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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

Repeated Corrected Simpson's 3/8 Quadrature Method for 
Solving Fredholm Linear Integral Equations of the Second 

Kind 
 

Wafaa  F. Mansoor 
Shaymaa  M. Shraida 

Dept. of Mathematics/College of Education for Pure Science(Ibn AL-Haitham) / 
University of Baghdad 

 
Received in:28 May  2012 , Accepted in:9  December  2012 

 
Abstract 
       In this paper, we use the repeated corrected Simpson's 3/8 quadrature method for 
obtaining the numerical solutions of Fredholm linear integral equations of the second kind. 
This method is more accurately than the repeated corrected Trapezoidal method and the 
repeated Simpson's 3/8 method. To illustrate the accuracy of this method, we give a numerical 
example. 
 
Keywords : Fredholm integral equations, repeated corrected Simpson's 3/8 method, 
numerical approximation.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

Introduction 
     Recall that the following linear integral equation of the second kind: 

b(x)

a
u(x) f (x) k(x, y)u(y)dy,= + λ ∫                                                                               …(1.1) 

where a x b,≤ ≤  λ  is areal number, f , k  are given continuous functions and u is the 
unknown function that must be determined. If b(x) = b then equation (1.1) is called Fredholm 
linear integral equation of the second kind. Also, if b(x) = x then equation (1.1) is called 
Volterra linear integral equation of the second kind. 
The mathematical modeling for many problems in different disciplines, such as engineering, 
chemistry, physics and biology leads to the above form of integral equations. Most 
differential equations can be expressed as integral equations, [1], [2]. There are some 
analytical methods for solving integral equations, [3], [4]. 
There are several numerical methods for solving the integral equations given by (1.1). For 
example, Repeated modified Trapezoid quadrature method [5], Runge-Kutta method [6], 
Taylor-series method [7], expansion methods [8] and decompostion method [9], [10].  
In this paper, we use the repeated corrected Simpson's 3/8 method to find the numerical 
solution for (1.1) in case b(x) = b, where b is a known constant. To do this, recall that the 
repeated corrected Simpson's 3/8 method, [11]: 

[ ] [ ]
b 2 8

(8)
0 1 2 3 0 2 1 3

a

3h 3h 81(b a)h
f (x) dx 31f  + 81(f  + f ) + 31f 19 f  + 27(f f ) 19f f ( ), a b

224 1120 4480
λ λ − −

′ ′ ′ ′= + − − + ξ ≤ ξ ≤∫   …(1.2) 

where 0f f (a)= , 1f f (a h)= + , 2f f (a 2h)= + , 3f f (b)= , 0f f (a)′ ′= , 1f f (a h)′ ′= + , 2f f (a 2h)′ ′= +  
and 3f f (b)′ ′= . 
 
Solution of Fredholm Linear Integral Equations of the second kind 
        In this section we use repeated corrected Simpson's 3/8 method to find the numerical 
solutions of the following integral equation: 

b

a
u(x) f (x) k(x, y)u(y)dy,= + λ∫                                                                                 …(2.1) 

To do this, we approximate the integral part that appeared in the right-hand side of equation 
(2.1) by the repeated corrected Simpson's 3/8 rule to get 

n 2 n 3

0 0 j j j 1 j 1 j j
j 1,4,7,... j 3,6,...

2

n n 0 0 0 0 j 1 j 1 j 1 j 1
j 1,4,7

3h
u(x) f (x) 31k(x, x )u 81 k(x, x )u k(x, x )u 62 k(x, x )u

224

3h
31k(x, x )u 19k(x, x )u 19J(x, x )u 27 k(x, x )u J(x, x )u

1120

− −

+ +
= =

+ + + +
=

λ    = + + + + +    



λ  ′ ′+ + + +
 

∑ ∑

n 2

j j j j n n n nk(x, x )u J(x, x )u 19k(x, x )u 19J(x, x )u

−


′ ′− − − − 


∑
 

…(2.2) 

where J(x, y) = 
k(x, y)

y
∂

∂
. Hence for 0 1 nx x , x ,..., x= , we get the following system equations: 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

n 2 n 3

i i i,0 0 i, j j i, j 1 j 1 i, j j i,n n
j 1,4,7,... j 3,6,...

2 n 2

i,0 0 i,0 0 i, j 1 j 1 i, j 1 j 1 i, j j i, j j i,n n
j 1,4,7

3h
u f + 31k u 81 k u k u 62 k u 31k u

224

3h
19k u 19J u 27 k u J u k u J u 19k u 1

1120

− −

+ +
= =

−

+ + + +
=

 λ    = + + + + +    
  

λ  ′ ′ ′ ′+ + + − − − −  


∑ ∑

∑ i,n n9J u , i 0,1,..., n


=


…(2.3) 
where ' 'i i i i i i i, j i j i,p i pu u(x ), u u (x ), f f (x ), k k(x , y ), J J(x , x ), i, j 0,1,..., n, p 0, n.= = = = = = =  

From equation (2.1) and setting 
k(x, y)

H(x, y)
x

∂
=

∂
 we obtain  

b

a
u (x) f (x) H(x, y)u(y)dy, a x b′ ′= + λ ≤ ≤∫                                                                 …(2.4) 

We note that if u is the solution of (2.1) it is a solution of (2.4) too. Now, for solving equation 
(2.4), we must consider two cases. 
 
Case (1): 
The partial derivative 

k(x, y)
L(x, y)

x y
∂

=
∂ ∂

 does not exists, in this case, we approximate the 

integral part that appeared in the right hand side of equation (2.4) with the repeated Simpson’s 
3/8 rule to obtain: 

n n
1

3 3
0 0 3 j 1 3 j 1 3 j 2 3 j 2 i 3 j 3 j n n

j 1 j 1

3h
u  (x) =f  (x)+ H(x, x )u 3 H(x, x )u H(x, x )u 2 H(x , x )u H(x, x )u

8

−

− − − −
= =

 λ    ′ ′ + + + +    
 

∑ ∑

…(2.5) 
By setting 0 1 nx x , x ,..., x= , in (2.5) one can get: 

n n
1

3 3
i i i,0 0 i,3 j 1 3 j 1 i,3 j 2 3 j 2 i,3 j 3 j i,n n

j 1 j 1

3h
u =f  + H u 3 H u H u 2 H u H u

8

−

− − − −
= =

 λ    ′ ′ + + + +    
 

∑ ∑              …(2.6) 

From equations (2.6) and (2.3) one can get the following system which consists of 
n

2n 3
3

− +  

equations with
n

2n 3
3

− + unknowns 'i 0 pu , i 0,1,..., n, u , u and for i=0, p, n,(p = 1,2,...,n-1)′=

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

2n 2 n 3

i i i,0 0 i, j j i, j 1 j 1 i, j j i,n n i,0 0 i,0 0
j 1,4,7,... j 3,6,...

n 2

i, j 1 j 1 i, j 1 j 1 i, j j i, j j i,n n
j 1,4,7

3h 3h
u f + 31k u 81 k u k u 62 k u 31k u 19k u 19J u

224 1120

27 k u J u k u J u 19k u 1

− −

+ +
= =

−

+ + + +
=

  λ λ    ′= + + + + + + +     
  

 ′ ′ ′+ − − − − 

∑ ∑

∑ i,n n
n n

1
3 3

0 0 0,0 0 0,3 j 1 3 j 1 0,3 j 2 3 j 2 0,3 j 3 j 0,n n
j 1 j 1

n
1

3
p p p,0 0 p,3 j 1 3 j 1 p,3 j 2 3 j 2 p,3 j 3 j p,n n

j 1

9J u , i 0,1,..., n,

3h
u =f  + H u 3 H u H u 2 H u H u ,

8

3h
u = f H u 3 H u H u 2 H u H u

8

−

− − − −
= =

−

− − − −
=


=



 λ    ′ ′ + + + +    
 

 λ    ′ ′ + + + + +    
 

∑ ∑

∑

n
3

j 1

n n
1

3 3
n n n,0 0 n,3 j 1 3 j 1 n,3 j 2 3 j 2 n,3 j 3 j n,n n

j 1 j 1

, p 1, 2,..., n 1,

3h
u = f H u 3 H u H u 2 H u H u .

8

=

−

− − − −
= =















 = −



 λ    ′ ′ + + + + +     
 

∑

∑ ∑

…(2.7) 
By solving the above system given by equation (2.7) the numerical solutions of equation 
(2.1), are obtained. 
 
Case (2): 
The partial derivative 

k(x, y)
L(x, y)

x y
∂

=
∂ ∂

 exists, in this case, we approximate the integral part 

that appeared in the right hand side of equation (2.4) with the repeated corrected Simpson’s 
3/8 rule to obtain: 

[

n 2 n 3

0 0 j j j 1 j 1 j j
j 1,4,7,... j 3,6,...

2

n n 0 0 0 0 j 1 j 1 j 1 j 1
j 1,4,7,.

3h
u  (x) =f  (x)+ 31H(x, x )u 81 H(x, x )u H(x, x )u 62 H(x, x )u

224

3h
31H(x, x )u 19H(x, x )u 19L(x, x )u 27 H(x, x )u L(x, x )u

1120

− −

+ +
= =

+ + + +
=

λ    ′ ′ + + + +    


 λ ′ ′+ + + + 


∑ ∑

n 3

..

j j j j n n n nH(x, x )u L(x, x )u 19H(x, x )u 19L(x, x )u

−
−


′ ′− − − 



∑

…(2.8) 
By setting 0 1 nx x , x ,..., x= , in (2.8) one can get: 

[

n 2 n 3

i i i 0 0 i j j i j 1 j 1 i j j i n n
j 1,4,7,... j 3,6,...

2 n 3

i 0 0 i 0 0 i j 1 j 1 i j 1 j 1 i j j i j j i n
j 1,4,7,...

3h
u  =f + 31H , u 81 H , u H , u 62 H , u 31H , u

224

3h
19H , u 19L , u 27 H , u L , u H , u L , u 19H , u

1120

− −

+ +
= =

−

+ + + +
=

 λ    ′ ′ + + + + +    
  

λ  ′ ′ ′ ′+ + + − − − 

∑ ∑

∑ n i n n19L , u


− 


 …(2.9) 

From equations (2.9) and (2.3) one can get the following system which consists of 
n

2n 3
3

− +  

equations with
n

2n 3
3

− +  unknowns iu , i 0,1,..., n,= 0u′ , 
'
pu and nu′ For i=0, p, n,                      

(p = 1,2,…,n-1). 
 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

 
 
 

2n 2 n 3

i i i,0 0 i, j j i, j 1 j 1 i, j j i,n n i,0 0 i,0 0
j 1,4,7,... j 3,6,...

n 2

i, j 1 j 1 i, j 1 j 1 i, j j i, j j i,n n
j 1,4,7

3h 3h
u f + 31k u 81 k u k u 62 k u 31k u 19k u 19J u

224 1120

27 k u J u k u J u 19k u 1

− −

+ +
= =

−

+ + + +
=

  λ λ    ′= + + + + + + +     
  

 ′ ′ ′+ − − − − 

∑ ∑

∑ i,n n

2n 2 n 3

0 0 0,0 0 0, j j 0, j 1 j 1 0, j j 0,n n 0,0 0 0,0 0
j 1,4,7,... j 3,6,...

0, j 1 j 1 0, j 1 j 1 0, j j 0, j j
j 1

9J u , i 0,1,..., n

3h 3h
u f 31H u 81 H u H u 62 H u 31H u 19H u 19L u

224 1120

27 H u L u H u L u

− −

+ +
= =

+ + + +
=




=



 λ λ ′ ′ ′= + + + + + + + +   
  

 ′ ′+ − − 

∑ ∑

n 2

0,n n 0,n n
,4,7,...

2n 2 n 3

p p p,0 0 p, j j p, j 1 j 1 k, j j p,n n p,0 0 p,0 0
j 1,4,7,... j 3,6,...

p, j 1 j 1 p, j 1 j 1 p, j j

19H u 19L u

3h 3h
u f 31H u 81 H u H u 62 H u 31H u 19H u 19L u

224 1120

27 H u L u H u L

−

− −

+ +
= =

+ + + +


′− − 



 λ λ  ′ ′ ′= + + + + + + + +   
  

′ ′+ − −

∑

∑ ∑

[

n 2

p, j j p,n n p,n n
j 1,4,7,...

2n 2 n 3

n n n,0 0 n, j j n, j 1 j 1 n, j j n,n n n0 0 n0 0
j 1,4,7,... j 3,6,...

nj 1 j 1

u 19H u 19L u , where p j, j 1

3h 3h
u f 31H u 81 H u H u 62 H u 31H u 19H u 19L u

224 1120

27 H u

−

=

− −

+ +
= =

+ +


  ′− − = + 



 λ λ ′ ′ ′= + + + + + + + +  
  

′ +

∑

∑ ∑

n 2

nj 1 j 1 nj j nj j nn n nn n
j 1,4,7,...

L u H u L u 19H u 19L u
−

+ +
=


































  ′ ′− − − −  


∑

  

…(2.10) 
By solving the above system given by equation (2.10) the numerical solutions of equation 
(2.1), are obtained. 
 
Numerical Example 
        In this section, we give some numerical examples to illustrate the above method for 
solving Fredholm linear integral equation of the second kind. Also, we compare this method 
with other methods such as repeated Simpson's 3/8 method and repeated corrected 
Trapezoidal method for solving Fredholm linear integral equation of the second kind.  
 
Consider the following Fredholm linear integral equation of the second kind: 

37 5 7 1
22 2 2

0

2 2 4
u(x) x (x 1) (x 1) x x (x y) u(y)dy

7 5 35
= − + + + − + +∫ , 0 ≤ x ≤ 1                    …(3.1) 

where 
7 5 7
2 2 22 2 4f (x) x (x 1) (x 1) x x ,

7 5 35
= − + + + −  

3
2

k(x, y) (x y)= + . This example is 

constructed such that the exact solution is u(x) = x. It is clear that 
2k(x, y)

x y
∂

=
∂ ∂

 
3

4 (x y)+
 

does not exist at x = y = [0, 1]. To do this, first, we divide the interval [0,1] into 9 intervals 

such that i
i

x ,
9

=  i 0,1,..., 9=  Therefore, the numerical solution of system (3.1) can be 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

obtained by equation (2.7), and illustrated in Table 1، Which contains the exact solutions and 
the numerical solutions by using the repeated Simpson's 3/8, repeated corrected trapezoidal 
and repeated corrected Simpson's 3/8 method are given for  h =0.112. Also, the least square 
error denoted by LSER and absolute error denoted by ABSER are presented in Table (1). 

Second, we divide the interval [0,1] into 18 subintervals that i
i

x ,
18

=  i 0,1,...,18=  Therefore, 

the numerical solution of system (3.1) can be obtained by equation (2.7), the results are 
presented in Table 2. The exact solutions and the numerical solutions using the repeated 
Simpson's 3/8, repeated corrected trapezoidal and repeated corrected Simpson's 3/8 method 
for  h =0.056. Also, in this table, the least square error denoted by LSER and absolute error 
denoted by ABSER are presented. 

Third, we divide the interval [0,1] into 36 subintervals that i
i

x ,
36

=  i 0,1,..., 36=  Therefore, 

the numerical solution of system (3.1) can be obtained by equation (2.7), the results of which 
being presented in Table 3. In that table, the exact solutions and the numerical solutions by 
using the repeated Simpson's 3/8, repeated corrected trapezoidal and repeated corrected 
Simpson's 3/8 method are given for  h =0.028. Also, the least square error denoted by LSER 
and absolute error denoted by ABSER are also included in table 3. 
 
Discussion        
      In this paper we proposed the repeated corrected Simpson's 3/8 method for solving 
Fredholm linear integral equation of the second kind. This method can be easily applied and it 
is efficient and accurate to estimate the solution of Fredholm linear integral equation when the 

function
k(x, y)

x
∂
∂

, 
k(x, y)

y
∂

∂
and f (x)′  exist.  

 
References 
1. Burton, T.A. (2005) Volterra integral and differential equation, second ed., Elsevier, 
Netherlands. 
2. kanwal, P. (1971) Linear integral equations, theory and technique, Academic press, Inc. 
3. Polyanin A. and Manzhirov, A. (1998) Handbook of integral equations, Crc press Llc. 
4. Atkinson, K. (1997) The Numerical solution of integral equations of the second kind, 
combridge University press. 
5.Saberi-Nadjafi, J. and Heidari, M. (2007) Solving Linear Integral Equations of the Second 
Kind with Repeated Modified Trapezoid Quadrature Method, Applied Mathematical and 
Computer, 189, No. 4, 980-985. 
6. Maleknejad, K. and Shahrezaee, M. (2004) Solving Runge-Kutta mthod for numerical 
solution of the Volterra integral equations, Applied Mathematical and Computer, 149, No. 2, 
399-410. 
7.Maleknejad, K.; Rabbani, M.; and Aghazadeh, N. (2006) Numerical solution of second kind 
Fredholm integral equations system by using a Taylor-series expansion method, Applied 
Mathematical and Computer, 175, No. 2, 1229-1234. 
8. Rabbani, M.; Maleknejad, K. and Aghazadeh, N. (2007) Numerical computational solution 
of the Volterra integral equations system of the second kind by using an expansion method, 
Applied Mathematical and Computer, 187, No. 2, 1143-1146. 
9. Babolian, E.; Biazar, J.and Vahidi, A. R. (2004) On the decomposition method for system 
of linear equations and system of linear Volterra integral equations, Applied Mathematical 
and Computer, 147, No. 1, 19-27. 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

10. Vahidi, A. R.; Mokhtari, M.and Vahidi, A. R. (2008) On the decomposition method for 
system of linear Fredholm integral equations of the second kind, Applied Mathematical 
Sciences, 2, No. 2, 57- 62. 
11.Al-Sa’dany, Sh. (2008) Some modified quadrature method for solving systems of Volterra 
linear integral equations, M.Sc. Thesis, University of Baghdad. 
 

Table (1): Comparison between the solution via corrected Trapezoidal, repeated 
Simpson's 3/8 and corrected Simpson's 3/8 with exact solution with N = 9 

Nodes 
Exact 

Solutio
ns 

Numerical solutions 
ui ，N=9 

xi  ui  corrected Trapezoidal repeated Simpson's 3/8 corrected Simpson's 3/8 
0 0 0.00000422427391 -0.00000804493901 0.00000074026838 

0.1 0.1 0.11111322328397 0.11111173431260 0.11111125469004 
0.2 0.2 0.22222443564682 0.22222707233601 0.22222240727548 
0.3 0.3 0.33333593424825 0.33334133788125 0.33333358046105 
0.4 0.4 0.44444756925228 0.44445521086227 0.44444476277767 
0.56 0.56 0.55555929602997 0.55556892483339 0.55555595228195 
0.67 0.67 0.66667109471999 0.66668258127157 0.66666714815178 
0.78 0.78 0.77778295432453 0.77779623077884 0.77777834986280 
0.89 0.89 0.88889486785395 0.88890990091898 0.88888955702285 

1 1 1.00000683043123 1.00002360760331 1.00000076931735 
LSER 1.8653e-010 2.0399e-009 2.5200e-012 

ABSER 4.04300e-005 1.2464e-004 4.522109e-006 
 

Table (2): Comparison between the solution via corrected Trapezoidal, repeated 
Simpson's 3/8 and corrected Simpson's 3/8 with exact solution with N = 18 

Nodes 
Exact 
Soluti

ons 

Numerical solutions 
ui ，N=18 

xi  ui  corrected Trapezoidal repeated Simpson's 3/8 corrected Simpson's 3/8 

0 0 0.00000036160574 -0.00000094631322 0.00000006419735 
0.1 0.1 0.11111124066062 0.11111112823436 0.11111112071068 
0.2 0.2 0.22222235617522 0.22222255247321 0.22222223531051 
0.3 0.3 0.33333349057657 0.33333387923624 0.33333335065382 
0.4 0.4 0.44444463362786 0.44444517497656 0.44444446649763 
0.56 0.56 0.55555578243406 0.55555645888978 0.55555558277799 
0.67 0.67 0.66666693572924 0.66666773862347 0.66666669945537 
0.78 0.78 0.77777809281810 0.77777901780483 0.77777781650046 
0.89 0.89 0.88888925326072 0.88889029834788 0.88888893389011 

1 1 1.00000041675099 1.00000158133359 1.00000005160523 
LSER 1.30200e-012 1.75118e-011 2.0854e-014 

ABSER 4.6354e-006 1.61007e-005 5.5708e-007 
 
 
 
 

334 | Mathematics 



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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

Table (3): Comparison between the solution via corrected Trapezoidal, repeated 
Simpson's 3/8 and corrected Simpson's 3/8 with exact solution with N = 36 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Nodes 
Exact 
Soluti

ons 

Numerical solutions 
ui ，N=36 

xi  ui  corrected Trapezoidal repeated Simpson's 3/8 corrected Simpson's 3/8 
0 0 0.00000003136368 – 0.00000009987493 0.00000000555524 

0.1 0.1 0.11111111905472 0.11111111207258 0.11111111176558 
0.2 0.2 0.22222223038488 0.22222224437474 0.22222222311676 
0.3 0.3 0.33333334291492 0.33333336952570 0.33333333450532 
0.4 0.4 0.44444445598728 0.44444449257726 0.44444444592461 
0.56 0.56 0.55555556941910 0.55555561485323 0.55555555737127 
0.67 0.67 0.66666668313083 0.66666673686136 0.66666666884296 
0.78 0.78 0.77777779707885 0.77777785883792 0.77777778033788 
0.89 0.89 0.88888891123562 0.88888898090618 0.88888889185461 

1 1 1.00000002558200 1.00000010313524 1.00000000339195 
LSER 9.53366e-015 1.4855e-013 1.71916e-016 

ABSER 5.488e-007 2.0724e-006 6.96503e-008 

335 | Mathematics 



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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

لحل معادالت تكاملیة خطیة  ةتكررالمعدلة الم  3/8 سمبسون لطریقة التربیعةا
 من نوع فریدھولم من النوع الثاني

 
 وفاء فیصل منصور
 شیماء مخلف شریدة

 جامعة بغداد / )بن الھیثما( للعلوم الصرفة كلیة التربیة /الریاضیات علوم قسم 
 

 2012كانون االول    9قبل البحث في:  ،  2012ایار    28استلم البحث في:
 

 الخالصة
الیجاد الحل العددي للمعادالت التكاملیة الخطیة من النوع الثاني. ھذة  المطورة 3/8 ھذا البحث نستخدم سمبسونفي         

لتوضیح دقة . 3/8الطریقة تحل المعادالت التكاملیة الخطیة بدقة أكثر من طریقة شبة المنحرف المطورة وطریقة سمبسون 
 .ھذه الطریقة تم اعطاء مثال عددي

 
 المطورة، التقریب العددي )3/8(  معادالت فرید ھولم التكاملیة، طریقة سمبسون:  المفتاحیةالكلمات 

  

336 | Mathematics 


	Recall that the following linear integral equation of the second kind:
	…(1.1)
	where   is areal number, ,  are given continuous functions and is the unknown function that must be determined. If b(x) = b then equation (1.1) is called Fredholm linear integral equation of the second kind. Also, if b(x) = x then equation (1.1) is ca...
	The mathematical modeling for many problems in different disciplines, such as engineering, chemistry, physics and biology leads to the above form of integral equations. Most differential equations can be expressed as integral equations, [1], [2]. Ther...
	There are several numerical methods for solving the integral equations given by (1.1). For example, Repeated modified Trapezoid quadrature method [5], Runge-Kutta method [6], Taylor-series method [7], expansion methods [8] and decompostion method [9],...
	In this paper, we use the repeated corrected Simpson's 3/8 method to find the numerical solution for (1.1) in case b(x) = b, where b is a known constant. To do this, recall that the repeated corrected Simpson's 3/8 method, [11]:
	Numerical Example
	Discussion
	In this paper we proposed the repeated corrected Simpson's 3/8 method for solving Fredholm linear integral equation of the second kind. This method can be easily applied and it is efficient and accurate to estimate the solution of Fredholm linea...