Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2580 
 

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Numerical Method for the Solution of Integro-Differential Equations of 

the Second Kind 

 

Dilmi. Mustapha 

College of Mathematics and Computer Science, 

University Pole, Road Bordj Bou Arreridj, M’sila 28000 Algeria 

moustafa.dilmi@univ-msila.dz 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: The purpose of this work is to search for an approximate solution to the Fredholm 

and Volterra integro-differential equations using Genocchi polynomials, replacing the initial 

conditions if necessary, where the integrals can be calculated using numerical methods, in 

order to obtain a variation problem and reduce it to a linear system, where its solution is to 

find the coefficients of the function. Unknowns and then solve the equation. The convergence 

and effectiveness of this method are confirmed by numerical examples that will be presented. 

Keywords: Integro-differential equations, Numerical method, Genocchi polynomials. 

 

1. Introduction 

Integro-differential equations are considered one of the most important fields in mathematical 

disciplines, for example pure mathematics and applied mathematics. Integro-differential equations 

linear and non linear have a very important role in modern science and technology such as heat 

transfer, diffusion processes, mechanics, biological species, and many other fields. To learn more 

about the sources in which these types of equations are studied in applications of physics, biology, 

and engineering, as well as in books on advanced integral equations. References can be found [4, 10, 

11, 15]. 

    The numerical solution of second order integro-differential equations with the boundary 

conditions of the Fredholm and Volterra equations and other equations related to this type of 

equations has been done by some authors. For example, the authors in [4] discussed the Chebyshev 

Collocation Method for the Solution of  Linear Integro-Differential Equations, which is the compact 

finite difference method, and the monotonic iterative sequence method for solving the second order 

Volterra integro-differential equation was implemented in [5, 27]. However, a sequential solution of 

second order integro-differential equations with boundary conditions of Fredholm and Volterra types 

by the homotopy analysis method was also considered in [16]. 

    Accordingly, this work aims to find approximate solutions to Fredholm and Volterra linear 

integro-differential equations of the second type using polynomials of the Genocchi type with the 

numerique method and then compare the approximate solutions with the exact solutions to see the 

effectiveness of the method through the examples that we will present. 

2. Genocchi polynomial method for I.D.E 

Consider the following integro-differential equation 

                   𝑦 ′(𝑥) = 𝑓(𝑥) + ∫ 𝑘(𝑥, 𝑡)𝑦(𝑡)
𝑥

𝑎
𝑑𝑡                                                                                      (1) 



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𝑦(𝑎) =∝ 

Where 𝑓(𝑥) and 𝑘(𝑥, 𝑡) are known functions, while 𝑦(𝑥) is the unknown function to be determined. 

The method under consideration employs Genocchi polynomials, as thoroughly discussed in 

references [12, 13, 14, 22]. 

These polynomials are used as a basis to approximate the solution over a closed and finite interval. It 

is assumed that 

𝑦𝑛(𝑥) = ∑ 𝛽𝑖
𝑛
𝑖=0 𝐺𝑖(𝑥) = ∑ 𝛽𝑖

𝑛
𝑖=0 𝐺𝑖 (

𝑥−𝑎

𝑏−𝑎
)                                                                                         (2) 

    Where 𝐺𝑖 (
𝑥−𝑎

𝑏−𝑎
) is shifted Genocchi polynomial at [𝑎, 𝑏] 

    Note that when we take the value 𝑥 = 𝑎 we get 
𝑥−𝑎

𝑏−𝑎
= 0, and when 𝑥 = 𝑏 we get 

𝑥−𝑎

𝑏−𝑎
= 1. 

    So we have  

                         𝑦 ′(𝑥) = 𝑦𝑛
′ (𝑥) = ∑ (

1

𝑏−𝑎
)𝑛

𝑖=0 𝛽𝑖𝐺𝑖
′ (
𝑥−𝑎

𝑏−𝑎
)                                                                       (3)  

 Substituting (2) and (3) into (1), results in               

∑ (
1

𝑏−𝑎
)𝑛

𝑖=0 𝛽𝑖𝐺𝑖
′ (
𝑥−𝑎

𝑏−𝑎
) = 𝑓(𝑥) + ∫ 𝑘(𝑥, 𝑡)

𝑥

𝑎
∑ 𝛽𝑖𝐺𝑖 (

𝑡−𝑎

𝑏−𝑎
)𝑛

𝑖=0 𝑑𝑡 = 𝑓(𝑥) +

∑ 𝛽𝑖
𝑛
𝑖=0 ∫ 𝑘(𝑥, 𝑡)𝐺𝑖

𝑥

𝑎
(
𝑡−𝑎

𝑏−𝑎
)𝑑𝑡                                                                                                  (4) 

To determine unknown coefficients 𝛽𝑖, we use the method technique by multiplying equation (4) by 

𝐺𝑗 (
𝑡−𝑎

𝑏−𝑎
) and then integrating with respect to 𝑥 from 0 to 1. So we have 

∑ (
1

𝑏−𝑎
)𝑛

𝑖=0 𝛽𝑖 ∫ 𝐺𝑖
′1

0
(
𝑥−𝑎

𝑏−𝑎
)𝐺𝑗 (

𝑥−𝑎

𝑏−𝑎
)𝑑𝑥 = ∫ 𝑓(𝑥)

1

0
𝐺𝑗 (

𝑥−𝑎

𝑏−𝑎
)𝑑𝑥 +

∫ [∑ 𝛽𝑖 ∫ 𝑘(𝑥, 𝑡)𝐺𝑖 (
𝑡−𝑎

𝑏−𝑎
) 𝑑𝑡

𝑥

𝑎
𝑛
𝑖=0 ]

1

0
𝐺𝑗 (

𝑥−𝑎

𝑏−𝑎
)𝑑𝑥                                              (5) 

for = 0,1, … , 𝑛 , or equivalently 

∑ (
1

𝑏−𝑎
)𝛽𝑖

𝑛
𝑖=0 ∫ 𝐺𝑖

′1

0
(
𝑥−𝑎

𝑏−𝑎
)𝐺𝑗 (

𝑥−𝑎

𝑏−𝑎
)𝑑𝑥 = ∫ 𝑓(𝑥)

1

0
𝐺𝑗 (

𝑥−𝑎

𝑏−𝑎
)𝑑𝑥 +

∑ 𝛽𝑖
𝑛
𝑖=0 ∫ [∫ 𝑘(𝑥, 𝑡)𝐺𝑖 (

𝑡−𝑎

𝑏−𝑎
) 𝑑𝑡

𝑥

𝑎
]

1

0
𝐺𝑗 (

𝑥−𝑎

𝑏−𝑎
)𝑑𝑥                                              (6)   

 If necessary, the integrals can be evaluated using numerical techniques. This process results in a 

system of linear equations involving the unknown coefficients {𝛽0, 𝛽1, … , 𝛽𝑛}. In many studies, 

researchers incorporate the initial condition by directly substituting it into the system. 

𝑦(𝑎) =∝ ⇒ ∑ 𝛽𝑖𝐺𝑖
𝑛
𝑖=0 (

𝑎−𝑎

𝑏−𝑎
) = ∑ 𝛽𝑖𝐺𝑖

𝑛
𝑖=0 (0) =∝                                                                              (7) 

Maintaining an equal number of equations in the previously constructed linear system, the unknown 

coefficients are determined by simultaneously solving equations (6) and (7). Once obtained, these 

values are substituted into equation (2) to derive an approximate solution to the original equation (1). 

 

 



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3. Genocchi Polynomials and Their Properties 

The classical Genocchi polynomial 𝐺𝑛(𝑥) is usually defined by means of the exponential generating 

functions 

2𝑡𝑒𝑥𝑡

𝑒𝑡 + 1
= ∑𝐺𝑛(𝑥)

+∞

𝑛=0

𝑡𝑛

𝑛!
 

Where 𝐺𝑛(𝑥) is the Genocchi polynomial of degree 𝑛 and is given by 

𝐺𝑛(𝑥)  = ∑(
𝑛
𝑘
)𝐺𝑛−𝑘(𝑥)

𝑛

𝑘=0

𝑥𝑘 

    𝐺𝑛−𝑘. Is the Genocchi number. 

    Some of the important properties of these polynomials include 

{
  
 

  
 
∫𝐺𝑝(𝑥)𝐺𝑞(𝑥)

1

0

𝑑𝑥 =
2(−1)𝑝𝑝! 𝑞!

(𝑝 + 𝑞)!
𝐺𝑝+𝑞, 𝑝, 𝑞 ∈ ℕ

∗

𝑑𝐺𝑝(𝑥)

𝑑𝑥
= 𝑛𝐺𝑛−1(𝑥),   𝑝 ∈ ℕ

∗

𝐺𝑝(1) + 𝐺𝑞(0) = 0,   𝑝 ∈ ℕ
∗

 

4. Numerical Examples 

In this section, we intend to show the efficiency of the method for solving Fredholm and Volterra 

integro-differential equations of the second kind by Genocchi polynomials by presenting six 

illustrative examples. The absolute error for this formulation is need by 𝐸(𝑥) = |𝑦(𝑥) − 𝑦𝑛(𝑥)|. 

Example 1. Let us consider the linear integro-differential equation of Volterra. 

𝑦′(𝑥) = 2 −
𝑥2

4
+
1

4
∫ 𝑦(𝑡)𝑑𝑡
𝑥

0

 

With the initial condition 𝑦(0) = 0 

Where the function 𝑦(𝑥) = 2𝑥 is the exact solution.  

The approximate solution 𝑦𝑛(𝑥) of 𝑦(𝑥) is obtained by the Genocchi polynomial method. 



Communications on Applied Nonlinear Analysis 

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Figure 1. Graph for example 1 

Example 2. Let us consider the linear integro-differential equation of Volterra. 

𝑦′(𝑥) = 1 − 2𝑥 sin(𝑥) + ∫ 𝑦(𝑡)
𝑥

0

𝑑𝑡 

With the initial condition 𝑦(0) = 0 

The exact solution is given by 

𝑦(𝑥) = 𝑥 cos(𝑥) 

The approximate solution 𝑦𝑛(𝑥) of 𝑦(𝑥) is obtained by the Genocchi polynomial method. 

 

Figure 2. Graph for example 2 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

y

 

 

exacte Solution

 approx. Solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

 

 

exacte Solution

 approx. Solution



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Example 3.  Let us consider the linear integro-differential equation of Fredholm 

𝑦′(𝑥) = 3𝑒3𝑥 −
1

3
(2𝑒3 + 1)𝑥 + ∫ 3𝑥𝑡𝑦(𝑡)

1

0

𝑑𝑡 

With the initial condition 𝑦(0) = 1 

The exact solution is given by 

𝑦(𝑥) = 𝑒3𝑥 

The approximate solution 𝑦𝑛(𝑥) of 𝑦(𝑥) is obtained by the Genocchi polynomial method. 

 

Figure 3. Graph for example 3 

Example 4. Let us consider the linear integro-differential equation of Fredholm. 

𝑦′(𝑥) = 3 + 6𝑥 + ∫𝑥𝑡

1

0

𝑦(𝑡)𝑑𝑡 

With the initial condition 𝑦(0) = 0 

The exact solution is given by 

𝑦(𝑥) = 3𝑥 + 4𝑥2 

The approximate solution 𝑦𝑛(𝑥) of 𝑦(𝑥) is obtained by the Genocchi polynomial method. 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

5

10

15

20

25

x

y

 

 

exacte Solution

 approx. Solution



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Figure 4. Graph for example 4 

Example 5. Consider linear Volterra integro-differential equation of second kind 

𝑦′′(𝑥) = 𝑥 + ∫(𝑥 − 𝑡)𝑦(𝑡)

𝑥

0

𝑑𝑡 

With the initial condition 𝑦(0) = 0, 𝑦′(0) = 1 

Where the function 𝑦(𝑥) = sinh (𝑥)  is the exact solution  

The approximate solution 𝑦𝑛(𝑥) of 𝑦(𝑥) is obtained by the Genocchi polynomial method. 

 

Figure 5. Graph for example 5 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

1

2

3

4

5

6

7

x

y

 

 

exacte Solution

 approx. Solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

0.2

0.4

0.6

0.8

1

1.2

1.4

x

y

 

 

exacte Solution

 approx. Solution



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Example 6. Consider the integro-differential equation 

𝑦′(𝑥) = − cos(2𝜋𝑥) − 2𝜋 sin(2𝜋𝑥) −
1

2
sin(4𝜋𝑥) + ∫ sin(4𝜋𝑥 + 2𝜋𝑡) 𝑦(𝑡)

1

0

𝑑𝑡 

With the initial condition 𝑦(0) = 1, 

Where the function  𝑦(𝑥) = cos (2𝜋𝑥) is the exact solution  

The approximate solution 𝑦𝑛(𝑥) of 𝑦(𝑥) is obtained by the Genocchi polynomial method. 

 

Figure 6. Graph for example 6 

 Conclusion 

    This article deals with the numerical solution of first order Fredholm and Volterra integro-

differential equations of the second kind, using the method technique by means of Genocchi 

polynomials. This technique was tested on six examples shown in the obtained figures, and the 

results were satisfactory and the method was quite effective. In addition, this method can be applied 

to high order Fredholm and Volterra integro-differential equations of the second kind, where the 

Matlab program is used to obtain approximate solutions. 

This technique will be applied in the future to fractional integro-differential equations and nonlinear 

integro-differential equations. 

Refrences 

[1] A. Adawi, F. Awawdeh, A numerical method for solving linear integral equations, Int. J. 

Contemp. Mathematics Sciences, 10, (2009) pp 485-496. 

[2] K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, the press 

Syndicate of the University of Cambridge, United Kingdom, 1997.  

[3] S. Aggarwal, N. Sharma, R. Chauhan. Solution of Linear Volterra Integro-Differential 

equations of Second Kind Using Mahgoub Transform. International Journal of Latest 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1

-0.5

0

0.5

1

1.5

x

y

 

 

exacte Solution

 approx. Solution



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ISSN: 1074-133X 

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Technology in Engineering, Management & Applied Science. Volume VII, Issue V, (2018) 

173-176.  

[4] A. Akyuz and M. Sezer. A Chebyshev Collocation Method for the Solution of Linear Integro-

Differential Equations, Int. J. Comput Math. Vol. 27, No. 4, (1999).  

[5] Q. M. Al-Mdallal, Monotone iterative sequences for nonlinear integro-differential equations 

of second order, Nonlinear Analysis, 12 (2011), no. 6, 3665-3673.  

[6] E. Babolian, A. Davari, Numerical implementation of Adomian decomposition method for 

linear Volterra integral equations of the second kind, App. Math. Comput, 165 (2005) 223-

227. 

[7] M. Djalil, H. Souidan, W. Souidan, Approximate Solution of Linear Volterra Integro-

Differential Equation by Touchard Polynomials Method, (WJSM) 11(2), (2018), 29-41 

[8] A. Jerri. Introduction to integral equations with applications. John Wiley and Sons, INC, 

1999. 

[9] J. He, Homotopy perturbation technique, Comput, Methods Appl. Mech. Engrg, 178 (1999) 

257-262. 

[10] A. Hossein, A New Analytical Method for Solving Systems of Linear Integro-Differential 

Equations, Journal of King Saud University- Science (2011), 23, 349-353. 

[11] S. M. Hosseini and S. Shamord, Numerical Piecewise Approximate Solution of Fredholm 

Integro-Differential Equations by the Tau Method, App. Math. Model 29(2005)1005-1021. 

Isah, C. Phang. Operational matrix based on Genocchi polynomials for solution of delay differential 

equations. Ain Shams Engineering Journal, 2017. 

[12] A. Isah, C. Phang. Operational matrix based on Genocchi polynomials for solution of delay 

differential equations. Ain Shams Engineering Journal, 2017. 

[13] A. Isah, C. Phang. Genocchi Wavelet-like operational matrix and its application for solving 

non-linear fractional differential equations. Open Physics, vol. 14, no. 1, pp. 463-472, 2016. 

[14] T. Kim. Some identities for the Bernoulli, the Euler and the Genocchi numbers and 

polynomials. Advanced Studies in Contemporary Mathematics, vol. 20, no. 1, pp. 23-28, 

2010. 

[15] M. Khanian, A. Davari. Solution of System of Fredholm Integro-Differential Equations by 

Adomain Decomposition Method, Australian Journal of Basic and Applied Science, Vol. 5, 

No. 12, (2011), 2356-2361. 

[16] S. J. Liao. Beyond Perturbation: Introduction to the Homotopy Analysis Methods, Chapman 

& Hall, Boca Raton, Fla, USA, 2003. 

[17] K. Maleknejad, N. Aghazadeh. Numerical solution of Volterra integral equations of the 

second kind with convolution kernel by using Taylor-series expansion method, Appl. Math. 

Comput, 161, (2005) 915-922. 

[18] K. Maleknejad, K. Nouri, M. Yousefi. Discussion on convergence of Legendre polynomial 

for numerical solution of integral equations, Applied Mathematics and Computation, 193, 

(2007), 335-339. 

[19] M. Nadir. Cours sur les équations intégrales, université M'sila 2008. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2588 
 

https://internationalpubls.com 

[20] M. Nadir. Solving Fredholm integral equations with application of the four Chebyshev 

polynomials, in Journal of Approximation Theory and Applied Mathematics, 4, ( 2014), pp 

37-44. 

[21] M. Nadir, D. Mustapha. Euler Series solutions for linear Integral equations AJMAA, Vol. 14, 

No. 2, Art. 11, (2017) pp 1-7. 

[22] C. Phang, N. F. Ismail, A. Isah, and J. R. Loh. A new efficient numerical scheme for solving 

fractional optimal control problems via a Genocchi operational matrix of integration. Journal 

of Vibration and Control, 2017. 

[23] M. Rahman. Mathematical Methods with Application, WIT Press, Southampton, UK, pp. 

456, 2000. 

[24] S. Roman, The umbral Calculus. 208 Dover Publications USA, 2005. 

[25] S. Wang, J.H. He. Variational iteration method for solving integro-differential equations. 

Phys. Lett., A 367 (2007) 188-191. 

[26] A.M. Wazwaz. A reliable algorithm for solving boundary value problems for higher-order 

integro-differential equations, Applied Mathematical and Computation, 118 (2001), 327-342. 

[27] L.H. Yang, Y. Lin. Reproducing kernel methods for solving linear initial boundary-value 

problems, Electronic Journal of Differential Equations, (2008), 1-11.