Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28), Spring 2011, pp 55-60
ISSN: 2008-594x
Discrete Collocation Method for Solving Fredholm–
Volterra Integro–Differential Equations
Mohsen Rabbani1*, S.H.Kiasoltani2
1
2
Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran
Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran
Received: 29 November 2010
Accepted: 31 January 2011
Abstract
In this article we use discrete collocation method for solving Fredholm–Volterra integro–
differential equations, because these kinds of integral equations are used in applied sciences and
engineering such as models of epidemic diffusion, population dynamics, reaction–diffusion in
small cells. Also the above integral equations with convolution kernel will be solved by discrete
collocation method. In this method we approximate solution of problem by no smooth
piecewise polynomial. Numerical results show a high accuracy and validity discrete collocation
method.
Keywords: Discrete Collocation Method, Fredholm–Volterra Integro, Differential Equations.
1 Introduction
For solving Valterra integral equations see [3, 4] . Also for chemical absorption
kinetics was used from Volterra integral equations in [5] . Several numerical methods for
solving Fredholm integral equations are in [1, 7] . Fredholm integro–differential equation
by using Petrove-Galerkin is solved in [8] also by spline wavelet and spline is solved
[6, 2] respectively. Fredholm-Volterra integro–differential equation particular with
convolution kernel is important and we consider this equation:
 y (t )  y (t )  f (t )  t k (t , ) y ( )d  1 k (t , ) y ( )d , t  [0, T ],

0 1
0 2


 y (0)   ,
(1)
where f (t ) and k i (t , ) for i  1,2 are known functions also k i (t , ) can be convolution
kernel, and y (t ) is unknown function.
*Corresponding author
E-mail address: mrabbani@iausari.ac.ir
55
M. Rabbani et al, Discrete Collocation Method for Solving Fredholm–Volterra Integro–Differential Equations
We decide to approximate y (t ) by Lagrange polynomial interpolation such that it
obtain by non- smooth piecewise polynomial space. So we use uniform mesh I h on
[0, T ],
I h  t n  n h ; n  0,1,..., N , N h  T .
(2)
Also,
 u0 ( ) , t 0    t1 ,
 u ( ) , t    t ,
 1
1
2
y ( )  u ( )  


u N 1 ( ) , t N 1    t N ,
(3)
where u ( ) is an element of the non-smooth piecewise polynomial with degree less than
m , in other words:
(4)
u ( )  S m( 11) ( h )  {u ( ) u ( )  un (t )   m1 ,   (t n , t n1 ], n  0,...N  1},
T
,..., TN . ( for simplifica tion, T  1).
N
For finding un (t ) , n  0,1,..., ( N  1) in (3) , we use Lagrange interpolation with
(ci , yn ,i ) points for n  0,1,..., ( N  1) and i  0,1,..., m, where yn , i is approximation
t 0  0 , t1 
y (tn ,i ) and ci , i  1,..., m are m -points Gauss as collocation parameters that they can
be found by roots of Legendre polynomial on [0,1] , for example in the case of m  4
they are obtained the following form:
p( s) 
c1 
7 (20s 3  30s 2  12s  1)  0 ,
5  15
,
10
c2 
1
,
2
c3 
(5)
5  15
,
10
also we choose c4  1 as end-point of [0,1].
By considering m -points Gauss, we define mesh points as follows:
t n , i  t n  ci h, i  1,2,..., m, n  0,1,..., ( N  1).
According to condition of interpolation we can write
u (t n ,i )  y n , i , i  1,..., m, n  0,1,..., ( N  1),
and,
m
u n (t n   h)   Li ( ) y n ,i ,
  [0,1] ,
n  0,1,..., ( N  1),
(6)
(7)
i 1
by choosing   tn   h then   (tn , tn 1 ] , in this way u (t ) is restricted to subintervals
such as (tn , tn1 ] .
For solving Eq.(1), it can be written the following form:
56
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28), Spring 2011, pp 55-60
tn
t
tn
0
tn
0
y (t )  y (t )  f (t )   k1 (t , ) y ( )d   k1 (t , ) y ( )d   k 2 (t , ) y ( )d 

1
tn
k 2 (t , ) y ( )d .
Now, we introduce discrete form Eq. (1) ,
 F (t )  f (t )  tn k (t , ) y ( )d  tn k (t , ) y ( ) d ,
0 1
0 2
 n

t
1
 y (t )  y (t )  Fn (t )   k1 (t , ) y ( ) d   k 2 (t , ) y ( ) d .
tn
tn

(8)
(9)
For numerical solution of Eqs. (8 9), we use (3  7) as follows:
 F (t )  f (t )  tn k (t ,  ) u ( )d  tn k (t ,  ) u ( )d ,
 n n, i
n, i
 0 1 n,i
 0 2 n, i


n  0,1,..., ( N  1), i  1,2,..., m ,
also,
tn , i
1
m
 Ln (t n , i ) y n ,i  y n ,i  Fn (t n , i )   t k1 (t n ,i ,  ) u ( )d   t k 2 (t n ,i ,  ) u ( )d ,
n
n
 i 1
n  0,1,..., ( N  1), i  1,2,..., m ,

(10)
(11)
where u ( ) is introduced by (3) . At first Fn (t n , i ) , n  0,1,..., ( N  1), i  1,2,..., m are
computed by (10) , then they are used in system (11) for obtaining y n , i s.
In (10) for n  j we have :
tj
F j (t j , i )  f (t j , i )   [k1 (t j , i ,  ) k 2 (t j , i ,  )] u ( ) d ,
0
j  0,1,2,..., ( N  1).
From (3) and (7), we can write
 F0 (t 0, i )  f (t 0, i ) , i  1,2,...m,

( j 1)
t p 1

F
(
t
)

f
(
t
)

[k1 (t p , i ,  )  k 2 (t p , i ,  )] u p ( )d ,
 j j, i

j, i

tp
p

0

 j  1,..., ( N  1),

(12)
By substituting Fn (t n , i ) , n  0,1,..., ( N  1) and i  1,2,..., m in (11) y n ,i s are obtained
by the following system :
( N 1)
tn , i
t p 1
m

L
(
t
)
y

y

F
(
t
)

k
(
t
,

)
u
(

)
d


k 2 (t n , i ,  ) u p ( )d , (13)
 n n , i n ,i

n, i
n n, i
1 n,i
n


tn
tp
p n
 i 1
n  0,1,..., ( N  1),

m
  tp
(  c i )
L
(

)

,   t p   h , so  
where q
and

h
i 1 (c q  c i )
iq
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M. Rabbani et al, Discrete Collocation Method for Solving Fredholm–Volterra Integro–Differential Equations
m
 tp
u p ( )   Lq ( ) y n , q   Lq (
) y n, q ,
h
q 1
q 1
m
p  0,1,..., ( N  1).
(14)
By solving algebraic system (13) we find yn , i S, i  1,2,..., m, n  0,1,..., ( N  1), then
by (14) , u p ( ), p  1,..., ( N  1) are found and also u (t ) as a approximation of y (t ) is
introduced.
3 Application
In this section, we solve two examples of Fredholm–Volterra integro–differential
equations by discrete collocation method.
Example 1
Consider Fredholm-Volterra integro–differential equations with convolution kernel :
t
1

5 3 t4
t 1
t
2
2
t 
 y (t )  y (t )  36e  15e  t   t  4t  7   0 (t   ) y ( )d   0  e y ( )d ,
3
6

 y (0)  5,

with exact solution y(t )  5  2t  t 2 .
1
, T  1, we use discrete collocation method and (12-14)
2
then interpolation is obtained the following form and numerical results are shown in
table1,
In the case of m  4 , h 
1

2
10 3
 5  2t  t  1.38584  10 t , 0  t  2
u (t )  
1
5  2t  t 2  3.91083  10 10 t 3 ,
 t  1,
2

Table 1. Numerical results for example 1.
u(t )  y(t )
ti
9
4.1  10
3.7  10 9
3.4  10 9
3.2  10 9
2.9  10 9
2.7  10 9
1.4  10 9
1.4  10 9
1.4  10 9
1.4  10 9
1.5  10 9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
58
2
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28), Spring 2011, pp 55-60
According numerical results, it is obvious that the proposed method has a high
accuracy.
Example 2
In this example we solve Fredholm-Volterra integro-differential equations as
follows:

t
1 sin( t )
cos t sin t
 2    e t  y ( )d  
y ( )d ,
 y (t )  y (t )  6  6e t  8t  4t 2  t 3 
0
0

t

t
 y (0)  0,

with exact solution y (t )  t 2 .
1
By choosing m  4, h  , T  1 in the discrete collocation method and use (12 14)
2
interpolation u (t ) is introduced by,
1

7
7
2
9 3
 2.44001  10  2.43645  10 t  t  9.22112  10 t , 0  t  2
u (t )  
1
 2.717  10 7  3.10075  10 7 t  t 2  2.08193  108 t 3 ,
 t 1
2

and absolute errors in some points are given in table 2
Table 2. Numerical results for example 2
u(t )  y(t )
ti
2
7
2.4  10
2..2  10 7
2..0  10 7
1.8  10 7
1.6  10 7
1.5  10 7
1.2  10 7
1.1  10 7
9.5  10 8
8.2  10 8
7.0  10 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
By considering numerical results are shown in the table 2, we conclude the discrete
collocation method has a high accuracy and it can be used for other problems.
4 Conclusion
In this work, we discrete integral equations to use collocation method, thus we could
obtain the numerical results with high accuracy. Also these results are shown ability and
validity the proposed method.
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M. Rabbani et al, Discrete Collocation Method for Solving Fredholm–Volterra Integro–Differential Equations
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