EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 2, 2024, 1070-1081
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Adomian Decomposition Method and Block by Block
Method for Solving Nonlinear Functional Volterra

Integral Equation in Two-Dimensions

Abeer M Al-Bugami

Department of Mathematics and Statistics, College of Sciences, Taif University,
P.O. Box 11099, Taif 21944, Saudi Arabia

Abstract. This work proposes a new definition of the nonlinear functional Volterra integral equa-
tion in two-dimensions (2D)of the second kind with continuous kernel. Furthermore, the work
is concerned with studying this new equation numerically. The existence of a unique solution
preposition by the equation is proven. In addition, the approximate solutions are obtained by two
powerful methods Adomian Decomposition method (ADM) and Block by block Method (BBM).
The given numerical examples showed the efficiency and accuracy of the introduced methods.

2020 Mathematics Subject Classifications: 34A12, 65A05,65D15

Key Words and Phrases: Functional Integral Equation, Adomian Decomposition Method, Block
by block Method

1. Introduction

The nonlinear functional integral equation is the result of numerous viscoelastic ma-
terial issues in the theories of elasticity and hydrodynamics. The references edited by
Tricomi[9], Hochastadt [11] and Green [10] contain many different methods to give us
the way for solving functional integral equation analytically. In practice, approximate
methods are needed. There are different methods available to guide us towards obtaining
the numerical solution. The excellent expositions by Atkinson [5], Linz [16], Delves and
Mohamed [7], Kumar [15], [14] are recommended reading for the interested reader. The
authors examined a group of nonlinear functional integral equations in [18]. The adequate
conditions for the existence of the Lp-solution of a Volterra functional-integral problem
in a Banach space were examined by Aldona in [8]. The variational iteration method
was utilized by the authors in [6] to derive the numerical solution for the one-dimensional
functional integral equations. Certain conclusions for a Volterra–Hammerstein integral
equation were established by the authors in [17].The authors used a numerical technique
based on the radial basis function in [12] to get numerical solutions of the two-dimensional

DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5087

Email address: abeer101aa@yahoo.com (A. M. Al-Bugami)

https://www.ejpam.com 1070 © 2024 EJPAM All rights reserved.



A. M. Al-Bugami / Eur. J. Pure Appl. Math, 17 (2) (2024), 1070-1081 1071

functional linear integral equations of Fredholm.AL-Bugami conducted a numerical study
of the two-dimensional integral equations in [3], [2]. The mixed integral equations with
continuous and singular kernels were examined by the authors in [1],[13], [4]. Using the
ADM and BBM, we investigate the novel equation for the nonlinear functional integral
equation in 2D in this work. Because it deals with the examination of numerical solutions
for NT-DFVIE, this work is noteworthy. Consider the NT-DFVIE.

µu(x, y) = g(x, y) + f(x, y,

∫ x

0

∫ y

0
p(x, y, t, s)θ(t, s, u(t, s))dtds) (1)

The functionsg(x,y),f(x,y,v(x,y))and θ(x,y,u(x,y))are given analytical functions defined, respec-
tively, andp(x,y,t,s) is the kernel of (1), p(x,y,t,s)≥0, and u (x, y)is the solution to be deter-
mined, while the constant parameter µdefines the kind of the (1).

2. Existences and uniqueness of a solution

This section will examine and use the Picard method to demonstrate that, under certain
circumstances, there is a unique solution to (1).
We assume the following conditions to be satisfied:
1- f(x, y, v (x, y))and θ(x, y, u (x, y)) in 0 ≤ x ≤ X , 0 ≤ y ≤ Y < ∞ , such that

{
∫ x

0
{
∫ y

0
|f(x, y, v(x, y))|2 dx}1/2dy}1/2 ≤ A1 ∥v∥ ,

and

{
∫ x

0
{
∫ y

0
|θ(x, y, w(x, y))|2 dx}1/2dy}1/2 ≤ A2 ∥w∥ ,

2- The kernel p(x,y,t,s)satisfies:

p(x, y, t, s) ≤ C, (C is a constant)

3- The two continuous function f(x, y, v (x, y))and θ(x, y, u (x, y)) satisfy the Lipschitz
condition:

|f(x, y, v1(x, y))− f(x, y, v2(x, y))| ≤ B1 |v1 − v2| ; ∀v1, v2 ∈ (−∞,∞),
|θ(x, y, w1(x, y))− θ(x, y, w2(x, y))| ≤ B2 |w1 − w2| ;∀w1, w2 ∈ (−∞,∞),

whereB1,B2are Lipchitz’s constants such that B1B2C=M2<1.

We consider the existence and uniqueness of the solution for the (1). Also, the continuity
and the normality of the integral operator are proved. We define the nonlinear integral
operators Sample theorem with citation:

(Wu)(x, y) = f(x, y,

∫ x

0

∫ y

0
p(x, y, t, s)θ(t, s, u(t, s))dtds) (2)

And
(W̄u)(x, y) = g(x, y) + (Wf)(x, y) (3)



A. M. Al-Bugami / Eur. J. Pure Appl. Math, 17 (2) (2024), 1070-1081 1072

As for the normality of the nonlinear integral operator Wu, we see that

∥Wu| = {
∫ x

0
{
∫ y

0

∣∣∣∣∣
∣∣∣∣f(x, y,∫ x

0

∫ y

0
p(x, y, t, s)θ(t, s, u(t, s))dtds)

∣∣∣∣2 dx
∣∣∣∣∣
2

}
1
2dy}

1
2 (4)

Applying condition (1), we obtain

∥Wu| = A1{
∫ x

0
{
∫ y

0

∣∣∣∣∣
∣∣∣∣∫ x

0

∫ y

0
p(x, y, t, s)θ(t, s, u(t, s))dtds)

∣∣∣∣2 dx
∣∣∣∣∣
2

}
1
2dy}

1
2 (5)

Then, we get

∥Wu| = A1{
∫ x

0
{
∫ y

0
|
∫ x

0

∫ y

0
|p(x, y, t, s)|2.|θ(t, s, u(t, s))|2dtds)|2dtdx}

1
2dsdy}

1
2 (6)

Which can be adapted in the form

∥Wu∥ = A1{
∫ x

0

∫ y

0
{
∫ x

0

∫ y

0
|p(x, y, t, s)|2dxdt}

1
2dyds}

1
2 .{

∫ x

0
{
∫ y

0
|θ(t, s, u(t, s))|2dtds}

1
2 }

1
2

(7)
If we use the boundary conditions (1), (2), we will have

∥Wu∥ = A1A2C ∥u∥ = M1 ∥u∥ (8)

To demonstrate the integral operator’s continuation, takeu1(x,y),u2(x,y)∈L2[0,x]×L2[0,y],to have:∥∥W̄u1 − W̄u2
∥∥

= {
∫ x
0 {

∫ y
0

∣∣∣∣f(x, y, ∫ x
0

∫ y
0 p(x, y, t, s)θ(t, s, u1(t, s))dtds)

−f(x, y,
∫ x
0

∫ y
0 p(x, y, t, s)θ(t, s, u2(t, s))dtds)

2dx2}
1
2dy}

1
2

(9)

Using the condition (3), we obtain

∥∥W̄u1 − W̄u2
∥∥ ≤ B1{{

∫ x

0

∫ y

0

∣∣∣∣∫ x

0

∫ y

0
p(x, y, t, s)[θ(t, s, u1(t, s))− θ(t, s, u2(t, s))]

∣∣∣∣2 dx} 1
2dy}

1
2

(10)
Then,∥∥W̄u1 − W̄u2

∥∥ ≤ B1{{
∫ x

0

∫ y

0

∫ x

0

∫ y

0
|p(x, y, t, s)}2.|θ(t, s, u1(t, s))−θ(t, s, u2(t, s))|2dx}

1
2dy}

1
2

(11)
From the conditions (1), (3), we have∥∥W̄u1 − W̄u2

∥∥ ≤ B1B2C ∥u1 − u2∥ = M2 ∥u1 − u2∥ (M2 < 1). (12)

Hence, W̄ is a contraction operator and W̄ has a unique fixed point, which is the unique
solution to (1), of course.



A. M. Al-Bugami / Eur. J. Pure Appl. Math, 17 (2) (2024), 1070-1081 1073

3. Numerical methods for solving NT-DFVIE.

3.1. The ADM

This section uses ADM to handle NT-DFVIE of the second kind:

µu(x, y) = g(x, y) + f(x, y,

∫ x

0

∫ y

0
p(x, y, t, s)θ(t, s, u(t, s))dtds),

(13)
The unknown solution is an infinite series of the following form:

u(x, y) =
∞∑
n=0

un(x, y) (14)

In addition, the term θ(t,s,u(t,s))in (13) is decomposed into an infinite series

θ(t, s, u) =

∞∑
n=0

An (15)

Moreover, the following equation defines Adomian’s polynomial, denoted asAn

An =
1

n!

dn

dλn
[θ(

∞∑
i=0

λiui)]λ=0, n = 0, 1, 2, ... (16)

Substituting from (15), (16) into(13), we get

u0(x, y) = g(x, y) (17)

ui(x, y) = f(x, y,

∫ x

0

∫ y

0
p(x, y, t, s)Ai−1(u0(t, s), ...., ui−1(t, s))dtds), i ≥ 1 (18)

3.2. The BBM

Consider

µu(x, y) = g(x, y) + f(x, y,

∫ x

0

∫ y

0
p(x, y, t, s)θ(t, s, u(t, s))dtds) (19)

Then, we get

U2(x, y) ≈ u(x2, y2) = g(x, y) + f(x, y,

∫ x

0

∫ y

0
p(x2, y2, t, s)θ(t, s, u2(t, s))dtds) (20)

or in the form

U2(x, y) ≈ u(x2, y2) = g(x, y) + f(x, y,

∫ x

0

∫ y

0
p(x2, y2, t, s)dtds) (21)



A. M. Al-Bugami / Eur. J. Pure Appl. Math, 17 (2) (2024), 1070-1081 1074

Using Simpson’s rule to approximate the integrals

U2 = g(x2, y2)+f(x, y,
h

3
{p(x2, y2, x0, y0, U0)+4p(x2, y2, x1, y1, U1)+p(x2, y2, x2, y2, U2)})

(22)
where

U0 = g(x0, y0). (23)

Also, we get

U1(x, y) ≈ u(x1, y1) = g(x1, y1) + f(x, y,

∫ x

0

∫ y

0
p(x1, y1, t, s, u1(t, s))dtds) (24)

U1 = g(x1, y1)+f(x, y,
h

3
{p(x1, y1, x0, y0, U0)+4p(x1, y1, x1/2, y1/2, U1/2)+p(x1, y1, x1, y1, U1)}

(25)
Therefore, we obtain:

U1/2 =
3

8
U0 +

3

4
U1 −

1

8
U2 (26)

Substituting (26) into (25), we obtain:

U1 = g(x1, y1) + f(x, y,
h

3
{p(x1, y1, x0, y0, U0) + 4p(x1, y1, x1/2, y1/2,

3

8
U0 +

3

4
U1 −

1

8
U2)})
(27)

In general, consider (20), where0 ≤ x ≤ a , 0 ≤ y ≤ b. Let0 = x0 < x1 < · · · < xN =
a , 0 = y0 < y1 < · · · < yN = b ,

U2m+1(x, y) ≈ u(x2m+1, y2m+1) = g(x2m+1, y2m+1)+f(x, y,

∫ x2m+1

0

∫ y2m+1

0
p(x2m+1, y2m+1, t, s, u(t, s))dtds)

(28)
Or equivalently

U2m+1(x, y) = g(x2m+1, y2m+1) + f(x, y,
∫ x2m

0

∫ y2m
0 p(x2m+1, y2m+1, t, s, u(t, s))dtds)+

f(x, y,
∫ x2m+1

0

∫ y2m+1

0 p(x2m+1, y2m+1, t, s, u(t, s))dtds)
(29)

Then,

U2m+1(x, y) = g(x2m+1, y2m+1) + f(x, y, h3 [p(x2m+1, y2m+1, x0, y0, U0)+
4p(x2m+1, y2m+1, x1, y1, U1) + ...+

p(x2m+1, y2m+1, x2m, y2m, U2m)] + h
6p(x2m+1, y2m+1, x2m, y2m, U2m)

+2h
3 p(x2m+1, y2m+1, x2m+ 1

2
, y2m+ 1

2
, 38U2m

+3
4U2m+1 − 1

8U2m+2) +
h
6p(x2m+1, y2m+1, x2m+1, y2m+1, U2m+1))

(30)

Also,

U2m+2(x, y) = g(x2m+2, y2m+2) + f(x, y,

∫ x2m+2

0

∫ y2m+2

0
p(x2m+2, y2m+2, t, s, u(t, s))dtds)

(31)



A. M. Al-Bugami / Eur. J. Pure Appl. Math, 17 (2) (2024), 1070-1081 1075

U2m+2 = g(x2m+2, y2m+2) + f(x, y, h3{p(x2m+2, y2m+2, x0, y0, U0)+
4p(x2m+2, y2m+2, x1, y1, U1) + ...

+ p(x2m+2, y2m+2, x2m+2, y2m+2, U2m+2)})
(32)

4. Numerical experiments and discussions

1.Consider

u(x, y) = g(x, y) + f(x, y,

∫ x

0

∫ y

0
(xys)(u(t, s))kdtds (33)

u (x, y) = xy is the exact solution. If we set k=1, in (33), one has

u(x, y) = g(x, y) + f(x, y,

∫ x

0

∫ y

0
(xys)u(t, s)dtds (34)

which is refereed to asLT-DFVIE, and if we set ϕ(x)in (33), we obtained the integral
equation, may consider the suggestion is called the NT-DFVIE of the second kind. Ad-
ditionally, the associated errors are calculated for both the linear and nonlinear cases. We
solve (33) using ADM and BBM. In the following Tables (1)-(2) we present the exact
solution (uExac) and the approximate solutions (uADM , uBBM ), and the corresponding
errors (ErrorADM ,ErrorBBM ) at N= 10.

Block-by-
block method

ADM uExact y x

ErrorBBM uBBM ErrorADM uADM

0.00000 0.000000 0.00000000 0.0000000 0.0000000 0.0 0.0

0.6950×10−8 0.0100000 0.277777×10−8 0.0100000 0.0100000 0.1 0.1

0.44804×10−5 0.0400044 0.355560×10−6 0.0400003 0.0400000 0.2 0.2

0.00004059 0.09004059 0.607559×10−5 0.0900006 0.0900000 0.3 0.3

0.00024690 0.16024690 0.000045529 0.1600455 0.1600000 0.4 0.4

0.00179211 0.25179211 0.000217285 0.2502172 0.2500000 0.5 0.5

0.01059498 0.34940502 0.000780024 0.3607800 0.3600000 0.6 0.6

0.01366165 0.50366165 0.002303060 0.4923030 0.4900000 0.7 0.7

0.03285243 0.67285243 0.005902298 0.6459022 0.6400000 0.8 0.8

0.04750594 0.85750594 0.013603387 0.8236033 0.8100000 0.9 0.9

0.08797644 1.08797645 0.028908955 1.0289089 1.0000000 1.0 1.0

Table (1) Numerical results by using BBM and ADM, N=10,k=1.



A. M. Al-Bugami / Eur. J. Pure Appl. Math, 17 (2) (2024), 1070-1081 1076

Fig. 1, the plot 3D errors for the value of errors by using BBM and ADM, N=10,
k=1.

Block-by-
block method

ADM uExact y x

ErrorBBM uBBM ErrorADM uADM

0.000000 0.0000000 0.000000 0.0000000 0.0000000 0.0 0.0

0.20760×10−7 0.01000002 0.16590×10−7 0.0100000 0.0100000 0.1 0.1

0.44804×10−5 0.04000448 0.20942×10−5 0.0400020 0.0400000 0.2 0.2

0.000040593 0.09004059 0.000034949 0.0900349 0.0900000 0.3 0.3

0.000246903 0.16024690 0.000253147 0.1602531 0.1600000 0.4 0.4

0.001792115 0.25179211 0.001154385 0.2511543 0.2500000 0.5 0.5

0.010594983 0.34940501 0.003908646 0.3639086 0.3600000 0.6 0.6

0.013661652 0.50366165 0.010721293 0.5007212 0.4900000 0.7 0.7

0.032852435 0.67285243 0.025066934 0.6650669 0.6400000 0.8 0.8

0.047505948 0.85750594 0.051535161 0.8615351 0.8100000 0.9 0.9

0.087976455 1.08797645 0.094913086 1.0949130 1.0000000 1.0 1.0

Table(2) Numerical results by using BBM and ADM, N=10,k=2.



A. M. Al-Bugami / Eur. J. Pure Appl. Math, 17 (2) (2024), 1070-1081 1077

Fig. 2, the plot 3D errors for the value of errors by using BBM and ADM, N=10, k=2.
2.Consider

u(x, y) = g(x, y) + f(x, y,

∫ x

0

∫ y

0
(xy)(u(t, s))kdtds (35)

u(x, y) = xy/2 is the exact solution, if we set k=1, in (35), one has

u(x, y) = g(x, y) + f(x, y,

∫ x

0

∫ y

0
(xy)u(t, s)dtds (36)

Block-by-
block method

ADM uExact y x

ErrorBBM UBBM ErrorADM uADM

0.0000000 0.0000000 0.0000000 0.0000000 0.00000000 0.0 0.0

0.9793×10−8 0.00500000 0.625003×10−7 0.0050000 0.00500000 0.1 0.1

0.224024×10−50.02000224 0.400040×10−5 0.0200040 0.02000000 0.2 0.2

0.000020296 0.04502029 0.000045585 0.0450455 0.04500000 0.3 0.3

0.000123451 0.08012345 0.000256410 0.0802564 0.08000000 0.4 0.4

0.000896057 0.12589605 0.000980387 0.1259803 0.12500000 0.5 0.5

0.0052974915 0.17470250 0.002939725 0.1829397 0.18000000 0.6 0.6

0.006830862 0.25183082 0.00746423 0.2524642 0.24500000 0.7 0.7

0.016426217 0.33642621 0.01680854 0.3368085 0.32000000 0.8 0.8

0.023752974 0.42875297 0.03460287 0.4396028 0.40500000 0.9 0.9

0.043988227 0.54398822 0.06651660 0.5665166 0.50000000 1.0 1.0

Table (3) Numerical results by using BBM and ADM, N=10,k=1.



A. M. Al-Bugami / Eur. J. Pure Appl. Math, 17 (2) (2024), 1070-1081 1078

Fig. 3, the plot 3D errors for the value of errors by using BBM and ADM, N=10, k=1.

Block-by-
block method

ADM uExact y x

ErrorBBM UBMM ErrorADM uADM

0.00000000 0.00000000 0.0000000 0.0000000 0.00000000 0.0 0.0

0.9793×10−7 0.004999902 0.12456×10−6 0.00500012 0.00500000 0.1 0.1

0.747110×10−5 0.020007471 0.789017×10−5 0.02000789 0.02000000 0.2 0.2

0.0000357169 0.045035716 0.000088335 0.04508833 0.04500000 0.3 0.3

0.0001830350 0.080183035 0.000484473 0.08048447 0.08000000 0.4 0.4

0.0011159424 0.126115942 0.001791559 0.12679155 0.12500000 0.5 0.5

0.0149879842 0.165012015 0.005149897 0.18514989 0.18000000 0.6 0.6

0.012010961 0.257010961 0.012413373 0.25741337 0.24500000 0.7 0.7

0.027208265 0.347208265 0.026247589 0.34624759 0.32000000 0.8 0.8

0.048275031 0.453275031 0.050109867 0.45510986 0.40500000 0.9 0.9

0.097513263 0.597513263 0.088059093 0.58805909 0.50000000 1.0 1.0

Table (4) Numerical results by using BBM and ADM, N=10,k=2.



A. M. Al-Bugami / Eur. J. Pure Appl. Math, 17 (2) (2024), 1070-1081 1079

Fig. 4, the plot 3D errors for the value of errors by using BBM and ADM, N=10,
k=2.

5. The conclusion

he new definition of the Functional Volterra Integral Equation in Two-Dimensional
(NT-DFVIE) was presented in this study. Effective numerical techniques are also sug-
gested in order to solve this equation. Error analysis and some numerical examples show
the accuracy and performance of the methods. Based on the preceding examples and
tables (1)-(4), we observe that:
1- As x andy are increasing in each interval [0, 1], the errors values for BBM, and ADM
are also increasing.
2- The error results by using ADM is smaller than the error results by using BBM. So,
the ADM is better than theBBM for solving nonlinear NT-DFVIE.
3- The error result by using ADM for linear case is larger than the nonlinear case, and by
using BBM the error results for nonlinear case is larger than the linear case.

Additional Points

Future Work. Other methods, such as the homotopyperturbation approach, homotopy
analysis method, Runge-Kutta method and the variational iterationapproach, will be used
to solve the the nonlinear functional Volterra integral equation in two-dimension.

Data Availability

All the data are available within the article and as the references that were cited.



REFERENCES 1080

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgements

The work was funded by the Deanship of Scientific Research at Taif University, which
the researcher would like to appreciate.

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