Matrix Approach to The Direct Computation Method for The Solution of Fredholm Integro-Differential Equations of The Second Kind With Degenerate Kernels


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 6(3) (2020), Pages 100-108 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: March 14, 2020 Reviewed: April 04, 2020 Accepted: November 01, 2020 
DOI: http://dx.doi.org/10.18860/ca.v6i3.8960  

Matrix Approach to The Direct Computation Method for The Solution of 
Fredholm Integro-Differential Equations of The Second Kind With 

Degenerate Kernels 

Nathaniel Kamoh1, Geoffrey Kumlengand2, Joshua Sunday3 

1,2,3Department of Mathematics, University of Jos, Jos, Nigeria 

Email: mahwash1477@gmail.com, gkumleng@gmail.com, 
joshuasunday2000@yahoo.com 

ABSTRACT 

In this paper, a matrix approach to the direct computation method for solving Fredholm Integro-
Differential Equations (FIDEs) of the second kind with degenerate kernels is presented. Our 
approach consists of reducing the problem to a set of linear algebraic equations by 
approximating the kernel with a finite sum of products and determining the unknown constants 
by the matrix approach. The proposed method is simple, efficient and accurate; it approximates 
the solutions exactly with the closed form solutions. The result of this research is the solution of 
the second type Fredholm integro-differential equation (FIDE) with a numerically accurate 
kernel degenerate. Some problems are considered using maple programme to illustrate the 
simplicity, efficiency and accuracy of the proposed method. 
 
Keywords: Fredholm; Matrix; Direct Solution; Integro-Differential Equation; Integral 

INTRODUCTION 

The subject Integro-Differential Equations (IDEs) is one of the most important 
mathematical tools in both pure and applied mathematics. In recent year’s mathematical 
modeling of real-life usually results in functional equations such as differential 
equations, integral and Integro-Differential Equations (IDEs) these equations play very 
important role in modern science and technology application such as the theory of signal 
processing, neutral networks, heat transfer, diffusion process, neutron diffusion and 
biological species. These equations can be classified into Fredholm equations and 
Volterra equations. The upper bound of the region for integral part of the Volterra type 
is a variable, while it is a fixed number for that of Fredholm type. More details and 
sources where these equations can be found are in the areas of Physics, biology, 
engineering and social sciences and have been extensively studied both at theoretical 
and practical level. Most integro-differential equations are usually very difficult to solve 
analytically and so accurate, acceptable and efficient numerical method is required to 
approximate the solution (see [1], [4], [5], [6], [13], [15] and [16]). 

In recent times, extensive efforts have been devoted to the numerical methods of 
solution for Fredholm integro-differential equations by many researchers. [2] 
considered non-standard finite difference method for the numerical solution of linear 
Fredholm integro-differential equations.  The method and the repeated composite 

http://dx.doi.org/10.18860/ca.v6i3.8960
mailto:mahwash1477@gmail.com
mailto:gkumleng@gmail.com
mailto:joshuasunday2000@yahoo.com


Matrix Approach to The Direct Computation Method for The Solution of Fredholm Integro-
Differential Equations of The Second Kind With Degenerate Kernels  

Nathaniel Kamoh 101 

trapezoidal quadrature method were used to transform the Fredholm integro-
differential equation into a system of non-linear algebraic equations and experiments on 
some linear model problems showed the simplicity and efficiency of the proposed 
method. The Wavelet method for the numerical solution of Fredholm integro-differential 
equation was used in [14]. [3] developed a finite difference hybrid method by a 
combination of power series and the shifted Legendre polynomial to solve Fredholm 
integro-differential equation. A new and efficient approach for the numerical solution of 
Fredholm integro-differential equations (FIDEs) of the second kind with an unbounded 
domain with degenerate kernel based on operational matrices with respect to 
generalized Laguerre polynomials (GLPs) was introduced in [8].The Adomian’s 
decomposition method which is a well-known method for solving functional equations 
in recent times was used to solve linear Fredholm integro-differential equations by [9]. 
The result obtained gives more accurate approximation as compared to two other 
methods. [10], applied the Legendre polynomials for the solution of the linear Fredholm 
integro-differential-difference equation of high order and the results obtained by the 
developed technique were more accurate than the results reported for the Taylor and 
the wavelet Galerkin methods 

The purpose of this paper is to solve Fredholm Integro-differential equation of 
the second kind with separable kernels by the approach of matrix method for the direct 
computation method, this approach is considered simple, accurate and easy to 
implement.  
 

 

METHODS 

Consider the standard form of the Fredholm integro-differential equation of the second 
kind given by  

𝜓(𝑛)(𝜃) = 𝜔(𝜃) + 𝜆 ∫ 𝑘(𝜃, 𝜉)𝜓(𝜉)𝑑𝜉
𝑏

𝑎

                                            (1) 

𝜓(𝑘)(0) = 𝑞𝑘 ,   0 ≤ 𝑘 ≤ 𝑛 − 1                                                              (2) 

where 𝜓(𝑛)(𝜃)indicates the 𝑛𝑡ℎ  derivative of 𝜓(𝜃) with respect to 𝜃. Because 
(1)combines differential operator and the integral operator, then it is necessary to 
define initial conditions given in (2) for the determination of the particular solution 
𝜓(𝜃)  of (1). Suppose that we wish to determine the approximate solution of the 
theoretical solution 𝜓(𝜃) of problem (1) at the domain  [𝑎, 𝑏]. Let the separable or 
degenerate kernel𝐾(𝜃, 𝜉)of (1) be approximated by∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉)

𝑛
𝑘=1 . Thus the integro-

differential equation (1) may be written as  

𝜓(𝑛)(𝜃) = 𝜔(𝜃) + 𝜆 ∫ ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉)

𝑛

𝑘=1

𝜓(𝜉)𝑑𝜉,   
𝑏

𝑎

                             (3) 

where 𝑘(𝜃, 𝜉) = ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉)
𝑛
𝑘=1  is a finite sum of products𝜏𝑘 (𝜃) and 𝜙𝑘 (𝜉), 𝜏𝑘 (𝜃) is a 

function of  𝜃 only and 𝜙𝑘 (𝜉) is a function of 𝜉 only. 
Equation (3) can be rewritten as  
 

𝜓(𝑛)(𝜃) = 𝜔(𝜃) + 𝜆 ∑ 𝜏𝑘 (𝜃)

𝑛

𝑘=1

∫ 𝜙𝑘 (𝜉)𝜓(𝜉)𝑑𝜉 
𝑏

𝑎

                                    (4) 

Discretizing the integral part of (4) by letting  𝜇𝑘 = ∫ 𝜙𝑘 (𝜉)𝜓(𝜉)𝑑𝜉 
𝑏

𝑎
, (4) becomes 



Matrix Approach to The Direct Computation Method for The Solution of Fredholm Integro-
Differential Equations of The Second Kind With Degenerate Kernels  

Nathaniel Kamoh 102 

𝜓(𝑛)(𝜃) = 𝜔(𝜃) + 𝜆 ∑ 𝜏𝑘 (𝜃)𝜇𝑘

𝑛

𝑘=1

                                                                (5) 

Now, multiplying both sides of (5) by the integral operator𝜒−𝑛 define by (𝜒−𝑛(⋆) =
∫(⋆)𝑑𝜃) with the application of the initial conditions (2), it follows that (5) can be 
written as 

𝜓(𝜃) − ∑
𝜃(𝑖−1)

(𝑖 − 1)!

𝑛

𝑖=1

𝜓(𝑖−1)(0) = 𝜒−𝑛 [𝜔(𝜃) + 𝜆 ∑ 𝜏𝑘 (𝜃)𝜇𝑘

𝑛

𝑘=1

]            (6)  

Simplifying (6), we obtain  

𝜓(𝜃) − ∑
𝜃 (𝑖−1)

(𝑖 − 1)!

𝑛

𝑖=1

𝜓(𝑖−1)(0) = 𝜔⋇(𝜃) + 𝜆 ∑ 𝜏𝑘
⋇(𝜃)𝜇𝑘

𝑛

𝑘=1

                    (7) 

Multiplying both sides of (7) by𝜙𝑚(𝜃), 𝑚 = 1,2, . . . , 𝑛  and integrating from 𝑎 to 𝑏 over 𝜃 
leads to a matrix equation that facilitates the determination of the𝜇𝑘 ’s in (5). Thus 
equation (7) becomes 

∫ 𝜓(𝜃)𝜙𝑚(𝜃)𝑑𝜃
𝑏

𝑎

− ∫ ∑
𝜃(𝑖−1)

(𝑖 − 1)!

𝑛

𝑖=1

𝑏

𝑎

𝜓(𝑖−1)(0)𝜙𝑚(𝜃)𝑑𝜃

= ∫ (𝜔⋇(𝜃) + 𝜆 ∑ 𝜏𝑘
⋇(𝜃)𝜇𝑘

𝑛

𝑘=1

)
𝑏

𝑎

𝜙𝑚(𝑥)𝑑𝜃                  (8) 

If we define𝜇𝑚,𝜔𝑚,𝑎𝑚𝑘 as∫ 𝜓(𝜃)𝜙𝑚(𝜃)𝑑𝜃
𝑏

𝑎
,∫ 𝜔⋆(𝜃)𝜙𝑚(𝜃)𝑑𝜃

𝑏

𝑎
 and∫ 𝜏 ∗𝑘 (𝜃)𝜙𝑚(𝜃)𝑑𝜃

𝑏

𝑎
 

respectively,then (8)can be written as 

𝜇𝑚 − 𝜆 ∑ 𝜇𝑘 𝑎𝑚𝑘 = 𝜔𝑚,

𝑛

𝑘=1

   𝑚 = 1,2, . . . , 𝑛                                           (9) 

Equation (9) gives a nonhomogeneous system of 𝑛 linear equations in 𝜇1,  𝜇2,  𝜇3, … , 𝜇𝑛 
unknown.𝜔𝑚 and 𝜏𝑚𝑘  are known since 𝜙𝑚(𝜃), 𝜔

⋆(𝜃) and𝜏 ∗𝑘 (𝜃) are all given. 
Putting (9) in matrix equation form, it follows that 

(𝐼 − 𝜆 ∑ 𝑎𝑚𝑘

𝑛

𝑘=1

) 𝜇𝑘 = 𝜔𝑚                                                                      (10) 

where 

𝜇𝑘 = [

𝜇1
⋮

𝜇𝑛
] , 𝜔𝑚 = [

𝜔1
⋮

𝜔𝑛
] , 𝑎𝑚𝑘 = [

𝑎11 ⋯ 𝑎1𝑛
⋮ ⋱ ⋮

𝑎𝑛1 ⋯ 𝑎𝑛𝑛
] 

Equation (10)can be used to determine the unknown 𝜇1,  𝜇2,  𝜇3, … , 𝜇𝑛 which are then 
substituted into 𝜓(𝜃) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘

⋇(𝜃)𝜇𝑘
𝑛
𝑘=1  for the particular solution of (1) 

RESULTS AND DISCUSSION  

We now apply the method presented in this paper to solved four model problems to 
illustrate the above mentioned approach and demonstrate its computational accuracy. 
This method differs from the direct computation method since the unknown constants 
are determined at once by the introduced matrix equation 

  



Matrix Approach to The Direct Computation Method for The Solution of Fredholm Integro-
Differential Equations of The Second Kind With Degenerate Kernels  

Nathaniel Kamoh 103 

Problem1 
Consider the linear Fredholm integro-differential equation given in [1] 

𝑦′′′(𝑥) = 5𝑙𝑛2 − 3 − 𝑥 + 4𝑐𝑜𝑠ℎ𝑥 − ∫ (𝑥 − 𝑡)𝑦(𝑡)𝑑𝑡,   𝑦(0) = 𝑦′′(0) = 0
𝑙𝑛2

0

, 𝑦′(0) = 4,

0 ≤ 𝜃 ≤ 𝜋 
The analytical solution to the problem is 𝑦(𝑥) = 4𝑠𝑖𝑛ℎ(𝑥) 
Let  

𝑘(𝜃, 𝜉) = ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉) = (𝜃 − 𝜉)

3

𝑘=1

 

𝜏1(𝜃) = 𝜃, 𝜏2(𝜃) = −1, 𝜙1(𝜉) = 1, 𝜙2(𝜉) = 𝜉 

Let the required solution be given as  

𝜓(𝜃) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘
⋇(𝜃)𝜇𝑘

3

𝑘=1

 

Multiplying the given FIDE through with the integral operator𝜒−3 to obtain 

𝜓(𝜃) − ∑
𝜃(𝑖−1)

(𝑖 − 1)!

3

𝑖=1

𝜓(𝑖−1)(0) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘
⋇(𝜃)𝜇𝑘

3

𝑘=1

 

or  

𝜓(𝜃) − ∑
𝜃(𝑖−1)

(𝑖 − 1)!

3

𝑖=1

𝜓(𝑖−1)(0) = 2𝑒𝜃 +
5

6
𝜃3𝑙𝑛2 −

1

2
𝜃3 −

1

24
𝜃4 −

2

𝑒𝜃
− (

1

24
𝜃4𝜇1 −

1

6
𝜃3𝜇2) 

where   

𝜔1 =
1

5
𝑙𝑛25 −

1

8
𝑙𝑛24 + 1, 𝜔2 = 5𝑙𝑛2 −

1

10
𝑙𝑛25 +

23

144
𝑙𝑛26 − 3 

𝑎11 =
1

120
𝑙𝑛25, 𝑎12 = −

1

24
𝑙𝑛24, 𝑎21 =

1

120
𝑙𝑛26,𝑎22 = −

1

30
𝑙𝑛25 

Using (10) to solve for𝜇𝑘,   𝑘 = 1, 2, we have  𝜇1 = 1,   𝜇2 = 5𝑙𝑛2 − 3 substituting in 

𝜓(𝜃) = 2𝑒𝜃 +
5

6
𝜃3𝑙𝑛2 −

1

2
𝜃3 −

1

24
𝜃4 −

2

𝑒𝜃
− (

1

24
𝜃4𝜇1 −

1

6
𝜃3𝜇2) 

we have 𝜓(𝜃) =  4sinh(𝜃)as the exact solution. Table 1 below reveals the performance 
of the proposed method for problems 1 
 

Table 1: The performance results of the proposed method for problem 1 
values  (𝑥 = 𝜃) Exact solution Proposed Method Absolute error 
0.00 
0.10 
0.20 
0.30 
0.40 
0.50 
0.60 
0.70 
0.80 
0.90 
1.00 

0.000000000 
0.400667000 
0.805344010 
1.218081174 
1.643009303 
2.084381222 
2.546614328 
3.034334807 
3.552423929 
4.106066904 
4.700804776 

0.000000000 
0.400667000 
0.805344010 
1.218081174 
1.643009303 
2.084381222 
2.546614328 
3.034334807 
3.552423929 
4.106066904 
4.700804776 

0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 



Matrix Approach to The Direct Computation Method for The Solution of Fredholm Integro-
Differential Equations of The Second Kind With Degenerate Kernels  

Nathaniel Kamoh 104 

Problem2 
Consider the linear Fredholm integro-differential equation in [1] given by 

𝑦′′′(𝑥) = 𝑒 − 2 − 𝑥 + 𝑒 𝑥 (3 + 𝑥) + ∫ (𝑥 − 𝑡)𝑦(𝑡)𝑑𝑡,   𝑦(0) = 0
1

0

, 𝑦′(0) = 1, 𝑦′′(0) = 2,

0 ≤ 𝜃 ≤ 𝜋 
The analytical solution to the problem is 𝑦(𝑥) = 𝑥𝑒 𝑥  
Let  

𝑘(𝜃, 𝜉) = ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉) = (𝜃 − 𝜉)

3

𝑘=1

 

𝜏1(𝜃) = 𝜃, 𝜏2(𝜃) = −1, 𝜙1(𝜉) = 1, 𝜙2(𝜉) = 𝜉 
The required solution is given as  

𝜓(𝜃) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘
⋇(𝜃)𝜇𝑘

3

𝑘=1

 

Multiplying the given FIDE with the integral operator𝜒−3, we have  

𝜓(𝜃) − ∑
𝜃(𝑖−1)

(𝑖 − 1)!

3

𝑖=1

𝜓(𝑖−1)(0) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘
⋇(𝜃)𝜇𝑘

3

𝑘=1

 

or  

𝜓(𝜃) − ∑
𝜃(𝑖−1)

(𝑖 − 1)!

3

𝑖=1

𝜓(𝑖−1)(0) =
1

6
𝑒𝜃3 −

1

3
𝜃3 −

1

24
𝜃4 + 𝜃𝑒𝜃 + (

1

24
𝜃4𝜇1 −

1

6
𝜃3𝜇2) 

where   

𝜔1 =
109

120
+

1

24
𝑒, 𝜔2 = −

1493

720
+

31

30
𝑒  

𝑎11 =
1

120
, 𝑎12 = −

1

24
,  𝑎21 =

1

144
, 𝑎22 = −

1

30
  

Using (10) to solve for  𝜇𝑘,   𝑘 = 1, 2 and substituting into  

𝜓(𝜃) =
1

6
𝑒𝜃3 −

1

3
𝜃3 −

1

24
𝜃4 + 𝜃𝑒 𝜃 + (

1

24
𝜃4𝜇1 −

1

6
𝜃3𝜇2) 

we obtain𝜓(𝜃) = 𝜃𝑒𝜃 giving the same result as the exact solution.  

Problem 3 
Consider the linear Fredholm integro-differential equation in [1] given as 

𝑦′′(𝑥) = 4𝑥 − sin(𝑥) + ∫ (𝑥 − 𝑡)2𝑦(𝑡)𝑑𝑡,   𝑦(0) = 0

𝜋

2

−
𝜋

2

, 𝑦′(0) = 1, −
𝜋

2
≤ 𝜃 ≤

𝜋

2
 

The analytical solution to the problem is 𝑦(𝑥) = 𝑠𝑖𝑛(𝑥) 
Let  

𝑘(𝜃, 𝜉) = ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉) = (𝜃 − 𝜉)
2

2

𝑘=1

 

𝜏1(𝜃) = 𝜃
2, 𝜏2(𝜃) = −2𝜃, 𝜏3(𝜃) = 1 𝜙1(𝜉) = 1, 𝜙2(𝜉) = 𝜉, 𝜙3(𝜉) = 𝜉

2 



Matrix Approach to The Direct Computation Method for The Solution of Fredholm Integro-
Differential Equations of The Second Kind With Degenerate Kernels  

Nathaniel Kamoh 105 

The required solution is given as  

𝜓(𝜃) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘
⋇(𝜃)𝜇𝑘

2

𝑘=1

 

Multiplying the given FIDE with the integral operator𝜒−2, we have 

𝜓(𝜃) − ∑
𝜃(𝑖−1)

(𝑖 − 1)!

2

𝑖=1

𝜓(𝑖−1)(0) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘
⋇(𝜃)𝜇𝑘

3

𝑘=1

 

or  

𝜓(𝜃) − ∑
𝜃(𝑖−1)

(𝑖 − 1)!

2

𝑖=1

𝜓(𝑖−1)(0) = 𝑠𝑖𝑛𝜃 +
2

3
𝜃³ + (

1

2
𝜃²𝜇₃ −

1

3
𝜃³𝜇₂ +

1

12
𝜃⁴𝜇₁) 

where 

𝜔1 = 0, 𝜔2 =
1

12
𝜋⁵ + 2, 𝜔2 = 0 

𝑎11 =  
1

960
𝜋⁵,  𝑎12 = 0,  𝑎13 =

1

24
𝜋³ 

𝑎21 = 0, 𝑎22 = −
1

240
𝜋⁵,  𝑎23 = 0 

𝑎31 =
1

5376
𝜋⁷, 𝑎32 = 0,  𝑎33 =

1

160
𝜋⁵ 

Using (10) to solve for  𝜇𝑘,   𝑘 = 1, 2, 3 and substituting in  

𝜓(𝜃) = 𝑠𝑖𝑛𝜃 +
2

3
𝜃³ + (

1

2
𝜃²𝜇₃ −

1

3
𝜃³𝜇₂ +

1

12
𝜃⁴𝜇₁) 

gives 𝜓(𝜃) = 𝑠𝑖𝑛(𝜃) having the same result as the exact solution.  
 
Problem 4 
Consider the linear Fredholm integro-differential equation given by 

𝑦(𝑖𝑣)(𝑥) = 2𝑥 − 𝜋 + sin(𝑥) + 𝑐𝑜𝑠(𝑥) − ∫ (𝑥 − 2𝑡)𝑦(𝑡)𝑑𝑡,   

𝜋

2

0

𝑦(0) = 𝑦′(0) = 1, 

𝑦′′(0) = 𝑦′′′(0) = −1, −
𝜋

2
≤ 𝜃 ≤

𝜋

2
 

The analytical solution to the problem is 𝑦(𝑥) = 𝑠𝑖𝑛(𝑥) + 𝑐𝑜𝑠(𝑥) 
Let  

𝑘(𝜃, 𝜉) = ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉) = (𝜃 − 2𝜉)

4

𝑘=1

 

𝜏1(𝜃) = 𝜃, 𝜏2(𝜃) = −2, 𝜏3(𝜃) = 0, 𝜏4(𝜃) = 0, 𝜙1(𝜉) = 1, 𝜙2(𝜉) = 𝜉, 𝜙3(𝜉) = 0, 𝜙4(𝜉) = 0 
The required solution is given as  

𝜓(𝜃) = 𝜔⋇(𝜃) + 𝛾 ∑ 𝜏𝑘
⋇(𝜃)𝜇𝑘

4

𝑘=1

 

Multiplying the given FIDE with the integral operator𝜒−4, we have 

𝜓(𝜃) − ∑
𝜃(𝑖−1)

(𝑖 − 1)!

4

𝑖=1

𝜓(𝑖−1)(0) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘
⋇(𝜃)𝜇𝑘

4

𝑘=1

 



Matrix Approach to The Direct Computation Method for The Solution of Fredholm Integro-
Differential Equations of The Second Kind With Degenerate Kernels  

Nathaniel Kamoh 106 

or  

𝜓(𝜃) − ∑
𝜃 (𝑖−1)

(𝑖 − 1)!

4

𝑖=1

𝜓(𝑖−1)(0) = 𝑐𝑜𝑠𝜃 + 𝑠𝑖𝑛𝜃 −
1

24
𝜋𝜃⁴ +  

1

60
𝜃⁵ + (

1

12
𝜃⁴𝜇₂ −

1

120
𝜃⁵𝜇₁) 

where   

𝜔1 = 2 −
1

4608
𝜋⁶, 𝜔2 =

1

2
𝜋 −

29

322560
𝜋⁷ 

𝑎11 =  −
1

46080
𝜋⁶,  𝑎12 =

1

1920
𝜋⁵  

𝑎21 = −
1

107520
𝜋⁷, 𝑎22 =

1

4608
𝜋⁶ 

Using (10) to solve for  𝜇𝑘,   𝑘 = 1, 2 and substituting in   

𝜓(𝑥) = 𝑐𝑜𝑠𝜃 + 𝑠𝑖𝑛𝜃 −
1

24
𝜋𝜃⁴ +  

1

60
𝜃⁵ + (

1

12
𝜃⁴𝜇₂ −

1

120
𝜃⁵𝜇₁) 

we obtain  𝜓(𝜃) = 𝑐𝑜𝑠(𝜃) + 𝑠𝑖𝑛(𝜃) giving the same result with the exact solution. 

CONCLUSIONS 

This paper deals with the solution of linear Fredholm integro-differential equations of 
the second kind with separable kernels. Our approach was based on the matrix 
approach which reduces the Fredholm integro- differential equation into a set of linear 
algebraic equations for the determination of the unknown constants. The advantage of 
this method over the direct computation method is that the constants in this method are 
obtained at once instead of the successive substitution approach inherent in the direct 
computation method. The method was tested on some model problems from the 
literature; the results obtained by the technique developed were the same with the exact 
solution revealing the effectiveness of the proposed method.  

ACKNOWLEDGMENTS  

The authors express their sincere thanks to the referees for the careful and details 
reading of their earlier version of the paper and for the very helpful suggestions. 

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Matrix Approach to The Direct Computation Method for The Solution of Fredholm Integro-
Differential Equations of The Second Kind With Degenerate Kernels  

Nathaniel Kamoh 107 

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https://doi.org/10.1108/MMMS-12-2017-0149


Matrix Approach to The Direct Computation Method for The Solution of Fredholm Integro-
Differential Equations of The Second Kind With Degenerate Kernels  

Nathaniel Kamoh 108 

The algorithm for the implementation of the given problems using the Maple 
programme is outline below: