EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 1, 2023, 304-313
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Mathematical Eigenfunctions analysis for 2nd Kind of
Fredholm Integral equations

Lamiaa Hazim Al-Taee1,∗, Ahmed A. Mohammed Fawze2

1 Mathematics Department, College of Education of Pure Sciences, University of Mosul,
Mosul, Iraq
2 Mathematics Department, College of Computer Science and Mathematics,
University of Mosul, Mosul, Iraq

Abstract. The current paper is a trial to investigate the Eigen functions appears through math-
ematical treatment of 2nd kind of Fredholm integral equations by making use a new developed an
Inverse Iterative Numerical Scheme (IINS). To test the applicability and accuracy of the proposed
IINS, two numerical examples were solved and compared with previous results. The discussion of
the results gave a good coincides up to some decimal places of results with the available ones.

2020 Mathematics Subject Classifications: 45Bxx, 45B05

Key Words and Phrases: Fredholm Integral equations, Eigen functions analysis, Iterative
methods

1. Introduction

Integral equations are a very complicated and old branch of mathematics, and solutions
for such equations need an appearing effort even after numerical methods and approaches
became excellent tools for accurate results. Integrals equations spread among wide range
of topics and applications exist in science, engineering and technology, e.g., continuum me-
chanics, potential theory, geo-physics, electricity, gas kinetic theory [1, 4, 5, 13, 16]. Also
in biology, theory of renewal energy, quantum mechanics, theory of radiation, theory of op-
timization, economic mathematics, population, the theory queuing, some medical applica-
tions, transport phenomena, acoustic applications, phase change problems [2, 3, 9, 11, 14].
When the various sciences became complicated as a result of the interactions between them
and developed greatly, scientists began to study natural phenomena, whether they were
physical, and to explain these phenomena and find different solutions to them, whether
analytical or numerical. Integrative equations have their own importance among the dif-
ferent types of mathematical sciences such as partial and differential equations, functional

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4647

Email addresses: blumiaa.h.s@uomosul.edu.iq (L. H. Al-Taee),
aahmedamer68@uomosul.edu.iq (A. A. M. Fawze)

https://www.ejpam.com 304 © 2023 EJPAM All rights reserved.



L. H. Al-Taee, A. A. M. Fawze / Eur. J. Pure Appl. Math, 16 (1) (2023), 304-313 305

analysis, the theory of operators, transformers and special functions. Therefore, it can
be said that there is no science from the different sciences except that the integrative
equations play a prominent role in it. Therefore, we find that many researchers have
been able to devise many different ways to solve the integrative equations, whether the
nucleus of the integrative equation is connected or unconnected and other methods that
are represented as ways Analytical or numerical. In one of the papers, a new analyti-
cal method was presented for solving systems of linear differential integrals, which is a
new and powerful method through which effective recursive relationships were obtained
to solve these systems, and it is an appropriate method to use it as an alternative to the
mathematical methods used in such problems. Fredholm integral equation is one of the
most important integral equations, and researchers look to the basis of integral equations
as a transformation of some points over given vector space that have specific criteria by
making use of certain specified operators to other points located in the same space [15].
There are wide ranges of numerical methods for implementing of Fredholm 2nd type, some
of these methods, the β-wavelet method [7], moments method based on β-wavelets [18],
and variational iteration method [12]. Some of the numerical trials for solving Fredholm
integral equation of 2nd kind that proposed by Maleknejad et al., [6]. The Homotopy
perturbation method is one of the iterative numerical methods that gave good approxi-
mate solutions for the problem underhand. In the present paper, certain type of integral
equations is analyzed. The analysis is to approximate Eigenvalues of 2nd type, Fredholm
equation using a proposed inverse numerical iterative scheme. The current paper is a trial
to investigate the Eigen functions appears through mathematical treatment of 2nd kind
of Fredholm integral equations by making use a new developed an (IINS). To test the
applicability and accuracy of the proposed NIS, two numerical examples were solved and
compared with previous results. The discussion of the results gave a good coincides up to
some decimal places of results with the available ones.

2. Mathematical Formulation

Starting by the so called (CFI) defined as:∫
Ω
I (ε1, ε2)ϕ (ζ2) dℵϖ2 = χϕ (ε1) (1)

In which, dℵx2 is Stieltjes measure, defined as:

dℵε2 = dε2 +
i=k∑
i=1

mkδ (υi) (2)

I (ε1, ε2): A positive symmetric kernel,
mk: CPM,
υi, i = 1, 2, .....k : Arbitrary points at which the massesmk are concentrated,
δ: DDF



L. H. Al-Taee, A. A. M. Fawze / Eur. J. Pure Appl. Math, 16 (1) (2023), 304-313 306

The CFI has wide applications in the field of vibrations of a string [8, 10, 17, 19],
the concepts of the so called INIS will be applied, and the general case of the string
with different densities ρ (ζ2) will be used herein in the current paper. The density will
be charged by a finite number of cursors; therefore, its measurement using the following
equation:

dℵε2 = ρ (ε2) dζ2 +

i=k∑
i=1

mkδ (υi) (3)

Therefore; ℓ2dℵ will be (HS) equipped by the scalar product:

[σ1, σ2]dN := (σ1, σ2)dε1ρ + (σ1, σ2)∆r(ε1)
(4)

Where

(σ1, σ2)dϖ1ρ
=

∫
Ω
σ1 (ε1)σ2 (ε1) ρ (ε1) dε1 (5)

(σ1, σ2)∆r(ε1)
:=

i=k∑
i=1

mkσ1 (υi)σ2 (υi) (6)

The ℓ2dℵ, has no singularity at the points υi and now considered the point is located
inside the space, therefore; Eigenvalue problem appears as:

τϕ = χϕ (7)

The term τζ is called (CF), defined as:

τϕ =

∫
Ω
I (., ε2)ϕ (ε2) dℵε2 (8)

By making use of equation (8) and equation (3), leads to:

(τϕ) (ε2) =

∫
Ω
I (ε1, ε2)ϕ (ε2) ρ (ε2) dε2 +

i=k∑
i=1

nkI (ε1, υi)ϕ (υi) (9)

(τϕ) (ε2) = (I (ε1, ε2) , ϕ (ε2))dxϖρ
+ (I (ε1, ε2) , ϕ (ε2))∆r(ε2)

(10)

And so;

[τσ1, σ2] dℵ = ((τσ1) (ε1) , σ2 (ε1)) dε1ρ + ((τσ1) (ε1) , σ2 (ε1))∆r(ε1)
(11)

[τσ1, σ2] dℵ =
(
(I (ε1, ε2)σ1 (ε2)) dε2ρ +

(
(I (ε1, ε2)σ1 (ε2))∆r(x2)

, σ2 (ε1)
))

dε1ρ



L. H. Al-Taee, A. A. M. Fawze / Eur. J. Pure Appl. Math, 16 (1) (2023), 304-313 307

+
(
(I (ε1, ε2)σ1 (ε2)) dε2ρ +

(
(I (ε1, ε2)σ1 (ε2))∆r(x2)

, σ2 (ε1)
))

∆r(ϖ1)
(12)

By making use of the kernel symmetry, one can get:

(
(I (ε1, ε2)σ1 (ε2))∆r(x2)

, σ2 (ε1)
)
∆r(x1)

=

i=k∑
i=1

l=k∑
l=1

ninlI (mi,ml)σ1 (mi)σ2 (ml) (13)

3. Inverse Iterative Numerical Scheme (IINS)

The kernel I (ε1, ε2) is assumed to be continuous function, over a square of unit length
in the positive and first quadrant, real, symmetric, and regular Additional assumptions to
Eigenvalues, they are real and positive.

Assuming Eigenvalues defined as:

0 < ..... ≤ γ3 ≤ γ2 ≤ γ1

Now let us consider some points inside the square in the 1st and +ve quadrant as
follows:

0 =: υ1 < υ2 < υ3 < .....υk < υk+1 := 1

Assume that in each interval [vi, vi+1] ∋ n− nodesof (MGLQ), as following:

ε
(i)
11 , ε

(i)
12 , ε

(i)
13 , ....., ε

(i)
1n (14)

Make use of equation (14) over [υk, υk+1], leads to:

υk := ε
(i)
10 < ε

(i)
11 < ε

(i)
12 < ..... < ε

(i)
1n < ε

(i)
1n+1 := υi+1, 0 ≤ i ≤ k (15)

There exists corresponding points over vertical axis, defined as:

υl := y
(l)
10 < y

(l)
11 < y

(l)
12 < ..... < y

(l)
1n < y

(l)
1n+1 := υl+1, 0 ≤ l ≤ k (16)

Assume subinterval R
(k,l)
i.j as:

R
(k,l)
i.j :=

{
(ε1, ε2)

∣∣∣ε1(k)i < ε1 < ε1
(k)
i+1 ;

∣∣∣y1(l)j < y1 < y1
(l)
j+1

}
i, j = 1, 2, ...&k, l = 0, 1, 2, ....,

(17)

Defining R̂
(k,l)
i.j within ℜ(k,l)

i.j and ℜ−(k,l)
i.j containing singular point for ℑ (ζ1, ζ2), there-

fore;

J (ε1, ε2) =

{
J (ε1, ε2j) when (ε1, ε2) ∈ R

(k,l)
i.j

ℵi,j when (ε1, ε2) ∈ R̂
(k,l)
i.j

(18)



L. H. Al-Taee, A. A. M. Fawze / Eur. J. Pure Appl. Math, 16 (1) (2023), 304-313 308

The ℵi,j , are constants satisfying the next condition:
So as the norm defined by equation (19), to < epsilon, the next inequality should be

verified: ∥∥T (ε1, ε2j)− J (ε1, ε2)
∥∥
ℓ2dℵ

≥
∣∣λω − λ̄ω

∣∣ (19)

λω Represents the exact Eigenvalues, while λω represents the approximate values.

4. Results & Discussion

Considering the following kernel:

I
(
ε1i , ε2j

)
=



ε1 (u− ε2) 0 ≤ ε1 ≤ 1
0 ≤ ε2 ≤ 1
ε1 ≤ ε2
ε2 (u− ε1) 0 ≤ ε1 ≤ 1
0 ≤ ε2 ≤ 1
ε2 ≤ ε1

And
u = u

or
u ̸= u

Assuming that:

dℵε2 = ρ (ε2) dε2

4.1. Problem-1

ρ (ϖ2) = u+ϖ2

&

u = u = 1

By following up the procedure (IINS) described in the above section, and comparing
the results with (RRM), the results are shown in table (1).



L. H. Al-Taee, A. A. M. Fawze / Eur. J. Pure Appl. Math, 16 (1) (2023), 304-313 309

Table 1: Comparison between approximate Eigenvalues

Eigenvalue
number

Eigenvalues
Absolute Error

IINS RRM

1 0.15271 0.15265 0.00006

2 0.03779 0.03777 0.00002

3 0.01676 0.01672 0.00004

4 0.00942 0.00940 0.00002

5 0.00603 0.00601 0.00002

6 0.00418 0.00417 0.00001

Figure 1: Comparison between IINS and RRM verses Eigenvalues for problem-1

Figure 2: Absolute error between IINS and RRM for Problem-1

From table (1) that there is a very small errors and can be neglected.



L. H. Al-Taee, A. A. M. Fawze / Eur. J. Pure Appl. Math, 16 (1) (2023), 304-313 310

4.2. Problem-2

dℵε2 = ρ (ε2) dε2 +
4∑

i=1

niδ (υi)

ρ (ε2) =


16ε22 − 4ε2 + 0.25, 0.00 ≤ ε2 ≤ 0.25
16ε22 − 12ε2 + 02.25, 0.25 ≤ ε2 ≤ 0.50
16ε22 − 20ε2 + 06.25, 0.50 ≤ ε2 ≤ 0.75
16ε22 − 28ε2 + 12.25, 0.75 ≤ ε2 ≤ 1.00

vi = 0.25i− 0.125, i = 1, 2, 3, 4

&

m1 = 0.25,m2 = 0.5,m3 = 0.5,m4 = 0.25

Table 2: Comparison between approximate Eigenvalues

Eigenvalue
number

Eigenvalues
Absolute Error

IINS RRM

1 0.20683 0.20677 0.00006

2 0.05254 0.05252 0.00002

3 0.02004 0.01999 0.00005

4 0.01936 0.01931 0.00005

5 0.00086 0.00084 0.00002

6 0.00085 0.00084 0.00001

Figure 3: Comparison between IINS and RRM verses Eigenvalues for Problem-2



L. H. Al-Taee, A. A. M. Fawze / Eur. J. Pure Appl. Math, 16 (1) (2023), 304-313 311

Figure 4: Absolute error between IINS and RRM for problem-2

5. Conclusion

The topics of the integral equations are so wide of their applications in science; engi-
neering and technology, therefore the attention from researchers do not stop. The present
paper, is just a step of these trials, in which, we tried to propose a technique based itera-
tive numerical procedure. As seen from the computation, that the error is very small and
can neglected.

Abbreviation

IINS Inverse Iterative Numerical Scheme
HS Hilbert space
CF Compact Factor
DDF Dirac-Delta Function
CPM Combination of Positive Masses
MGLQ Modified Gauss-Legendre Quadrature Rule
RRM Rayleigh-Ritz Method

Acknowledgements

This paper was supported by the College of Education for Pure Sciences, University of
Mosul, Iraq and the College of Computer Science and Mathematics, University of Mosul,
Mosul, Iraq.



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