𝑥1 , 𝑦1, 𝑧1). 

3.4  Hermite–Galerkin Method. .Consider the which is represented the approximate solution: 

𝑢𝑁(𝑥1, 𝑦1, 𝑧1) ≈ ∑ ∑ ∑ 𝑐𝑖𝑗𝑘𝐻𝑖(𝑥1)𝐻𝑗(𝑦1)𝐻𝑘(𝑧1)

𝑁+1

𝑘=1

𝑁+1

𝑗=1

𝑁+1

𝑖=1

.                                                                               (5)      

Where 𝐻𝑖  (𝑥1), 𝐻𝑗(𝑦1) and 𝐻𝑘(𝑧1) are Hermite polynomial, 𝑐𝑖𝑗𝑘  the coefficients are determined. 

First substituting Eq.(5) into three-dimensional nonlinear Fredholm Integral Equations Eq.(1) and applying the Galerkin 

condition leads to the following equation: 

∫ ∫ ∫ ∑ ∑ ∑ 𝑐𝑖𝑗𝑘𝐻𝑖(𝑥1)𝐻𝑗(𝑦1)𝐻𝑘(𝑧1)𝐻𝑞(𝑥1)𝐻𝑝(𝑦1)𝐻𝑣(𝑧1) 𝑑𝑥1𝑑𝑦1𝑑𝑧1

𝑁+1

𝑘=1

𝑁+1

𝑗=1

𝑁+1

𝑖=1

𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

− ∫ ∫ ∫ (𝑊)𝐻𝑞(𝑥1)𝐻𝑝(𝑦1)𝐻𝑣(𝑧1) 𝑑𝑥1 𝑑𝑦1 𝑑𝑧1

𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

= ∫ ∫ ∫ 𝑔(𝑥1, 𝑦1, 𝑧1)𝐻𝑞(𝑥1)𝐻𝑝(𝑦1)𝐻𝑣(𝑧1) 𝑑𝑥1 𝑑𝑦1 𝑑𝑧1

𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

                                                                  (6) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No.10S Y (2025) 

 

                                                                                                                                                                                                                                                                                                                                                                                          

                                                                                                                                    3313 

where  

𝑊 = 𝜆 ∫ ∫ ∫ 𝑘(𝑥1, 𝑦1 , 𝑧1, 𝑠1, 𝑡1, 𝑟1)
𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

 𝑄 (∑ ∑ ∑ 𝑐𝑖𝑗𝑘𝐻𝑖(𝑥1)𝐻𝑗(𝑦1)𝐻𝑘(𝑧1).

𝑁+1

𝑘=1

𝑁+1

𝑗=1

𝑁+1

𝑖=1

 ) 𝑑𝑠1 𝑑𝑡1 𝑑𝑟1 

For all 𝑞, 𝑝, 𝑣 = 1, 2, … , 𝑁 + 1. The coefficients 𝑐𝑖𝑗𝑘  are determined by solving the nonlinear system of matrix iteratively: 

𝐴  ⋅ 𝑐 = 𝐵 

Where 𝐴 is a matrix of size (𝑁 + 1)3 × (𝑁 + 1)3, 𝐵 is a matrix of size (𝑁 + 1)3 × 1 and 𝑐 is a matrix of size (𝑁 + 1)3 × 1 

defined by [𝑐111 𝑐112  … 𝑐(𝑁+1)3  ]
𝑇
.Thus, we have obtained a nonlinear system of equations which contains (𝑁 + 1)2  

unknown coefficients. solving this system to obtain the values of these coefficients, we arrive at the approximate solution 

𝑢𝑁(𝑥1 , 𝑦1, 𝑧1). 

4. Results 

4.1 Existence and Uniqueness of Solution for Three-Dimensional Nonlinear Fredholm Integral 

Equation 

We consider the nonlinear Fredholm integral equation in three dimensions Eq(1): 

𝑢(𝑥1, 𝑦1, 𝑧1) − 𝜆 ∫ ∫ ∫ 𝑘(𝑥1, 𝑦1, 𝑧1, 𝑠1, 𝑡1, 𝑟1) 𝑄(𝑢(𝑠1, 𝑡1, 𝑟1))𝑑𝑠1 𝑑𝑡1 𝑑𝑟1

𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

= 𝑔(𝑥1, 𝑦1, 𝑧1), 

Where the domain 𝐷 = [𝑎, 𝑏] × [𝑐, 𝑑] × [𝑒, 𝑓], and 𝑄 is a nonlinear function. 

Theorem 1. Let 𝑋 = 𝐶([𝑎, 𝑏] × [𝑐, 𝑑] × [𝑒, 𝑓]) be the Banach space of continuous functions 𝑢(𝑥1, 𝑦1, 𝑧1), equipped with 

the supremum norm: 

‖𝑢‖ = 𝑠𝑢𝑝(𝑥1,𝑦1,𝑧1)∈[𝑐,𝑑]×[𝑒,𝑓]|𝑢(𝑥1, 𝑦
1
, 𝑧1)|.  

Assume that: 

(1) The kernel 𝑘(𝑥1, 𝑦1, 𝑧1, 𝑠1, 𝑡1, 𝑟1) is continuous on 𝐷 × 𝐷, 

(2) The function 𝑄: 𝑅 ⟶ 𝑅 satisfies a Lipschitz condition: 

|𝑄(𝑢) − 𝑄(𝑣)| ≤ 𝐿|𝑢 − 𝑣|        ∀𝑢, 𝑣 ∈ 𝑅, 

(3) The constant 𝐶 = 𝐿𝑀 < 1, where 

𝑀 = 𝑠𝑢𝑝(𝑥1,𝑦1,𝑧1)∈D ∫ ∫ ∫ |𝑘(𝑥1, 𝑦1, 𝑧1, 𝑠1, 𝑡1, 𝑟1)|𝑑𝑠1 𝑑𝑡1𝑑𝑟1.
𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

  

 

Then, the integral equation has a unique solution 𝑢∗ ∈ 𝑋. 

Proof. Define the operator 𝑇 ∶ 𝑋 → 𝑋 by 

(𝑇𝑢)(𝑥1, 𝑦1, 𝑧1) = 𝑔(𝑥1, 𝑦1, 𝑧1) + ∫ ∫ ∫ 𝑘(𝑥1, 𝑦1 , 𝑧1, 𝑠1, 𝑡1, 𝑟1) 𝑄(𝑢(𝑠1, 𝑡1, 𝑟1))𝑑𝑠1 𝑑𝑡1 𝑑𝑟1

𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

 

Let 𝑢, 𝑣 ∈ 𝑋. Then  

|(𝑇𝑢)(𝑥1, 𝑦1, 𝑧1) − (𝑇𝑣)(𝑥1, 𝑦1, 𝑧1)|

= |∫ ∫ ∫ 𝑘(𝑥1, 𝑦1 , 𝑧1, 𝑠1, 𝑡1, 𝑟1)
𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

[ 𝑄(𝑢(𝑠1, 𝑡1, 𝑟1)) −  𝑄(𝑣(𝑠1, 𝑡1, 𝑟1))]𝑑𝑠1 𝑑𝑡1 𝑑𝑟1| 

≤ 𝐿 ∫ ∫ ∫ |𝑘(𝑥1, 𝑦1, 𝑧1, 𝑠1, 𝑡1, 𝑟1)|
𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

⋅ |𝑢(𝑠1, 𝑡1, 𝑟1) − 𝑣(𝑠1, 𝑡1, 𝑟1)|𝑑𝑠1 𝑑𝑡1𝑑𝑟1 

≤ 𝐿 ∫ ∫ ∫ |𝑘(𝑥1, 𝑦1, 𝑧1, 𝑠1, 𝑡1, 𝑟1)|
𝑓

𝑒

𝑑

𝑐

𝑏

𝑎

𝑑𝑠1 𝑑𝑡1𝑑𝑟1 ⋅ ‖𝑢 − 𝑣‖. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No.10S Y (2025) 

 

                                                                                                                                                                                                                                                                                                                                                                                          

                                                                                                                                    3314 

Taking the supremum over (𝑥1, 𝑦1, 𝑧1) ∈ 𝐷, 𝑤𝑒 𝑔𝑒𝑡: 

‖𝑇𝑢 − 𝑇𝑣‖ ≤ 𝐿𝑀‖𝑢 − 𝑣‖ = 𝐶‖𝑢 − 𝑣‖. 

Since 𝐶 < 1, the operator T is a contraction. By Banach's Fixed point Theorem, there exists a unique fixed point 𝑢∗ ∈ 𝑋  

Such that 𝑇𝑢∗ = 𝑢∗, i.e, 𝑢∗is the unique solution of the integral equation. □ 

4.2 Numerical examples 

We present the numerical results for the three-dimensional nonlinear Fredholm integral equations. Tables containing the 

approximate solution values are provided, along with three-dimensional graphical representations of the solutions obtained 

using MATLAB. The results are derived by employing the proposed Fibonacci and Hermite-Galerkin methods and further 

compared with the other methods discussed earlier. 

 

 

 

 

 

 

 

 

Example1. Consider the following 3D-NFIE [7,8]: 

𝑢(𝑥1, 𝑦1, 𝑧1) = 𝑔(𝑥1, 𝑦1, 𝑧1) + ∫ ∫ ∫
1

100

1

0

1

0

1

0

 𝑥1
2𝑠1(𝑡1

2 + 𝑦1)𝑟1𝑧1𝑢2(𝑠1, 𝑡1, 𝑟1)  𝑑𝑠1 𝑑𝑡1 𝑑𝑟1,      (𝑥1, 𝑦1, 𝑧1) ∈ [0,1]3     (7) 

Where 𝑔(𝑥1, 𝑦1, 𝑧1) = 𝑥1
2𝑦1

2𝑧1 −
1

16800
𝑥1

2𝑧1 −
1

12000
𝑥1

2𝑦1𝑧1, and exact solution is 𝑢(𝑥1, 𝑦1, 𝑧1) = 𝑥1
2𝑦1

2𝑧1. 

The comparison of absolute errors of equation(7) for different values of (𝑥1, 𝑦1, 𝑧1) ,between Fibonacci collocation (FC), 

Hermite–Galerkin (H-G) methods with N=2, the Successive Approximations Method at N=10 [8], and the Haar Wavelet 

Method at 2N=8 [7] ,are presented in Table (1). Figures (1) and (2) show the absolute error distribution obtained by both 

methods. 

TABLE 1. Numerical results for Example 1, 𝑧1 = 0.5 

(𝑥1, 𝑦1, 𝑧1) Fibonacci collocation 

method 𝑁 = 2 

Hermite-Galerkin 

method 𝑁 = 2 

Method of [8] with 

𝑁 = 10 

Method of [7] with 

2𝑁 = 8 

(0,0,0) 1.4727 × 10−11 8.729 × 10−9 0. 0. 

(0.1,0.1,0.1) 3.9781 × 10−12 2.3279 × 10−9 1.549 × 10−9 3.51 × 10−9 

(0.2,0.2,0.2) 7.7772 × 10−13 5.6042 × 10−11 1.345 × 10−8 3.075 × 10−8 

(0.3,0.3,0.3) 3.7868 × 10−13 4.2694 × 10−10 4.9181 × 10−8 1.127 × 10−7 

(0.4,0.4,0.4) 6.0653 × 10−13 1.6048 × 10−9 1.2552 × 10−7 2.884 × 10−7 

(0.5,0.5,0.5) 1.1211 × 10−12 2.2105 × 10−9 2.6264 × 10−7 6.0474 × 10−7 

(0.6,0.6,0.6) 2.6918 × 10−12 1.7484 × 10−9 4.8404 × 10−7 1.116 × 10−6 

(0.7,0.7,0.7) 6.4935 × 10−12 6.5454 × 10−10 8.1658 × 10−7 1.886 × 10−6 

(0.8,0.8,0.8) 1.3411 × 10−11 3.3782 × 10−11 1.2905 × 10−6 2.9856 × 10−6 

(0.9,0.9,0.9) 2.3367 × 10−11 6.1231 × 10−10 1.9393 × 10−6 4.4936 × 10−6 

(1.0,1.0,1.0) 3.4658 × 10−11 2.3253 × 10−9 2.8000 × 10−6 6.4936 × 10−6 

. 

Figure 1. Absolute error of Example 1 by FC method, 𝑁 = 2, and 𝑧1 = 0.5. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No.10S Y (2025) 

 

                                                                                                                                                                                                                                                                                                                                                                                          

                                                                                                                                    3315 

 

 

 

 

 

 

 

 

 

Example 2. Consider the following 3D-NFIE [8]: 

𝑢(𝑥1, 𝑦1, 𝑧1) = 𝑔(𝑥1, 𝑦1, 𝑧1) +  ∫ ∫ ∫ 𝑥1
2

1

0

1

0

1

0

𝑦1𝑧1𝑡1𝑟1 𝑠𝑖𝑛(𝑠1)𝑢3(𝑠1, 𝑡1, 𝑟1)  𝑑𝑠1 𝑑𝑡1 𝑑𝑟1,      (𝑥1, 𝑦1, 𝑧1) ∈ [0,1]3             (8) 

Where 𝑔(𝑥1, 𝑦1, 𝑧1) = 𝑦1𝑧1𝑐𝑜𝑠(𝑥1)  −
1

100
𝑥1

2𝑦1𝑧1 −
1

100
𝑥1

2𝑦1𝑧1𝑐𝑜𝑠4(1) , and the exact solution is 

 𝑢(𝑥1, 𝑦1 , 𝑧1) = 𝑦1 + 𝑧1𝑐𝑜𝑠 (𝑥1). 

Table (2) presents the absolute error of equation (8) for different values of  (𝑥1, 𝑦1, 𝑧1), between Fibonacci collocation 

(FC), Hermite–Galerkin (HG) methods with N=6, the Successive Approximations Method at N=10  [8] Figures (3)–(4) 

illustrate the absolute errors obtained by both methods. 

TABLE 2. Numerical results for Example 2, 𝑧1 = 0.5 

(𝑥1, 𝑦1, 𝑧1) Fibonacci collocation 

method 𝑁 = 4 

Fibonacci collocation 

method 𝑁 = 6 

Hermite-Galerkin 

method 𝑁 = 4 

Hermite-Galerkin 

method 𝑁 = 6 

Method of [8] with 

𝑁 = 10 

(0,0,0) 3.3736 × 10−10  2.5886 × 10−7  2.5387 × 10−2  1.0534 × 10−1  0.  

(0.1,0.1,0.1) 3.863 × 10−7  2.6904 × 10−11  1.367 × 10−3  2.822 × 10−3  1.9452 × 10−9  

(0.2,0.2,0.2) 2.6101 × 10−9  2.9308 × 10−9  1.5613 × 10−3  6.1957 × 10−3  3.1124 × 10−8  

(0.3,0.3,0.3) 3.9605 × 10−7  1.2532 × 10−8  2.1344 × 10−3  2.6484 × 10−3  1.5756 × 10−7  

(0.4,0.4,0.4) 4.1756 × 10−8  3.9765 × 10−8  1.5719 × 10−2  2.2452 × 10−2  4.9799 × 10−7  

(0.5,0.5,0.5) 6.1579 × 10−7  9.7949 × 10−8  2.4358 × 10−2  4.4104 × 10−2  1.2158 × 10−6  

(0.6,0.6,0.6) 2.1135 × 10−7  2.0196 × 10−7  1.4499 × 10−2  1.369 × 10−2  2.521 × 10−6  

(0.7,0.7,0.7) 6.6599 × 10−7  3.7446 × 10−7  6.0005 × 10−4  2.5634 × 10−3  4.6706 × 10−6  

(0.8,0.8,0.8) 6.6795 × 10−7  6.4132 × 10−7  3.7639 × 10−3  8.2663 × 10−3  7.9678 × 10−6  

(0.9,0.9,0.9) 5.3352 × 10−6  1.0188 × 10−6  5.2081 × 10−3  8.7402 × 10−3  1.2763 × 10−5  

(1.0,1.0,1.0) 1.6307 × 10−6  1.5608 × 10−6  7.4386 × 10−3  1.1347 × 10−1  1.9452 × 10−5   

 

 

Figure 2. Absolute error of Example 1 by HG method, 𝑁 = 2, and 𝑧1 = 0.5. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No.10S Y (2025) 

 

                                                                                                                                                                                                                                                                                                                                                                                          

                                                                                                                                    3316 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 3. Consider the following 3D-NFIE [7]: 

𝑢(𝑥1, 𝑦1, 𝑧1) = 𝑔(𝑥1, 𝑦1, 𝑧1) + ∫ ∫ ∫ 𝑥1
2

1

0

1

0

1

0

𝑡1 𝑐𝑜𝑠(𝑧1) 𝑠𝑖𝑛(𝑠1)𝑢3(𝑠1, 𝑡1, 𝑟1)  𝑑𝑠1 𝑑𝑡1 𝑑𝑟1,      (𝑥1, 𝑦1, 𝑧1) ∈ [0,1]3,        (9) 

where 𝑔(𝑥1, 𝑦1, 𝑧1) = 𝑦1𝑧1𝑐𝑜𝑠2(𝑥1)  +
1

140
𝑥1

2 𝑐𝑜𝑠 (𝑧1)(𝑐𝑜𝑠7(1) − 1) , and the exact solution is 

 𝑢(𝑥1, 𝑦1 , 𝑧1) = 𝑦1 + 𝑧1𝑐𝑜𝑠2(𝑥1). 

Table (3) shows the comparison of the absolute error of equation (9) using Hermite–Galerkin (HG) and Fibonacci 

collocation (FC) methods for various values of (𝑥1, 𝑦1 , 𝑧1). Figures (5)&(6) illustrate the  absolute errors obtained by both 

methods. A comparison with the Haar Wavelet Method at 2𝑁 = 32 [7]. 

TABLE 3. Numerical results for Example 3, 𝑧1 = 0.5 

(𝑥1, 𝑦1, 𝑧1) Fibonacci collocation 

method 𝑁 = 6 

Hermite-Galerkin 

method 𝑁 = 6 

Method of [7] with 

2𝑁 = 32 

(0,0,0) 6.3588 × 10−7  2.7131 × 10−1  0.  

(0.1,0.1,0.1) 1.0895 × 10−7  7.1995 × 10−3  1.1244 × 10−8 

(0.2,0.2,0.2) 2.9425 × 10−9  1.5717 × 10−2  4.4304 × 10−8 

Figure 3. Absolute error of Example 2 by FC method, 𝑁 = 6, and 𝑧1 = 0.5. 

Figure 4. Absolute error of Example 2 by HG method, 𝑁 = 6, and 𝑧1 = 0.5. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No.10S Y (2025) 

 

                                                                                                                                                                                                                                                                                                                                                                                          

                                                                                                                                    3317 

(0.3,0.3,0.3) 5.5898 × 10−8  6.4034 × 10−3  9.7169 × 10−8 

(0.4,0.4,0.4) 6.3907 × 10−8  5.7392 × 10−2  1.6654 × 10−7 

(0.5,0.5,0.5) 1.4021 × 10−8  1.1352 × 10−1  2.4794 × 10−7 

(0.6,0.6,0.6) 1.2999 × 10−7  3.6224 × 10−2  3.3578 × 10−7 

(0.7,0.7,0.7) 1.9685 × 10−7  3.5896 × 10−3  4.2354 × 10−7 

(0.8,0.8,0.8) 4.7301 × 10−8  1.4785 × 10−2  5.0391 × 10−7 

(0.9,0.9,0.9) 8.6554 × 10−7  1.0595 × 10−2  5.6902 × 10−7 

(1.0,1.0,1.0) 2.3595 × 10−7  3.1193 × 10−1  6.1061 × 10−7 

 

 

 

 

 

5. Discussion 

In this article, we have obtained the numerical solutions of 3D-NFIEs using two different polynomial-based approaches: 

the Fibonacci collocation method and the Hermite–Galerkin method. Adetailed comparison between the two techniques 

has been carried out, and the results demonstrate that the Fibonacci approach provides better accuracy than the Hermite–

Galerkin method. Furthermore, when compared with other methods reported in the literature [7,8]. In the third example, 

the Haar wavelet method reaches its maximum error of 1.1244 × 10−8 at 2N=32 [8],  

whereas the Fibonacci method yields a significantly smaller error of  2.9425 × 10−9  at 𝑁 = 6, In the second example, the 

Fibonacci method also achieves an error of 2.6904 × 10−11  at 𝑁 = 6, which is lower than that obtained by the Successive 

Approximations method, which records an error of 1.9452 × 10−9  at 𝑁 = 10. In the first example, the Fibonacci method 

outperforms both methods, achieving the smallest error 3.7868 × 10−13 at 𝑁 = 2. The Fibonacci method still outperforms 

them, achieving higher accuracy with a smaller number of basis functions 𝑁. This highlights the efficiency and 

effectiveness of the Fibonacci collocation scheme in solving such classes of integral equations. All computations and 

implementations are performed using MATLAB. 

References 

[1] Basseem, B. (2015). Numerical solution of three-dimensional nonlinear integral equations using degenerate kernel 

technique. University Journal of Integral Equations, 3, 6–15.  

Figure 5. Absolute error of Example 3 

by FC method, 𝑁 = 6, and 𝑧1 = 0.5. 

 

Figure 6. Absolute error of Example 3 

by HG method, 𝑁 = 6, and 𝑧1 = 0.5. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No.10S Y (2025) 

 

                                                                                                                                                                                                                                                                                                                                                                                          

                                                                                                                                    3318 

[2] El-Mikkawy, M., & Sogabe, T. (2010). A new family of k-Fibonacci numbers. Applied Mathematics and 

Computation, 215(12), 4456–4461. 

[3] Falcon, S., & Plaza, ´ A. (2007). The k-Fibonacci sequence and the Pascal 2-triangle. Chaos, Solitons & Fractals, 

33(1), 38–49. 

[4] Falcon, S., & Plaza, A. (2009). On k-Fibonacci sequences and polynomials and their derivatives. Chaos, Solitons 

& Fractals, 39(3), 1005–1019. 

[5] Grimm, R. E. (1973). The autobiography of Leonardo Pisano. The Fibonacci Quarterly, 11(1), 99–104. 

[6] Hursan, G., Zhdanov, M. S. (2002). Contraction integral equation method in three-dimensional electromagnetic 

modeling. Radio Science, 37(6), 1-13. 

[7] Kazemi, M., Torkashvand, V., Ezzati, R. (2021). A new method for solving three-dimensional nonlinear Fredholm 

integral equations by Haar wavelet. International Journal of Nonlinear Analysis and Applications, 12(2), 115-133.  

[8]  Kazemi, M. (2021). Approximating the solution of three-dimensional nonlinear Fredholm integral equations. 

Journal of Computational and Applied Mathematics, 395, 113590. 

[9] Mahdy, A. M., Nagdy, A. S., Hashem, K. M., Mohamed, D. S. (2023). A computational technique for solving 

three dimensional mixed Volterra–Fredholm integral equations. Fractal and Fractional, 7(2), 196.  

[10] Maleknejad, K., Almasieh, H., & Hashemizadeh, E. (2012). Bernstein polynomials for solving a class of nonlinear 

two-dimensional Fredholm integral equations. International Journal of Nonlinear Analysis and Applications, 

12(2), 115–133.  

[11] Ogola, J. A. (2020). Numerical solutions of Fredholm integral equations of the second kind. Research Report in 

Math ematics, No. 30, University of Nairobi.  

[12] Rahman, M. (2007). Integral equations and their applications. WIT Press.  

[13] Rahman, M. M. (2013, July). Numerical solutions of Volterra integral equations using Galerkin method with 

Hermite polynomials. In Proceedings of the International Conference on Applied Mathematics and Computational 

Methods in Engineering.  

[14] Sadri, K., Amini, A., Cheng, C. (2017). Low cost numerical solution for three-dimensional linear and nonlinear 

integral equations via three-dimensional Jacobi polynomials. Journal of Computational and Applied Mathematics, 

319, 493-513.  

[15] Tsitsas, N. L. (2021). Second-kind Fredholm integral-equation analysis of scattering by layered dielectric gratings. 

IET Microwaves, Antennas Propagation, 15(10), 1194-1205.  

[16] Wongyat, T., Sintunavarat, W. (2017). The existence and uniqueness of the solution for nonlinear Fredholm and 

Volterra integral equations together with nonlinear fractional differential equations via w-distances. Advances in 

Dif ference Equations, 2017(1), 211.  

[17] Yal¸cınba¸s, S., & Aynig¨ul, M. (2011). Hermite series solutions of linear Fredholm integral equations. 

Mathematical and Computational Applications, 16(2), 497–506.