TX_1~ABS:AT/ADD:TX_2~ABS:AT


41 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 41-45

ReseaRch aRticle

Modification of Adomian Decomposition Method to Solve Two 
Dimensions Volterra Integral Equation Second Kind
Talhat I. Hassan1, Chiman I. Hussein2

1Department of Automative Technology Engineering, Erbil Technology College, Erbil Polytechnic University, Kurdistan Region, Iraq,  
2Department of General Sciences, College of Basic Education, Salahaddin University-Erbil, Kurdistan Region, Iraq

ABSTRACT

In this work, the adomian decomposition method is proposed for the first time to solve problems in two-dimensional Volterra Integral 
equations of the second kind. In addition, modification of this method is introduced to address this problem. Several new theorems 
are introduced and proven regarding contractive mapping and uniform convergence to support the method. Numerical problems are 
solved using the software Matlab to obtain results and demonstrate the efficiency and simplicity of the technique. The modified Adomian 
decomposition method performs faster than the adomian decomposition method for obtaining results.

Keywords: Adomian decomposition method (ADM), two-dimensional Volterra Integral equations numerical solution, least square error, 
uniformly convergence

INTRODUCTION

Integral equations are a branch of mathematics that plays 
a significant role in various fields. Over recent decades, 
many problems involving integral equations have been 

formulated from different scenarios in applied sciences such 
as mathematical physics, engineering, and electromagnetic 
waves [1], [3], [7]. Varies problems have evolved in two – 
dimensional (2D) integral equations, with one type being the 
2D Volterra equation of the second kind (TDVIE-2nd).

The TDVIE-2nd arises in various phenomena, physics, and 
engineering areas as the Electric field integral equation, and 
partial differentiation equations defined in closed regions can be 
transformed into this type [5], [6], [9]. Usually, finding an analytic 
solution for TDVIE-2 is very difficult. Therefore, to address these 
problems, we must find the numerical solution for it. In scientific 
research of applied sciences, the numerical method is an important 
tool for treating integral equations [10], [12], [13], 14].

Another example is the use of numerical techniques 
for analyzing the radiation of dipole antenna. Evaluating 
radiation patterns for TD segments in three-dimensional (3D) 
patterns [2], [13].

The numerical method was used to estimate 
electromagnetically features for element-loaded media such as 
electromagnetic waves and sea ice. An example in physics is the 
solid angle beam of an antenna, which is described in equation 
(1) showing the integral for normalizing power patterns on 
a sphere (4π, s, r). The mathematical representation of it is 
given by equations [8], [16]:

( )
2

0 0

 A P Ø, )sin(Ø)dØd      
π π

Ω = Θ Θ∫∫
 

and

 

4

A P(Ø, )d
s r

π

Ω = Θ Ω∬  (1)

Polar coordinates increments are displayed. Solid angle 
dA = r2dΩ over the surface of a sphere of radius r, where dΩ 
was the solid angle that was subtended by area.

DEFINITIONS AND THEOREMS

Introducing some definitions and theorems.

Definition 1. The following equation

( ) ( ) ( )( )
h e

a c

u e,h w e,h G e,h, y,z)u y,z dydz.              = + τ∫∫
 

(2)

Corresponding Author: 
Talhat I. Hassan, Department of Automative Technology Engineering, 
Erbil Technology College, Erbil Polytechnic University, Kurdistan 
Region, Iraq. E-mail: talhat.hassan@epu.edu.iq

Received: February 02, 2024 
Accepted: April 03, 2024 
Published: June 10, 2024

DOI: 10.24086/cuesj.v8n1y2024.pp41-45

Copyright © 2024 Talhat I. Hassan, Chiman I. Hussein. This is an open-access 
article distributed under the Creative Commons Attribution License.

Cihan University-Erbil Scientific Journal (CUESJ)

https://creativecommons.org/licenses/by-nc-nd/4.0/


Hassan and Hussein: Modification of adomian decomposition method to solve two dimensions

42 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 41-45

Is named TDVIE-2nd, such that w(e,h) and 
G(e,h,y,z) ≠ 0 were given continuous functions 
on region 1  {(e,h) : 0 h c ,   a e b}= ≤ ≤ ≤ ≤S , and 

1{(e,h, y,z) : a y e b,0 z h c } = ≤ ≤ ≤ ≤ ≤ ≤P  respectively, τ is a 
scalar while u(x,t) is the unknown functions.[4,14]

SOLVING TDVIE-2ND BY ADM.

Here, reformulating and applying ADM to solve TDVIE-2nd, 
let

u e h u
i

n
i

, ,� � � � �
�
�
0

e h  (3)

be the numerical treatment for equation (2), then 
substituting it in equation (1) we get

( ) ( ) ( ) ( )
t x

0 0a c

e,h w e,h G e,h, y,z y,z dydz. 
n n

i i

i i

u u
= =

= + τ∑ ∑∫∫  (4)

After those powers for ui (e,h) equaling, we get

u0
 (e,h) = w(e,h) is initial solution. Now u1 (x,t) formulate as

( ) ( ) ( )( )
h e

1 0

a c

e,h w e,h G e,h, y,z) y,z dydz.              u uτ= + ∫∫
Integrating it for variables. Then u2 (e,h) formulate as

( ) ( ) ( )( )
h e

2 1

a c

e,h w e,h G e,h, y,z) y,z dydz.              u u= + τ∫∫
Then the second formula of iterative. We find u3 (e,h) by 

using the form, as

( ) ( ) ( )( )
h e

3 2

a c

e,h w e,h G e,h, y,z) y,z dydz.              u u= + τ∫∫
And the general formula of the iterative method is

( ) ( ) ( )( )
h e

1

a c

e,h w e,h G e,h, y,z) y,z dydz.              n nu u −= + τ∫∫
 

(5)

After computing these components, we get the numerical 
solution for TDVIE-2nd as

u u
i

n
i

e h e h, ,� � � � �
�
�
0

Note that, we chose Y0 = u0 (e,h) = w(e,h), and the 
iteration

Y T Yn n� �
( )

1  (6)

Where Y un n� � �e h, and

( )( )
h e

1

a c

( ) G e,h, y,z) y,z dydz.              n nT Y u −= τ∫∫
Theorem 1: Let the smooth function u (e,h) and un(e,h) is nth 

numerical solution for z (x,t) then ( ) ( )u e,h e,h
(2 )

n
n

Eu
π
λ

− ≤ , where 

E > 0 independence on n and bounded by partial derivatives 
of u(e,h) where π = 3.14287 [14].

Theorem 2: Assume u (e,h) is the actual solution for 
TDVIE-2nd, such that w(e,h) defined on regions 

1{(e,h) : a h b,   c e c }= ≤ ≤ ≤ ≤S  and G (e,h,y,z) on 
( ){ }1e,h, y,z : a h e b,c z h c   = ≤ ≤ ≤ ≤ ≤ ≤P are bounded 

functions, � then L M N: →  is contractive mapping.

Proof: The nth numerical solution defined in equation (5) 
as,

( ) ( )( )
h e

1

a c

e,h G e,h, y,z) y,z dydz.              n nu u −= τ∫∫

Now

 

( ) ( ) ( )( )

( )( )

h e

a c

h e

1

a c

L(u e,h) ( e,h) G e,h, y,z)u y,z dydz

G e,h, y,z) y,z dydz  

n

n

L u

u −

− = τ

−τ

∫∫

∫∫

� � � � � �� ��� ��
a

h

c

e

G e h y z u y z y z dydz�, , , )( , ,un 1

� � � � � �� ��� ��
a

h

c

e

G e h y z u y z y z dydz�, , , ) , ,un 1

Since G(e, h, y, z) bounded function then G(e, h, y, z ≤ d) 

� � � � � �� � � � � � � �� ��� ��� �� �
a

h

c

e

a

h

c

e

d u y z y z dydz d u y z y z, , , ,u un n1 1
ddydz

By using theorem (1), getting

( ) ( )1
1u e,h e,h

(2 )
n

n

Eu
π

−
−

τ
− ≤

h e h e

1 1

a c a c

d dydz d dydz 
(2 ) (2 )n n

E E
π π− −

 τ τ
≤ λ = λ 

 ∫∫ ∫∫

( ) ( )
h e

1

a c

L(u e,h) ( e,h) d dydz 
(2 )

n
n

EL u
π −

τ
− ≤ τ ∫∫

Then for �n �� , we get
h e

1

a c

 d dydz 0
(2 )n

E
π −

λ
τ →∫∫

,

Therefore L(u(e,h)) – L u e h e h( , ) ( , ) .� � � � � �L un 0

Note: Er is the error and a,t,n,e,k1 are constants.

Algorithm steps of the technique.

Input: a,t,n,e,Er,k1

1. Let u u
i

n
i

e h e h, ,� � � � �
�
�
0

be the approximate solution 

for TDVIE-2nd.



Hassan and Hussein: Modification of adomian decomposition method to solve two dimensions

43 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 41-45

2. Put u0 e h w e h, ( , )� � �
for i =1 to n

3. In equation (5) calculate ui (e,h).

4. Absolute error calculating by r u e h u e hn i

i

n

� �
�
� ( , ) ( , )

0

.

If rn ≤ Er Go to output

End if

End for

5. Continues in this way, till obtaining the numerical 
solution for TDVIE-2nd.

Output: Numerical results and rn.

Theorem 3: The TDVIE-2nd satisfies these conditions [13].

1. G(e, h, y, z) is real and continuous function, and 
|G(e, h,y,z)| ≤ m in

1{(e,h, y,z) : a y e b,c z h c } = ≤ ≤ ≤ ≤ ≤ ≤P  where G(e, h, 
y,z) ≠ 0

2. w(e,h) is real and continuous in the region 

1{(e,h) : a h b,   c e c }= ≤ ≤ ≤ ≤S , |w(e,h)| ≤ m in S  and w 
(e,h) ≠ 0. Then it has one and only one continuous solution u 
(e,h) in  .S

Theorem 4: Let u (x,t) be a function defined over 
{(e,h) : a h ,   a e t}b= ≤ ≤ ≤ ≤S , G (e, h, y, z) be a 

continuous function over the region 
1{(e,h, y,z) : a y e t,c z h c } = ≤ ≤ ≤ ≤ ≤ ≤P  where |G(e, h, y, z) 

≤ M|, M is positive constant, w(e,h) is a continuous function 
on S  where|w(e,h) ≤ m|, then {un (e,h)} given as

( ) ( )
h e

n n 1

a c

u (e,h) G e,h, y,z u y,z dydz,     −= τ∫∫
with u0 (e,h) = w(e,h) is uniformly convergent to u(x,t)

Proof: Since u0 (e,h) = w (e,h), then

|u0 (e,h)| = |w(e,h)|≤ m, and

( ) ( ) ( )
h e

1 0

a c

u e,h G e,h, y,z u y,z dydz             = τ∫∫

( )mM h a)(e c                                                        ≤ λ − −

Also,

( ) ( )
h e

2 1

a c

u (e,h) G e,h, y,z u y,z dydz                    = τ∫∫

( )
h e

a c

G e,h, y,z mM(y a)(z c)dydz      ≤ τ − −∫∫

� � ��
�
�

�
�
�� 2 2 2 21

2 2
m

*
e cM h a( ) ( )

and

( ) ( )

( )

h e

3 2

a c

h e

2 2

a c

u (e,h) G e,h, y,z u y,z dydz 

1G e,h, y,z ( ) (z c) )dydz
2*2

y a

= τ

 ≤ τ − − 
 

∫∫

∫∫
3 3 3 3

2 2

1m ( ) (e a)                 
2 3

M h a ≤ τ − − 
 

Carrying in a similar manner, we get 
n n n n

n 2 2 2 2

mu (e,h) M (h a) (e c) ]               
2 3 4 n

≤ τ − −
…

As n ��  we have

( ) ( )k k kk
1 22 2 2 2k

mu (x, t) lim M c a c c
2 3 4 k∞ ∞→

≤ τ − −
…

By using the ratio test.

( ) ( )

( ) ( )

n 1

k
n

k 1 k 1 k 1k 1
1 22 2 2 2 2

k k k kk
1 22 2 2 2

Rlim
R

1 M c a c c .  
2 3 4 k (k 1)lim                         1 M c a c c .  

2 3 4 k

∞

∞

+

→

+ + ++

→

τ − −
… +=

τ − −
…

( )1 2
2k

M (c a)(c c )
lim 0.                                    

(k 1)∞→

λ − −
= =

+

Then, it is convergence ∀  of τ, M, m, (c1 – a) and (c2 – c)

Hence, it is absolutely and uniformly convergent
� �( , )e h S .

SOME EXAMPLES

In integral equation problems, we often make approximations 
of the unknown function to facilitate numerical computations 
and obtain solutions. This approach also helps in writing 
programs for implementing these techniques on a computer. 
Discussing the use of ADM for solving TDVIE-2nd through 
numerical examples.

Example 1. Find the approximate solution of TDVIE-2nd.

u e h eh e yz u y z dydz

h e

, ) , .� � � � � � �� ���h e h
5 4

0 0

2

9

where the exact solutions u(e,h) = eh.

Solution: Using ADM, put

u eh h e0

5 4

9
e h w e h, ,� � � � � � �

Hence, the first and second approximations can be 
obtained

u h e h e1

5 4 3 4
42

378
e h,

( )� � � � �



Hassan and Hussein: Modification of adomian decomposition method to solve two dimensions

44 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 41-45

u h e h e2

8 8 4 3
100

3762
e h,

( )� � � � �

Note that after two iterations the absolute error is zero.

e h, ( . , . )� � � 0 1 0 2
Example 2. Find the approximate solution of TDVIE-2nd.

u e h,
( ) ( )� � � � �
� � �e h he h e h he e2 2

2 2 2
3 3

12

( )( )
h e

0 0

e )(y z)u y,z dydz.              h+ + +∫∫
With the exact solution u(e,h) = e2 +h2.

Solution: Using ADM, put

u e h he h e h he e0 2 2

2 2 2
3 3

12
e h w e h, ,

( ) ( )� � � � � � � �
� � �

Hence, the first and second approximations obtained,

u he t e h e h e h e1

2 5 4 2 3 3
270 290 277

15120
e h,

( ) ( )� � � � � � �

� � � � �290 3780 270 1260 3780

15120

2 4 2 5 2h e h he he e )

u h e h e h e h e h e h e2

2 2 7 6 2 5 3 4 4
3240 7760 9853 9527

5443
e h,

( )( )� � � � � � � �
2200

�
� � � �( )97200 9853 104400 7760 99720

5443200

4 3 5 3 2 6 3 2h h e h e h e t x

MODIFICATION FOR ADM TO SOLVE 
TDVIE-2ND

In the modification, ADM assumes that w(e,h) written as

w e h w e h w e h, , ,� � � � � � � �1 2
 (7)

for minimizing the steps for computations, therefore 
fasting the convergence process.

Put u0 (e,h) = w1(e,h) and

( ) ( ) ( )
h

1 0
2

0  

e,h w e,h  G e,h, y,z (y,z)dydz,         u u= + τ∫∫
Ω  

( ) ( ) ( )τ+ = + ∫∫
Ω

h
1

2

0  

  e,h w e,h   G e,h, y,z (y,z)dydz, k ku u  (8)

If the opposite terms accurse between the functions 
u0 ( , )e h  and u1( , )e h , deleting these terms between these two 

functions, then remain terms in u e h
0
( , )  probably given the 

actual solution for the problem.

Example 3: Solve example 1 by modifying ADM.

Solution: Using modified ADM.

since w e h eh,� � � �
h e5 4

9
, then

Table 4: Results of example (2) for different iterations

First 
iteration

Third 
iteration

Fifth 
iteration

Seventh 
iteration

ADM

LSE 2.56×10–7 3.43×10–10 6.67×10–11 3.68×10–14

RT 0:1.6532 0:19564 0:27754 0:3.2243

Table 2: A comparison of numerical and exact solutions for 
the ADM in the point (e, h) = (0.1, 0.2) with the exact solution 
0.05000000

Iteration Approximate 
solution by ADM

Absolute 
error for ADM

0 0.0798933333 2.9893×10–2

1 0.0499999999 2.0471×10–9

2 0.0499999999 3.6320×10–12

3 0.0500000000 0.0000000000

Table 3: Results of example (1) for different iterations

First 
iteration

Third 
iteration

Fifth 
iteration

Seventh 
iteration

ADM

LSE 4.77×10–5 3.43×10–8 2.65×10–12 6.16×10–15

RT 0:12675 0:17832 0:2.4473 0:3.0542

Table 1: Comparison of numerical and exact solutions for the 
ADM in the point (e, h) = (0.1, 0.2) with the exact solution 
0.02000000.

Iterations Approximate 
solution by ADM

Absolute error 
for ADM

0 0.0199999644 3.5664×10–8

1 0.0199999983 1.6732×10–9

2 0.0200000000 1.4000×10–11

3 0.0200000000 0.0000000000 

( ) ( ) ( )1 2w e,h  w e,h w e,h= +  Put w e h
1
,� � � eh  and 

w e h
2

5 4

9
,� � � � h e , then u eh0

1
e h w e h, ,� � � � � � ,

then first approximation getting by

( ) ( ) ( ) ( )

( )( )

( )( )

h e

1 2 0

0 0

h e4 5 4
2

0

0 0

h e

2

0 0

e,h w e,h  G e,h, y,z u y,z dydz   

e yz)u y,z dydz
9 9

e yz) yz dydz 0              

u

he h eh

h

= +

= − + λ = −

+ =

∫∫

∫∫

∫∫
Then, by computing these two components, we get the 

exact solution for the problem.



Hassan and Hussein: Modification of adomian decomposition method to solve two dimensions

45 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 41-45

Example 4: Solve example 2 by modifying ADM.

Solution: Using modified ADM.

since w e h,
( ) ( )� � � � �
� � �e h he h e h he e2 2

2 2 2
3 3

12
 then

w e h w e h w e h, , ,� � � � � � � �1 2
 put w e h

1

2 2
,� � � �e h  and 

w e h
2

2 2 2
3 3

12
,

( ) ( )� � � � � � �he h e h he e

( ) ( )0 2 2
1e,h w e,h  , u e h= = + then first approximation 

getting by

( ) ( ) ( )
h e

1 2 0

0 0

e,h w e,h  G e,h, y,z u (y,z)dydz              u = + ∫∫

( )( )

2 2 2

h e

2 2

0 0

( ) (3 3 )
12

e )(y z) dydz 0            

he h e h he e

h y z

+ − −
= −

+ + + + =∫∫

Then by computing these two components, we get the 
exact solution for the problem.

CONCLUSIONS

This study proposes numerical solutions for TDVIE-
2nd utilizing the ADM method, suggesting suitable algorithms, 
and presenting results through a Matlab software program. 
The numerical results were compared based on LSE and RT to 
facilitate a discussion. Finally, we may infer that:

1. The ADM demonstrated their effectiveness for numerically 
solving TDVIE-2nd and producing precise answers.

2. From tables (3) and (4) the LSE decreases when the 
number of iterations increases.

3. Modified ADM needs only one iteration for getting the 
exact solution for the problem.

REFERENCES
1. M. A. Abdou and S. Ahmed. The stability of the solution of 

nonlinear integral equation in two dimensional problems. The 
International Journal of Engineering and Science, vol. 9, no. 4, 
pp. 1-12, 2020.

2. M. A. Abdou and G. M. Abdul-Kader. Mixed Fredholm-Volterra 
integral equation. Journal Le Matematiche, vol. LX- Fasc. I, 
pp. 41-58, 2005.

3. K. E. Atkinson. The Numerical Solution of Integral Equations of the 
Second Kind. Cambridge University Press, Cambridge, 1997.

4. E. Babolian and N. Dastani. He’s homotopy perturbation 
method: An effective tool for solving a nonlinear system of two-
dimensional Volterra- Fredholm integral equations. Journal of 
Mathematical and Computer Modelling, vol. 55, no. 7, pp. 1233-
1244, 2012.

5. F. D. Bazrafshan, A. H. Mahbobi, A. Sousaraie and M. Ebrahim. 
Solving two- dimensional integral equations. Journal of King 
Saud University (Science) vol. 3, no. 2, pp. 111-114, 2011.

6. A. H. Borzabadi and M. Heidari. A successive numerical scheme 
for some classes of Volterra-Fredholm integral equations. Iranian 
Journal of Mathematical Sciences and Informatics, vol. 10, no. 2, 
pp. 1-10, 2015.

7. H. Bronner and J. P. Kauthen. Numerical solutions of two 
dimensional Volterra integral equations of the second kind by 
collection and iterated collection. IMA Journal of Numerical 
Analysis, vol. 9, no. 3, pp. 49-59, 1989.

8. C. H. Chin. Broadband patch antenna with a folded plate pair 
as a differential feeding scheme. IEEE Transactions Antennas 
Propagation, vol. 55, no. 9, pp. 57-69, 2007.

9. T. I. Hassan, N. A. Sulaiman and S. Saleh. Aitken method on 
iterative kernel method for solving system of Volterra- Fredholm 
integral equations of the second kind. Journal of Pure and Applied 
Sciences, vol. 28(6), pp. 79-88, 2016.

10. T. I. Hassan. An approximate solutions of two dimension linear 
mixed Volterra-Fredholm integral equation of the second kind 
via iterative kernel method. Journal of Raparin University, vol. 6, 
no. 2, pp. 101-110, 2019.

11. H. Ibrahim, F. Attah and T. Gyegwe. On the solution of Volterra 
Fredholm and mixed Volterra-Fredholm integral equations using 
the new iterative method. Journal of Applied Mathematics, vol. 6, 
no. 1, pp. 1-5, 2016.

12. H. M. Mehdi and R. K. Saeed. Numerical solution of system 
of two-dimensional linear fredholm integral equations of the 
second kind by adomian decomposition method. Journal of Pure 
and Applied Sciences, vol. 1, no. 4, pp. 30-34, 2011.

13. T. I. Hassan. Iterative kernel technique to solve system fredholm 
integral equation first kind for degenerate kernel. Journal of Erbil 
Polytechnics University, vol. 12, no. 1, pp. 150-158, 2023.

14. Z. Qian and W. Chew. Fast Full-wave surface integral equation 
solver for multi-scale structure modeling. IEEE Transactions 
Antennas Propagation, vol. 57, no. 11, pp. 3594-3602, 2009.

15. H. I. Talaat I. H. Chiman. Proposing Adomian Decomposition Method 
to Treat Two Dimensional Inhomogeneous Mixed Volterra- Fredholm 
Integral Equation. IEEE, New Jersey, pp. 49-53, 2022.

16. H. I. Talaat. Studying the Stability for two Dimension Volterra 
Fredholm integral Equations of the Second Kind. In: AIP 
Conference Proceedings, 2023.

17. J. Zhao and W. C. Chew. Integral equation solution of Maxwell’s 
equations from zero frequency to microwave frequencies. IEEE 
Transactions Antennas Propagation, vol. 48, no. 10, pp. 1635-
1646, 2000.