Modified Decomposition Method for Solving Nonlinear
Fredholm-Volterra Integral Equations
1
1,2,3
F. S. Zulkarnain, 2Z.K. Eshkuvatov,3N. M. A. Nik Long
Department of Mathematics, Faculty of Science, Universiti Putra Malaysia (UPM), Malaysia
2,3
Institute for Mathematical Research (INSPEM),UPM, Malaysia
Abstract. In this note, an approximate solution of nonlinear Volterra-Fredholm integral equation is obtained by using
modified decomposition method. General cases of nonlinear terms in the equation are considered. Finally some
numerical examples are presented to validate the accuracy and efficiency of the method.
Keywords: Volterra-Fredholm integral, decomposition method, numerical method.
PACS: 02.30.Rz, 02.60.Lj
u  x, t   f  x, t 
INTRODUCTION
Recently, Adomian decomposition method [1,2] has
been focus on many stochastic and deterministic
problems in physics, biology and chemical reactions.
In mathematical problems, this method has been
applied for solving a large class of linear and nonlinear
equations including differential and integral equations,
integro-differential equations, and so on.
The Volterra-Fredholm integral equation arises
from parabolic boundary value problems, the
mathematical modeling of the spatio-temporal
development of an epidemic and biological problems.
The essential features of these models are of wide
applicability. Detailed descriptions and formulation of
these models can be found in [3,4] and references
therein.
Particularly, Bildik and Inc (2007) have solved
the following nonlinear Volterra-Fredholm integral
equation by using modified decomposition method.
u  x   f  x   1  K1  x, t  u  t   dt
a
x
p
 2  K 2  x, t  u  t   dt
b
(1)
q

t

0 
F  x, t ,  , , u  ,   d  d ,
(2)
 x, t   0, T  and ux, t  is an unknown
function, f x, t  and F x, t ,  , , u ,  are analytic
where
functions on D    0, T  , where  is a closed
n
subset of R , n  1,2,3. Wazwaz (2002) has treated
Eq. (2) using modified decomposition method.
In this paper, we applied the modified
decomposition method (MDM) to obtain approximate
solution of nonlinear Fredholm-Volterra integral
Equation(9) for any p, q  1 . Moreover, we have
shown 4 Examples to validate the method proposed.
2. MODIFIED DECOMPOSITION
METHOD (MDM)
2.1 General Scheme for Operator
Equation
a

where p is a positive, q  1 , f x is the given function
K1 x, t  and K 2 x, t  are square integrable kernels, 1 ,
2 , a and b are constants. Yalçinbaş (2002) have
obtained an approximation solution of Eq. (1) by using
Kanwal and Liu method. Maleknejad and Hadizadeh
(1999) have presented the solution for following
nonlinear mixed Volterra-Fredholm integral equation
by using the standard decomposition method
Let us consider the operator equation of the form
Lu  Ru  Nu  g x,
(3)
where L is the operator of highest order derivative
which is assumed to be invertible, R is the remaining
linear operator collecting the lower order derivatives,
Nu represents the nonlinear terms and g is a source
term. Multiplying inverse function
of Eq. (3) we get
L1 to both sides
u  f x   L1 Ru   L1 Nu ,
(4)
where f x   L1 g x  .In standard decomposition
method, the unknown function u x  is searched as a
series of the form

2.2 Application of Modified Decomposition
Method (MDM) for Nonlinear
Fredholm-Volterra Integral Equations
We consider nonlinear Fredholm-Volterra equation
u  x   f  x   1  K1  x, t  F1  u  t   dt
b
u  x    u n  x ,
(5)
a
n 0
 2  K 2  x, t  F2  u  t   dt ,
(9)
x
a
the components
u0 , u1 , u2 , are usually determined
where f (x ) is the given function, K1 ( x, t ) and
recursively by
K 2 ( x, t ) are square integrable kernels of the equation,
u0  x   f  x ,
u n1  x    L1 Ru n  x   L1  Nun  x ,
(6)
n  0.
It is note that the decomposition method suggest that
the initial component u0 x  as the function f x . To

apply modified decomposition method, the function
f x  on the right side of the Eq. (2) is divided into
two parts
f x  f1 x  f 2 x.
(7)
1 and  2 are parameters while F1 (u(t )) and F2 (u(t ))
are nonlinear terms and u (x) is unknown function to
be determined.
The nonlinear term F1 (u(t )) and F2 (u(t )) are
represented by the Adomian polynomial series.
Wazwaz (2002) introduced the procedure to calculate
the Adomian polynomials

F1 u t    An t ,
n 0

(10)
F2 u t    Bn t .
n 0
Based on the expression above, Wazwaz (1999)
propose a slight variation only on the components
u0 x  and u1 x . He proposed that f1 ( x) contain the

zeroth component
u 0 while f 2 ( x) is the remainder
terms. Thus, the following recursive relations for the
modified decomposition method are formulated as
u0  x   f1  x ,
C0  F  u0  ,
C1  u1 F   u0  ,
1 2
u1 F   u0  ,
2!
C3  u3 F   u0   u1u2 F   u0 
C2  u 2 F   u0  
u1  x   f 2  x   L1 Ru 0 x 
 L1 Nu0 x 
(8)
u n1 x    L1 Ru n  x   L1  Nun  x ,
n 1
We can conclude that the zeroth component of
standard decomposition method in (4) is defined by
the function f while in modified decomposition
method, the zeroth component is defined by selecting
some terms f1 x of function f x . The remaining

where An (t ) and Bn (t ) are Adomian polynomials
defined by

part will be added to the next component u1 .Easy
calculations and fast solutions in many cases depend
on our proper choice of the parts f1 and f 2 of f .

(11)
1 3
u1 F   u0  ,
3!
Here Ai  Ci if F  F1 and Bi  Ci if F  F2 .The
nonlinear terms can be considered in different cases.
p
For example for the case of F  u    u  x   , the first
few Adomian polynomials C n are given by
C0  u0p ,
3. EXAMPLES
C1  pu0p 1u1 ,
Example 1. Consider the nonlinear Fredholm integral
equation of the second kind
p ( p  1) p  2 2
C2  pu0p 1u2 
u0 u1 ,
2!
C3  pu0p 1u3  p ( p  1)u0p  2u1u2
(12)
u x   1 
p ( p  1)( p  2) p  3 3

u0 u1 ,
3!

For the case of F (u )  sin u ( x) , few terms of
Adomian polynomials An are defined by
C0  sin u 0 ,
1
5
x   3xt u 2 t  dt ,
0
4
(16)
Solution: It is clear that f  x   1  5 , 1  3, 2  0
4
and K1 x, t   xt . Since 2  0 the Adomian
Bn  0, n  0,1,... and An for nonlinear
term F1 ut   u 2 t  are given in (12) with p  2 .
polynomials
A0  u02 ,
C1  u1 cos u 0 ,
(13)
1
C 2  u 2 cos u 0  u12 sin u 0 ,
2!
A1  2u0u1 ,
A2  2u0u2  u12 ,
(17)
A3  2u0u3  2u1u2 ,
1
C3  u3 cos u 0  u1u 2 sin u 0  u13 cos u 0 ,
3!

For the case of F u   e u t  , the first few Adomian
polynomials Cn are given in the form
Now, we divided
f x  into two parts such as
1
f1 x   1  x and f 2  x    x . Using the modified
4
recursive relation (15) we obtain
C 0  e uo ,
1
1
u1  x    x   3xt A0  t  dt  0,
0
4
C1  u1e uo ,
1 

C 2   u 2  u12 e uo ,
2
! 

1 

C3   u3  u1u 2  u13 e uo ,
3! 

(14)
Consequently
u k x   0,

k  1.
Then, the solution is
Thus, the components u 0 , u1 , u 2 , of u x  substitute
into series in Eq (5) and each components are defined
by
u0  x   f1  x  ,
ux   1  x.
Example 2. Consider the nonlinear Volterra integral
equation of the second kind
u  x   4x2 
u1  x   f 2  x   1  K1  x, t  A0  t  dt
b
a
 2  K 2  x, t  B0  t  dt ,
x
(15)
a


x
cos  4 x 2   1
2
 4  xt sin u  t  dt ,
x
0
un 1  x   1  K1  x, t  An  t  dt
b
Solution: From Eq. (18) we found that
a
 2  K 2  x, t  Bn  t  dt ,
x
a
n  1.
(18)
   
f x   4 x 2 
x
cos 4 x 2  1 , 1  0 ,
2
K1 x, t   0 , K 2 x, t   xt.
u0  x   x ,
2  4 ,
u1  x  
The nonlinear term F2 (u)  sin ux  are given in
Eq. (13). Again we divide f  x  into two parts by
2
x
selecting f1 x   4x and f 2  x   cos 4 x 2  1 .
2
Then, it follows that
   x  t  t 2 dt   et dt ,

x
Consequently
u k x   0,
u t  
 0.
So that
k  1.
Its exact solution is
Solution: Note that
Thus, the solution is
Example 3. Consider the nonlinear Fredholm-Volterra
integral equation.
5 2
 x  ex
4 3
(19)
   x  t  u  t  dt   e
0
x
0
u t 
dt

(20)
u x   2 x .

1
1
1
20 x  12 x 6 , 1  ,  2  ,
2
4
15
K1  x, t   x , K2  x, t   x .
f x  
u ( x)  4 x 2 .
2
k  1.
1
 20 x  12 x6 
15
1 1
1 x
  x u 2  t  dt   x u 4  t  dt
0
2
4 0
0
1
0
Example 4. Consider the nonlinear Fredholm-Volterra
integral equation.
 4  xt sin  4t 2  dt ,
u  x 
0
ux   x.
x
u1  x   cos  4 x 2   1
2
uk ( x)  0,
x
Then, the solution is
Consequently, u1 x  is defined by

1
 0.
   
u 0 x   4 x 2 ,
5 1
 x  ex
4 3
The Adomian polynomials for F1 t   u 2 t  are given
in (17) and F2 t   u 4 t  are given in (12) with p  4 .
To use the MDM we compare two initial guess:
u0 
4
999
x and u 0 
x.
3
500
Solution: Noted that
5 2
f  x    x  e x , 1  1 , 2  1,
4 3
K1  x  t , K 2  1 .
The Adomian polynomials for
given in (17) while for
F1 t   u 2 t  are
F2 t   e u t  are given in (14).
We choose f1 ( x)  x as the initial value. Then it
follows from Eq. (15) that
By using the recursive relation in Eq. (15) we present
the numerical results in Table 1. The results in Table 1
are obtained by substituting the 5 terms into the series
Eq. (5).
Based on Table 1, initial value u 0  x   999 x show
500
that MDM gives more approximate to the exact
solution than initial value u 0  x   4 x .
3
Table 1. Numerical results for Example 4.
MDM (5
MDM (5
x Exact
solution
iterations)
iterations) with
with initial guess
initial guess
u x 
u0  x  
0
0
0.1
0.2
0.3
0.4
0.5
0.2
0.4
0.6
0.8
1.0
4
x
3
u0  x  
x
Exact
solution
u x 
999
x
500
MDM (5
iterations)
with initial guess
u0  x  
4
x
3
MDM (5
iterations) with
initial guess
u0  x  
999
x
500
0
0
0.6
1.2
1.00712624
1.19805306
0.17106581
0.34210702
0.51290891
0.68263034
0.84902789
0.19970755
0.39941485
0.59911980
0.79881408
0.99847536
0.7
0.8
0.9
1.0
1.4
1.6
1.8
2.0
1.14667287
1.24685921
1.26622664
1.12955769
1.39743720
1.59637529
1.79420977
1.98889004
CONCLUSION
In this paper, we have solved general nonlinear
Fredholm-Volterra integral equations by using
modified decomposition method. From the numerical
Examples 1-3 we can see that MDM is identical with
the exact solution for the given problems considered.
In Example 4 (Table 1) developed MDM is not exact
but rate of convergence of MDMcan be accelerated by
the choice of initial guess.
ACKNOWLEDGMENTS
This work was supported by University Putra Malaysia
under Fundamental Research GrandScheme (FRGS).
Project code is 01-12-10-989FR.
REFERENCES
1. G. Adomian, Solving Frontier Problems of Physics: The
Decomposition Method, Boston: Kluwer. (2004)
2. G. Adomian, Comp. and Math.with Appl., 21, 101-127
(1991).
3. O. Diekmann, Thresholds and travelling waves for the
geographical spread infection, J. Math Biol., 6, 109-130.
(1978).
4. H. R. Thieme, A model for the spatial spread of an
epidemic, J. Math. Biol., 4, 337-351 (1977).
5. N. Bildik, M. Inc, Modified decomposition method for
nonlinear method Volterra-Fredholm integral equations,
Chao.Solit and Fract.,33, 308-313 (2007).
6. S. Yalçinbaş, Taylor polynomial solutions of nonlinear
Volterra-Fredholm integral equations.Appl. Math.and
Comp.,127, 195-206 (2002).
7. K. Maleknejad and M. Hadizadeh, A new computational
method for Volterra-Fredholm integral equations, Comp.
and Math.with Appl., 37, 1-8 (1999)
8. A. M. Wazwaz, A new algorithm for calculating adomian
polynomials for nonlinear operators.Appl. Math.and
Comp.,111, 53-69 (2000).
9. A. M. Wazwaz, A realiable treatment for mixed
Volterra-Fredholm integral equations, Appl. Math. and
Comp., 127, 405-414 (2002).
10. A. M. Wazwaz, A reliable modification of Adomian
decomposition method, Appl. Math. and Comp., 102, 7786 (1999).
11. S. H. A. M. Shah, A. W. Shaikh, S. Sandilo, S., H.
(2010). Modified decomposition method for nonlinear
Volterra-Fredholmintegrodifferentialequation.Journal of
Basic and Applied Sciences.,6, 13-16 (2010).