EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1982-2000
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Adomian Modification Methods via Orthogonal
Polynomials: A Comparative Study

Mariam AL-Mazmumy1, Huda Bakodah1, Aishah Alsulami1, Nawal Alzaid1,∗

1 Department of Mathematics and Statistics, College of Science, University of Jeddah,
P.O. Box 80327, Jeddah, Saudi Arabia.

Abstract. The present manuscript proposes different modification procedures for the standard
Adomian Decomposition Method (ADM). These procedures are based on the application of or-
thogonal polynomials that play vital parts in approximation theories. Moreover, the study also
scrutinizes four nonlinear inhomogeneous initial-value problems, and distinctively examines their
respective absolute error differences. Remarkably, different computational benefits of the proposed
modification are noted with regard high-level of accuracy and fewer computational steps.

2020 Mathematics Subject Classifications: 33c45,34A34,34A12

Key Words and Phrases: ADM, Adomian modification methods, Adomian polynomials, or-
thogonal polynomials, ordinary differential equations (ODEs), Initial-Value Problems (IVPs)

1. Introduction

The celebrated Adomian Decomposition Method (ADM) [2] has in the past and present
decades been greatly utilized to solve a variety of functional equations. The method that
was proposed by George Adomian (in the 1980s) has further undergone different stages
of reformations, modifications, and improvements. Indeed, there exist a huge number of
related literature with regards to the development of ADM associated with its applicability
in solving various forms of IVPs of both the ordinary and partial differential equation types
[1, 3, 4, 13]. On the other hand, orthogonal functions are regarded with high admiration
in the fields of numerical methods, and approximation theories among others. However,
in line with their applications, Hosseini [7] demonstrated the relevance of Chebyshev’s
polynomials in improving the known accuracy of the standard ADM. In fact, different
nonlinear and linear models were examined via the method to have good approximate
solutions. We mention also the excellent work of Liu [8] where Legendre’s polynomials
were coupled in the ADM instead of the ordinary Adomian procedure. Additionally, ADM

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5080

Email addresses: mhalmazmumy@uj.edu.sa (M. Al-Mazmumy),hobakodah@uj.edu.sa (H. Bakodah),
aalsulami1183.stu@uj.edu.sa (A. Alsulami), naalzaid@uj.edu.sa (N. Alzaid)

https://www.ejpam.com 1982 © 2024 EJPAM All rights reserved.



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1983

was equally enhanced using the Gegenbauer’s and Jacobi’s orthogonal polynomials [6] to
solve some important models of mathematical physics; one may in the same fashion read
about the relevance of Laguerre’s and Hermite’s orthogonal polynomials in optimizing the
standard ADM procedure in [11, 12].

However, the present manuscript proposes different modification procedures for the
standard ADM. These procedures are based on the application of orthogonal polynomials
that play vital parts in approximation theories as rightly mentioned. More specifically,
the following orthogonal polynomials: Legendre’s, Chebyshev’s, Laguerre’s, Hermite’s,
Gegenbauer’s, and lastly the Jacobi’s polynomials will be considered to devise modifica-
tion methods for the standard ADM. Moreover, the present study will scrutinize four test
problems and distinctively examines their respective absolute error differences. Addition-
ally, we organize the paper in the following manner: Section 2 gives the standard ADM
procedure; while its modifications based on orthogonal polynomials are presented in Sec-
tion 3. Section 4 makes consideration to certain illustrative test examples; while Section
5 gives certain concluding comments.

2. Standard ADM procedure

The present section gives a generalized derivation procedure for tackling nonlinear
Initial-Value Problems (IVPs) based on the ADM. To do so, let us consider the following
differential equation

G(u(x)) = g(x), (1)

with G representing a generalized ordinary (or partial) differential operator, and g(x) as
a source term. This operator being general, it can equally be expressed to involve both
linear and nonlinear operators. Thus, we decompose the operator further, and rewrite the
above equation as follows

Lu+Ru+Nu = g, (2)

where L is the highest linear operator that is invertible, with R < L; while N is specifically
the nonlinear operator. More so, we rewrite the latter equation as follows

Lu = g −Ru−Nu, (3)

such that applying the inverse linear operator L−1 to both sides of the above equation
yields

u = ϕ(x) + L−1g − L−1Ru− L−1Nu. (4)

where ϕ(x) is the function emanating from the prescribed initial data.
Further, the iterative procedure by the name ADM decomposes the solution u(x) using
an infinite series of the following form

u(x) =

∞∑
n=0

un(x), (5)



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1984

while the nonlinear component Nu is equally decomposed using the following infinite series

N(u) =
∞∑
n=0

An(u0, u1, ...), (6)

where An’s are polynomials devised by Adomian, and recursively determined using the
following scheme

An(u0, u1, ...) =
1

n!

dn

dλn

N
 n∑

j=0

λjuj


λ=0

, n = 0, 1, 2, ... (7)

Therefore, upon substituting Eqs. (5) and (6) into Eq. (4), one gets

∞∑
n=0

un(x) = ϕ(x) + L−1g(x)− L−1R

∞∑
n=0

un(x)− L−1
∞∑
n=0

An(u0, u1, ...), (8)

Furthermore, the ADM procedure swiftly reveals the generalized recursive solution for
the problem from the above equation as follows

u0 = ϕ(x) + L−1g(x),

un+1 = −L−1Run − L−1An(u0, u1, ...), n ≥ 0,
(9)

where An’s are the Adomian polynomials computed from Eq. (7). Expressing few of these
terms, we get

A0(u0) = N (u0) ,

A1(u0, u1) =
dN (u0)

du0
u1,

A2(u0, u1, u2) =
dN (u0)

du0
u2 +

1

2

d2N (u0)

du20
u21,

A3(u0, u1, u2, u3) =
dN (u0)

du0
u3 +

d2N (u0)

du20
u1u2 +

1

3!

d3N (u0)

du30
u31,

...

Remarkable, it is obvious that the Adomian polynomials An’s depend on the solution
components un. For instance, A0 relies merely on u0; A1 relies merely on u0 and u1; A2

relies merely on u0, u1 and u2 , and so on.
Finally, a realistic solution is obtained by considering the following m-term approxi-

mations as

Ψn =
n−1∑
j=0

uj , (10)

where

u(x) = lim
n→∞

Ψn(x) =

∞∑
j=0

uj(x). (11)



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1985

3. Adomian modification methods via orthogonal polynomials

The present section gives some important modifications of the standard ADM that
are based on the application of orthogonal polynomials. One could easily recall the im-
portance of orthogonal functions in approximation theory and numerical methods. Thus,
these functions/polynomials are equally used in the present study to further optimize the
exactness of the standard ADM. To begin with, let us make use of the Taylor’s series ex-
pansion to expand the given source term g(x) in Eq. (2) for an arbitrary positive integer,
say m as follows

g(x) =

m∑
n=0

gn(0)

n!
xn. (12)

Therefore, in what follows, we have obtained series of Adomian modification methods
via orthogonal polynomials. More specifically, we have utilized the following orthogonal
polynomials including the Legendre’s, Chebyshev’s, Laguerre’s, Hermit, Gegenbauer’s and
Jacobi’s polynomials [5].

3.1. Adomian modification via Legendre’s polynomials

To present a modification method based on the application of the Legendre’s polyno-
mials, we express the source term g(x) given in Eq. (2) as a series of Legendre’s polynomial
as follows [8, 10]

g(x) =

m∑
n=0

cnPn(x), (13)

where Pn(x) are the orthogonal Legendre’s polynomials, and the coefficients of Legendre’s
expansion ci are determined through

ci =
2i+ 1

2

∫ 1

−1
g(x)Pi(x)dx, i = 0, 1, · · ·

Thus, substituting Eq. (13) into Eq. (9), we get the following recursive solution
u0 = ϕ(x) + L−1[c0P0(x) + c1P1(x) + c2P2(x) + · · ·+ cmPm(x)],

un+1 = −L−1Run − L−1An, n ≥ 0.
(14)

Finally, a realistic solution via the application of Legendre’s polynomial is thus obtained
in this regard by considering the following m-term approximations using u(x) =

∑m
n=0 un,

where m is the order of the solution.

3.2. Adomian modification method via Chebyshev’s polynomials

(i) First kind Chebyshev’s polynomials In Hosseini [7], the source term g(x) is
suggested to be decomposed using Chebyshev’s series as follows

g(x) =
m∑

n=0

cnTn(x), (15)



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1986

where Tn(x) are the orthogonal Chebyshev’s polynomial of the first kind; while the
coefficient of Chebyshev’s expansion ci are expressed as follows

c0 =
1
π

∫ 1
−1

g(x)T0(x)√
1− x2

dx,

ci =
2
π

∫ 1
−1

g(x)Ti(x)√
1− x2

dx, i = 1, 2, · · ·
(16)

Thus, upon using Eqs. (9) and (15), we get the following recursive solution{
u0 = ϕ(x) + L−1[c0T0(x) + c1T1(x) + c2T2(x) + · · ·+ cmTm(x)],

un+1 = −L−1Run − L−1An, n ≥ 0.
(17)

(ii) Second kind Chebyshev’s polynomials In the same fashion, we make use of the
second kind Chebyshev’s polynomials in approximating the source term g(x) instead
of the first kind Chebyshev’s polynomials [9, 14] as follows

g(x) =
m∑

n=0

cnUn(x), (18)

where Un(x) are the orthogonal Chebyshev’s polynomial of the second kind; while
the coefficients of the Chebyshev’s expansion ci are given by

ci =
2
π

∫ 1
−1

√
1− x2g(x)Ui(x)dx, i = 0, 1, 2, · · · (19)

Now, on using Eqs. (9) and (18), we get the following recursive solution{
u0 = ϕ(x) + L−1[c0U0(x) + c1U1(x) + c2U2(x) + · · ·+ cmUm(x)],

un+1 = −L−1Run − L−1An, n ≥ 0.
(20)

Thus, realistic solutions via the application of the Chebyshev’s polynomials of the first
and second kinds are thus obtained in this regard by considering the following m-term
approximations using u(x) =

∑m
n=0 un, where m is the order of the solution.

3.3. Adomian modification method via Laguerre’s polynomials

In the same way, it is suggested that the source term g(x) to be decomposed using
Laguerre’s series [11] as follows

g(x) =
m∑

n=0

cnLn(x) (21)

where Ln(x) are orthogonal Laguerre’s polynomials, and ci are given by

ci =

∫ ∞

0
e−xLi(x)g(x)dx, i = 0, 1, · · · (22)



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1987

Accordingly Eqs.(9) and (21), we get the recursive solution as in the preceding method
as follows{

u0 = ϕ(x) + L−1[c0L0(x) + c1L1(x) + c2L2(x) + · · ·+ cmLm(x)],
un+1 = −L−1Run − L−1An, n ≥ 0,

(23)

where the closed-form solution u(x) is obtained upon summing the individual components
as suggested by ADM.

3.4. Adomian modification method via Hermite’s polynomials

In the same way, it is suggested that the source term g(x) to be decomposed using
Hermite’s series [12] as follows

g(x) =
m∑

n=0

cnHn(x), (24)

where Hn(x) are orthogonal Hermite’s polynomials, and the coefficient ci are determined
using

ci =
1

2ii!
√
π

∫ ∞

−∞
e−x2

Hi(x)g(x)dx, i = 0, 1, · · · (25)

What’s more from Eqs.(9) and (21), we get the following recursive solution as explained
earlier as follows{

u0 = ϕ(x) + L−1[c0H0(x) + c1H1(x) + c2H2(x) + · · ·+ cmHm(x)],
un+1 = −L−1Run − L−1An, n ≥ 0,

(26)

where the closed-form solution u(x) is obtained upon summing the individual components
as suggested by ADM.

3.5. Adomian modification methods via Gegenbauer’s and Jacobi’s poly-
nomials

(i) Gegenbauer’s polynomials

Firstly, we express the source term g(x) via the Gegenbauer’s series [6] as follows

g(x) =

m∑
n=0

cnC
α
n (x), (27)

where Cα
n (x) are the orthogonal Gegenbauer’s polynomials, and the coefficients of

Gegenbauer’s expansion ci are defined as follows

ci =

∫ 1
−1 g(x)C

α
i (x)(1− x2)α−1/2dx∫ 1

−1[C
α
i (x)]

2(1− x2)α−1/2dx
, i = 0, 1, 2, · · · (28)



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1988

where the normalization of the functions are done using as (1 − x2)α−1/2 as the
weight function.

Lastly, substituting Eq. (27) into Eq. (9), we get the recursive solution without
further delay as follows{

u0 = ϕ(x) + L−1[c0C
α
0 (x) + c1C

α
1 (x) + c2C

α
2 (x) + · · ·+ cmCα

m(x)],
un+1 = −L−1Run − L−1An, n ≥ 0.

(29)

(ii) Jacobi’s polynomials
Considering Jacobi’s orthogonal polynomials over [–1, 1], we in the same way decom-
pose the source term g(x) as follows

g(x) =
m∑

n=0

cnP
(α,β)
n (x), (30)

where α, β > −1 and P
(α,β)
n (x) are the Jacobi’s polynomials that are orthogonal,

and the coefficients ci of Jacobi expansion are defined as follows

ci =

∫ 1
−1 g(x)P

(α,β)
i (x)(1− x)α(1 + x)βdx∫ 1

−1[P
(α,β)
i (x)]2(1− x)α(1 + x)βdx

, i = 0, 1, 2, · · · (31)

where the weight function (1 − x)α(1 + x)β is used for the normalization in this
equation.

Thus, the following recursive solution is obtained from Eqs. (30) and (9) as follows{
u0 = ϕ(x) + L−1[c0P

(α,β)
0 (x) + c1P

(α,β)
1 (x) + c2P

(α,β)
2 (x) + · · ·+ cmP

(α,β)
m (x)],

un+1 = −L−1Run − L−1An, n ≥ 0.

(32)

Moreover, realistic solutions via the application of the above polynomials could be obtained
in the same manner by considering the following m-term approximations using u(x) =∑m

n=0 un.

4. Illustrative examples

The current section demonstrates the application of the Adomian modification methods
via orthogonal polynomials to comparatively examine different forms of IVPs of ODEs as
test examples. Moreover, we shall utilize seven-term approximations via the Maple 18
package programmer for the computational simulation.

Example 1. Consider the IVP of Duffing’s equation [6]

u′′ + 3u− 2u3 = sin(2x) cos(x), 0 ≤ x ≤ 1,
u(0) = 0, u′(0) = 1,

(33)

that admits the following exact solution u(x) = sin(x).



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1989

Firstly, we express the given equation in operator form as follows

u = x+ L−1(sin(2x) cos(x))− 3L−1(u) + 2L−1(u3), (34)

where L−1(.) is the inverse operator defined by L−1(.) =
∫ x
0

∫ x
0 (.)dxdx and N(u) = u3

Substituting Eqs.(5) and (6) into Eqs. (34) , we get the following recursive solution

u0 = x+ L−1(sin(2x)cos(x)),
un+1 = −3L−1(un) + 2L−1An(u0, u1, · · · ), n ≥ 0,

From Eq.(7), the nonlinear tearm N(u) = u3 requires the following Adomian polynomials

A0 = u30,
A1 = (3u20u1),
A2 = (3u20u2 + 3u0u

2
1),

A3 = (3u20u3 + 6u0u1u2 + u31),
...

(35)

In what follows, we shall be utilizing the proposed modification methods to treat the
governing Duffing’s equation. More so, we shall be starting with the classical Taylor’s
series before the proposed schemes. Additionally, we denote the solution u(x) based on
the respective modifications as follows: ut(x) via the Taylor’s series expansion; uP (x)
via the Legendre’s series expansion; uT (x) via the Chebyshev’s series expansion; uL(x)
via the Laguerre’s series expansion; uH(x) via the Hermite’s series expansion; u1g(x) via

the Gegenbauer’s series expansion (α = 1); and lastly u
(1,1)
j (x) via the Jacobi’s series

expansion (α = 1, β = 1).
Modification method via Taylor’s series will be used for the expansion of the

source term g(x) for m = 6 as follows

g(x) = 2x− 7

3
x3 +

61

60
x5 +O(x7), (36)

Then, we get the following iterative components

u0 = u(0) + xu′(0) + L−1(2x− 7
3x

3 + 61
60x

5) = x+ 1
3x

3 − 7
60x

5 + 61
2520x

7,
u1 = −3L−1(u0) + 2L−1A0 = −1

2x
3 + 1

20x
5 + 47

840x
7 − 89

60480x
9 + · · · ,

u2 = −3L−1(u1) + 2L−1A1 =
3
40x

5 − 3
40x

7 − 523
20160x

9 + · · · ,
u3 = −3L−1(u2) + 2L−1A2 = − 3

560x
7 + 29

960x
9 + · · · ,

u4 = −3L−1(u3) + 2L−1A3 =
1

4480x
9 + · · · ,

...

such that upon summing the above components yields the following series solution

ut(x) =

6∑
n=0

un(x) = x− 1

6
x3 +

1

120
x5 − 1

5040
x7 +

73

24192
x9 + · · · (37)



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1990

Modification method via Legendre’s polynomials is up now. Expanding of the
source term g(x) via Legendre’s polynomials for m = 6 gives

g(x) =
6∑

n=0

cnPn(2x− 1), 0 ≤ x ≤ 1, (38)

where Pn(.) are orthogonal Legendre’s polynomials, and ci are given by

ci =
2i+ 1

2

∫ 1

−1
g(0.5x+ 0.5)Pi(x)dx, i = 0, 1, · · · (39)

This means that

g(x) ≈ −0.00001047 + 2.000674384x− 0.106599082x2 + · · · − 0.4690686000x6. (40)

Thus, we get the following solution components

u0 = u(0) + xu′(0) + L−1(−0.00001047 + 2.000674384x+ · · · − 0.4690686000x6),

= x− 0.000005237x2 + 0.3334454093x3 + · · · − 0.008375467814x8,

u1 = −0.5x3 + 0.00000130940075x4 + · · · ,
u2 = 0.075x5 − 1.307582000× 10−7x6 + · · · ,
u3 = −0.005357142858x7 + · · · ,

...

such that their summation yields

uP (x) =
6∑

n=0

un(x) = x−0.000005237603000x2−0.1665545908x3−0.00088541019923x4+· · ·

(41)
Modification method via Chebyshev’s polynomials goes off by expanding the

source term g(x) as follows

g(x) =
6∑

n=0

cnTn(2x− 1), 0 ≤ x ≤ 1, (42)

where Tn(.) are orthogonal Chebyshev’s polynomials, and ci are given by where

c0 =
1
π

∫ 1
−1

g(0.5x+ 0.5)T0(x)√
1− x2

dx,

ci =
2
π

∫ 1
−1

g(0.5x+ 0.5)Ti(x)√
1− x2

dx, i = 1, 2, · · ·
(43)

such that

g(x) ≈ −0.000004054169+ 2.000464751x− 0.0088519738x2 + · · · − 0.4661302071x6. (44)



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1991

Therefore, the solution components are as follows

u0 = u(0) + xu′(0) + L−1(−0.000004054169 + 2.000464751x− 0.0088519738x2 + · · · − 0.4661302071x6),

= x− 0.0000020270845x2 + 0.3334107920x3 + · · · − 0.008323753699x8,

u1 = −0.5x3 + 5.06771124910−7x4 + · · · ,
u2 = 0.075x5 − 5.06771124910−8x6 + · · · ,
u3 = −0.005357142858x7 + · · · ,

...

that leads to the following series solution

uT (x) =
6∑

n=0

un(x) = x− 0.0000020270845x2 − 0.1665892081x3 − 0.0007371577121x4 + · · ·

(45)
Modification method via Laguerre’s polynomials starts off by expanding the

source term g(x) in the following form

g(x) =

6∑
n=0

cnLn(x), 0 ≤ x ≤ 1, (46)

where Ln(.) are orthogonal Laguerre’s polynomials, and ci are given by

ci =

∫ ∞

0
e−xLi(x)g(x)dx, i = 0, 1, · · · (47)

such that

g(x) ≈ 148321

625000
+

150161

156250
x− 219727

250000
x2 + · · · − 25849

450000000
x6. (48)

This gives the following iterative solutions

u0 = u(0) + xu′(0) + L−1(
148321

625000
+

150161

156250
x− 219727

250000
x2 + · · · − 25849

450000000
x6),

= x+ 0.118656800x2 + 0.160171733x3 − 0.07324233332x4 + · · · − 0.000001025753968x8,

u1 = −0.5x3 − 0.02966420000x4 − 0.07597424000x5 + · · · ,
u2 = 0.075x5 + 0.002966420000x6 − 0.07685530286x7 + · · · ,
u3 = −0.005357142857x7 + · · · ,

...

that sums to the following

uL(x) =

6∑
n=0

un(x) = x+ 0.1186568000x2 − 0.3398282667x3 − 0.1029065333x4 + · · · (49)



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1992

Modification method via Hermite’s polynomials expands the source term g(x)
as follows

g(x) =

6∑
n=0

cnHn(x), 0 ≤ x ≤ 1, (50)

where Hn(.) are orthogonal Hermite’s polynomials, and ci are given by

ci =
1

2ii!
√
π

∫ ∞

−∞
e−x2

Hi(x)g(x)dx, i = 0, 1, · · · (51)

such that
g(x) ≈ 1.412928152x− 0.8518569111x3 + 0.1099617181x5. (52)

We, therefore, obtain

u0 = u(0) + xu′(0) + L−1(1.412928152x− 0.8518569111x3 + 0.1099617181x5),

= x+ 0.2354880253x3 − 0.04259284556x5 + 0.002618136146x7,

u1 = −0.5x3 + 0.06467679622x5 + · · · ,
u2 = 0.075x5 − 0.07604834258x7 + · · · ,
u3 = −0.005357142858x7 + · · · ,
...

and leading to the following series solution

uH(x) =
6∑

n=0

un(x) = x− 0.2645119748x3 + 0.09708395066x5 + · · · (53)

Modification method via Gegenbauer’s polynomials equally starts off by ex-
panding the function g(x) as follows

g(x) =
6∑

n=0

cnC
α
n (2x− 1), 0 ≤ x ≤ 1, (54)

where Cα
n (.) are orthogonal Gegenbauer’s polynomials, and ci are given by

ci =

∫ 1
−1 g(0.5x+ 0.5)Cα

i (x)(1− x2)α−1/2dx∫ 1
−1[C

α
i (x)]

2(1− x2)α−1/2dx
, i = 0, 1, 2, · · · (55)

such that for α = 1, we get

g(x) ≈ −0.000017465407+ 2.000848250x− 0.0119838208x2 + · · · − 0.4708946817x6. (56)



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1993

Accodingly, we get

u0 = u(0) + xu′(0) + L−1(−0.000017465407 + 2.000848250x− 0.0119838208x2 + · · · − 0.4708946817x6),

= x− 0.0000087327035x2 + 0.3334747083x3 + · · · − 0.008408833601x8,

u1 = −0.5x3 + 0.000002183175875x4 + · · · ,
u2 = 0.075x5 − 2.183175875× 10−7x6 + · · · ,
u3 = −0.005357142858x7 + · · · ,
...

that leads to the following series solution

ug(x) =
6∑

n=0

un(x) = x− 0.0000087327035x2 − 0.1665252918x3 − 0.0009964685573x4 + · · ·

(57)
Modification method via Jacobi’s polynomials also goes as explained by expand-

ing g(x) as

g(x) =
6∑

n=0

cnP
(α,β)
n (2x− 1), 0 ≤ x ≤ 1 (58)

where Pn(.) are orthogonal Jacobi’s polynomials, and ci are given by

ci =

∫ 1
−1 g(0.5x+ 0.5)P

(α,β)
i (x)(1− x)α(1 + x)βdx∫ 1

−1[P
(α,β)
i (x)]2(1− x)α(1 + x)βdx

, i = 0, 1, 2, · · · (59)

such that for α = β = 1, one gets

g(x) ≈ 0.0001041764 + 1.997243414x+ 0.020167913x2 + · · · − 0.3992660100x6. (60)

In the same manner, we get the following solution components

u0 = u(0) + xu′(0) + L−1(0.0001041764 + 1.997243414x+ 0.020167913x2 + · · · − 0.3992660100x6),

= x+ 0.0000520882000x2 + 0.332873902x3 + · · · − 0.007129750179x8,

u1 = −0.5x3 − 0.00001302205000x4 + · · · ,
u2 = 0.075x5 + 1.30220500× 10−6x6 + · · · ,
u3 = −0.00535714285x7 + · · · ,
...

that sums to the following

uj(x) =
6∑

n=0

un(x) = x+ 0.00005208820000x2 − 0.1671260977x3 + 0.00166763736x4 + · · ·

(61)



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1994

Finally, we report the absolute error differences between the exact solution u(x) and the
respective modification solutions in Table 1. In this table, ut(x) stands for the modification
via the Taylor series; uP (x) via the Legendre’s series; uT (x) via the Chebyshev’s series;
uL(x) via the Laguerre’s series; uH(x) via the Hermite’s series; ug(x) via the Gegenbauer’s
series (α = 1); and finally uj(x) via the Jacobi series (α = 1, β = 1).

Table 1: Absolute error comparisons via the proposed modifications for Example 1

x |u(x)− ut(x)| |u(x)− uP (x)| |u(x)− uT (x)| |u(x)− ug(x)| |u(x)− uj(x)|

0 0 0 0 0 0
0.25 1.130× 10−8 1.50× 10−9 6.80× 10−9 2.750× 10−8 3.852× 10−7

0.50 5.743× 10−6 5.10× 10−9 2.680× 10−8 5.050× 10−8 8.532× 10−7

0.75 2.154× 10−4 1.508× 10−7 1.710× 10−7 7.830× 10−8 1.199× 10−6

1 2.790× 10−3 1.365× 10−6 1.394× 10−6 1.272× 10−6 2.724× 10−6

x |u(x)− uL(x)| |u(x)− uH(x)|

0 0 0
0.25 4.465× 10−3 1.444× 10−3

0.50 6.491× 10−3 9.756× 10−3

0.75 3.965× 10−3 2.484× 10−2

1 2.387× 10−2 3.959× 10−2

Example 2. Consider the following nonlinear IVP of ODE [7]

u′′ + uu′ = 2 cos(x2) + x sin(2x2)− 4x2 sin(x2), 0 ≤ x ≤ 1,
u(0) = 0, u′(0) = 0,

(62)

admiting the exact solution u(x) = sin(x2).

As in the preceding example 1, we start off by expressing the governing model in the
ADM operator form

u = L−1(2 cos(x2) + x sin(2x2)− 4x2 sin(x2))− L−1(uu′), (63)

where the inverse operator takes the expression L−1(.) =
∫ x
0

∫ x
0 (.)dxdx and N(u) =

uu′. Substituting Eqs. (5) and (6) into Eq. (63), we get the following recursive solution

u0 = L−1(2cos(x2) + xsin(2x2)− 4x2 sin(x2)),
un+1 = −L−1An(u0, u1, · · · ), n ≥ 0,

From Eq. (7), the nonlinear term N(u) = uu′ is also expressed through the following
Adomian polynomials



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1995

A0 = u0u
′
0,

A1 = u1u
′
0 + u0u

′
1,

A2 = u2u
′
0 + u1u

′
1 + u0u

′
2,

A3 = u3u
′
0 + u2u

′
1 + u1u

′
2 + u0u

′
3,

...

Thus, without further delay, by the same procedure as in Example 1, we present the
respective solution via the application of the proposed modification methods for m = 6 as
follows

ut(x) =
6∑

n=0

un(x) = x2 − 1

6
x6 +

1

54
x9 − 1

648
x12 + · · · ,

uP (x) =
6∑

n=0

un(x) = 1.000731859x2 − 0.01306146233x3 + 0.08294338582x4 + · · · ,

uT (x) =
6∑

n=0

un(x) = 1.000309776x2 − 0.009680124167x3 + 0.07289912332x4 + · · · ,

uL(x) =
6∑

n=0

un(x) = 1.964424074x2 − 1.747651757x3 + 0.5839657700x4 + · · ·

uH(x) =
6∑

n=0

un(x) = 1.501201152x2 + 0.1619524511x3 − 0.7135433355x4 + · · · ,

ug(x) =
6∑

n=0

un(x) = 1.001214606x2 − 0.01625100983x3 + 0.0916858975x4 + · · ·

uj(x) =

6∑
n=0

un(x) = 1.001718846x2 − 0.01916817000x3 + 0.09913448668x4 + · · · ,

(64)

Similarly, we report in Table 2 the absolute error differences between the exact solution
u(x) and the respective solutions by the proposed modification methods.

Table 2: Absolute error comparisons via the proposed modifications for Example 2

x |u(x)− ut(x)| |u(x)− uP (x)| |u(x)− uT (x)| |u(x)− ug(x)| |u(x)− uj(x)|

0 0 0 0 0 0
0.25 6.259× 10−8 1.174× 10−6 2.881× 10−6 4.032× 10−6 1.150× 10−5

0.50 2.754× 10−5 2× 10−7 3.136× 10−6 9.808× 10−6 2.402× 10−5

0.75 8.542× 10−4 1.10× 10−6 3.980× 10−6 1.448× 10−5 3.430× 10−5

1 8.087× 10−3 3.201× 10−7 5.459× 10−6 1.631× 10−5 3.972× 10−5



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1996

x |u(x)− uL(x)| |u(x)− uH(x)|

0 0 0
0.25 3.494× 10−2 3.088× 10−2

0.50 5.352× 10−2 9.762× 10−2

0.75 2.793× 10−2 1.235× 10−1

1 1.948× 10−1 5.025× 10−2

Example 3. Let us consider the following nonlinear IVP of ODE [11]

u′′ + u′ + 2xu3 = 2xe−3x, 0 ≤ x ≤ 1,
u(0) = 1, u′(0) = −1,

(65)

having the exact solution u(x) = e−x.

Firstly, we express the equation in the following operator form

u = 1− x+ L−1(2xe−3x)− L−1(u′)− 2L−1(xu3) (66)

where L−1(.) =
∫ x
0

∫ x
0 (.)dxdx and N(u) = u3. Substituting Eqs. (5) and (6) into Eq.

(66), we get the following recursive solution

u0 = 1− x+ L−1(2xe−3x),

un+1 = −L−1(u
′
n)− 2L−1xAn(u0, u1, · · · ), n ≥ 0,

The nonlinear term N(u) = u3 is given as in Example 1.
Thus, by the same procedure as in Example 1, we present the respective solution via

the application of the proposed modification methods for m = 6 as follows

ut(x) =

6∑
n=0

un(x) = 1− x+
1

2
x2 − 1

6
x3 +

1

24
x4 − 1

120
x5 +

1

720
x6 − 1

5040
x7 +

81

1120
x8 + · · · ,

uP (x) =
6∑

n=0

un(x) = 1− x+ 0.5000639003x2 − 0.1679247192x3 + 0.05074785700x4 + · · · ,

uT (x) =
6∑

n=0

un(x) = 1− x+ 0.5000267045x2 − 0.1675800359x3 + 0.04949342753x4 + · · · ,

uL(x) =
6∑

n=0

un(x) = 1− x+ 0.5778656006x2 − 0.4966684977x3 + 0.5997034709x4 + · · · ,

ug(x) =
6∑

n=0

un(x) = 1− x+ 0.5001073089x2 − 0.1682601554x3 + 0.05188023933x4 + · · · ,

uj(x) =

6∑
n=0

un(x) = 1− x+ 0.500154036x2 − 0.1685812115x3 + 0.0529062453x4 + · · · ,

(67)



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1997

Accordingly, we report in Table 3 the absolute error differences between the exact solution
u(x) and the respective solutions by the proposed modification methods.

Table 3: Absolute error comparisons via the proposed modifications for Example 3

x |u(x)− ut(x)| |u(x)− uP (x)| |u(x)− uT (x)| |u(x)− ug(x)| |u(x)− uj(x)|

0 0 0 0 0 0
0.25 9.764× 10−7 6.150× 10−8 1.585× 10−7 3.051× 10−7 8.615× 10−7

0.50 2.223× 10−4 9.90× 10−8 3.338× 10−7 5.373× 10−7 1.497× 10−6

0.75 5.086× 10−3 6.693× 10−6 6.967× 10−6 5.916× 10−6 4.718× 10−6

1 4.545× 10−2 2.439× 10−4 2.443× 10−4 2.431× 10−4 2.417× 10−4

x |u(x)− uL(x)|

0 0
0.25 1.424× 10−3

0.50 4.071× 10−4

0.75 2.418× 10−3

1 4.348× 10−3

Example 4. Consider the following nonlinear IVP [11]

u′′ + u′ − uu′ = (−2 + 4x2 − 2x)e−x2
+ 2xe−2x2

, 0 ≤ x ≤ 1,
u(0) = 1, u′(0) = 0,

that admits the exact solution u(x) = e−x2
.

We start by expressing the model in an operator notation as follows

u = 1 + L−1((−2 + 4x2 − 2x)e−x2
+ 2xe−2x2

)− L−1(u′) + L−1(uu
′
), (68)

where L−1(.) =
∫ x
0

∫ x
0 (.)dxdx and N(u) = uu′. Substituting Eqs. (5) and (6) into Eq.

(68), we get the following recursive solution

u0 = 1 + L−1(−2 + 4x2 − 2x)e−x2
+ 2xe−2x2

),

un+1 = −L−1(u
′
n) + L−1An(u0, u1, · · · ), n ≥ 0,

The nonlinear term N(u) = uu′ is given as in Example 2.
Also, without lost of generality, by the same procedure as in Example 1, we present the

respective solutions via the application of the proposed modification methods for m = 6



N. Alzaid et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1982-2000 1998

as follows

ut(x) =

6∑
n=0

un(x) = 1− x2 +
1

2
x4 − 1

6
x6 +

7

216
x9 + · · · ,

uP (x) =

6∑
n=0

un(x) = 1− 0.9999424490x2 − 0.0009236505467x3 + 0.5047677208x4,

uT (x) =

6∑
n=0

un(x) = 1− 0.9999762780x2 − 0.0006668175867x3 + 0.504082822x4 + · · · ,

uL(x) =
6∑

n=0

un(x) = 1− 1.224806499x2 + 0.7301686677x3 − 0.1966821240x4 + · · · ,

uH(x) =
6∑

n=0

un(x) = 1− 0.8811213405x2 − 0.07972377693x3 + 0.2955485375x4 + · · · ,

ug(x) =
6∑

n=0

un(x) = 1− 0.9999022015x2 − 0.001176415607x3 + 0.505394132x4 + · · · ,

uj(x) =
6∑

n=0

un(x) = 1− 0.9998581770x2 − 0.001420987833x3 + 0.5059695492x4 + · · · ,

(69)

Therefore, we report in Table 4 the absolute error differences between the exact solution
u(x) and the respective solutions by the proposed modification methods.

Table 4: Absolute error comparisons via the proposed modifications for Example 4

x |u(x)− ut(x)| |u(x)− uP (x)| |u(x)− uT (x)| |u(x)− ug(x)| |u(x)− uj(x)|

0 0 0 0 0 0
0.25 5.058× 10−7 1.208× 10−7 2.654× 10−7 3.305× 10−7 1.001× 10−6

0.50 9.578× 10−5 2.950× 10−8 2.344× 10−7 8.493× 10−7 2.121× 10−6

0.75 1.663× 10−3 1.045× 10−7 3.259× 10−7 1.272× 10−6 3.070× 10−6

1 1.011× 10−2 6.420× 10−8 5.574× 10−7 1.371× 10−6 3.558× 10−6

x |u(x)− uL(x)| |u(x)− uH(x)|
0 0 0

0.25 5.183× 10−3 5.497× 10−3

0.50 2.399× 10−3 1.126× 10−2

0.75 9.282× 10−3 3.958× 10−3

1 1.642× 10−2 1.438× 10−2



REFERENCES 1999

5. Conclusion

The present study proposed different modification methods for the standard Adomian
Decomposition Method (ADM) to tackle a variety of problems of mathematical physics.
These modification methods were based on the application of orthogonal polynomials that
play vital parts in numerical methods, as well as in approximation theories. Furthermore,
the study also scrutinized four different test nonlinear inhomogeneous IVPs, and distinc-
tively examined their respective absolute error differences. Notably, the proposed mod-
ification methods based on the application of Legendre’s orthogonal polynomials uP (x)
was noted to have the least error among its contending companions in three of the test
problems; it also performed outstandingly in the remaining problem. Finally, as certain
computational benefits of the proposed modification are realized with regards high-level of
accuracy and fewer computational steps in the study, it is therefore recommended to im-
plement these methods on high-order IVPs arising in the general science and engineering
applications.

References

[1] G Adomian. A review of the decomposition method and some recent results for
nonlinear equations. Mathematical and Computer Modelling, 13(7):17–43, 1990.

[2] George Adomian. Solving frontier problems of physics: the decomposition method,
with a preface by yves cherruault. Fundamental Theories of Physics, Kluwer Academic
Publishers Group, Dordrecht, 1, 1994.

[3] M Almazmumy, FA Hendi, HO Bakodah, and H Alzumi. Recent modifications of
adomian decomposition method for initial value problem in ordinary differential equa-
tions. 2012.

[4] HO Bakodah, MA Banaja, BA Alrigi, A Ebaid, and R Rach. An efficient modification
of the decomposition method with a convergence parameter for solving korteweg de
vries equations. Journal of King Saud University-Science, 31(4):1424–1430, 2019.

[5] William Wallace Bell. Special functions for scientists and engineers. Courier Corpo-
ration, 2004.

[6] Yücel Çenesiz and Aydın Kurnaz. Adomian decomposition method by gegenbauer and
jacobi polynomials. International Journal of Computer Mathematics, 88(17):3666–
3676, 2011.

[7] Mohammad Mahdi Hosseini. Adomian decomposition method with chebyshev poly-
nomials. Applied Mathematics and Computation, 175(2):1685–1693, 2006.

[8] Yucheng Liu. Adomian decomposition method with orthogonal polynomials: Legen-
dre polynomials. Mathematical and Computer Modelling, 49(5-6):1268–1273, 2009.



REFERENCES 2000

[9] Yucheng Liu. Adomian decomposition method with second kind chebyshev polyno-
mials. In Proceedings of the Jangjeon Mathematical Society, volume 12, pages 57–67.
The Society, 2009.

[10] Yucheng Liu. Application of legendre polynomials in adomian decomposition method.
In Proceedings of 2012 International Conference on Computer, Electrical, and Sys-
tems Sciences, Amsterdam, pages 13–14, 2012.

[11] Y Mahmoudi, M Abdollahi, N Karimian, and H Khalili. Adomian decomposition
method with laguerre polynomials for solving ordinary differential equation. Journal
of Basic and Applied Scientific Research, 2(12):12236–12241, 2012.

[12] Y Mahmoudi, N Karimian, and M Abdollahi. Adomian decomposition method with
hermite polynomials for solving ordinary differential equations. Journal of Basic and
Applied Scientific Research, 3(3):255–258, 2013.

[13] Abdul-Majid Wazwaz. A reliable modification of adomian decomposition method.
Applied mathematics and computation, 102(1):77–86, 1999.

[14] Yingying Xie, Lingfei Li, and Mancang Wang. Adomian decomposition method with
orthogonal polynomials: Laguerre polynomials and the second kind of chebyshev
polynomials. Mathematics, 9(15):1796, 2021.