DOI : 10.17577/IJERTV14IS110408
- Open Access
- Authors : Hasan A. Qrewi, Karri Sanjay Kumar
- Paper ID : IJERTV14IS110408
- Volume & Issue : Volume 14, Issue 11 , November – 2025
- DOI : 10.17577/IJERTV14IS110408
- Published (First Online): 04-12-2025
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
A Simplified Adomian Decomposition Framework Based on the Sumudu Transform for Differential Equation Models
Hasan A. Qrewi,
Faculty of Mechatronics Engineering, Maaref University of Applied Sciences, Syria
Karri Sanjay Kumar
Marwadi Education Foundation, Gujarat, IN
Abstract – This paper presents the use of the Sumudu Transform combined with the modified Adomian Decomposition Method (ADM) to obtain approximate analytical solutions for different types of mathematical models. The method is applied to both linear and nonlinear ordinary differential equations, and numerical results are provided to demonstrate its accuracy by comparing the approximate solutions with exact ones whenever available.
Keywords – Sumudu Transform ; non-linear ; linear ; ADM; ODF.
-
INTRODUCTION
Recent research has shown a growing interest in enhancing the capabilities of homotopy methods through integration with integral transforms, particularly the Sumudu transform, which has proven effective in simplifying the mathematical structure of both fractional and nonlinear differential equations. The work of Bulut and Belgacem (2013) laid the foundation for applying the Sumudu transform to fractional differential equations, followed by Kumar and Daftardar-Gejji (2018), who developed the iterative Sumudu Transform Method (STIM) to extend its application to fractional partial differential equations. Subsequently, Prakash et al. (2018) and Rehman et al. (2018) introduced hybrid models combining Sumudu transforms with homotopy perturbation and double Sumudu transform methods to efficiently solve Burgers and KdV equations. Moreover, Mustafa (2023) enhanced this approach by improving the iterative schemes associated with the Sumudu transform, resulting in higher accuracy and stability for complex nonlinear systems.
In parallel, other integral transforms, such as the Zakai-type transform, have also been utilised, as highlighted in the study Mathematical Modelling in Engineering via MADM, which demonstrates the advantages of combining these transforms with modified Adomian Decomposition Methodsparticularly MADM (Reliable Modification of Adomian Decomposition Method) and TADM (Modified Adomian Decomposition Method by expressed in Taylor series ) ( see Al-Mazmumy et al
)to improve solution accuracy and reduce the number of required iterations.
In the present work, we have modified the homotopy method by integrating it with the Sumudu transform to enhance the efficiency of analytical solutions for both linear and nonlinear
equations. This modification reformulates the homotopy steps within the Sumudu transform space, allowing for a gradual reduction of nonlinearity and a smoother extraction of solution components compared to traditional approaches. Application of the modified method to a set of differential equationsboth homogeneous and non-homogeneousdemonstrated a significant reduction in the number of iterations and an increase in solution accuracy.
Furthermore, a detailed comparison with exact solutions confirmed that the modified method rapidly converges to the true solution, with minimal error, even in nonlinear cases that typically challenge standard perturbation techniques. This confirms that integrating the Sumudu transform with the homotopy framework provides a powerful approach for solving complex mathematical models without requiring excessive simplifications or additional assumptions.
Overall, this integration reflects a promising research trend toward developing hybrid methods that combine the advantages of integral transforms with the flexibility of homotopy techniques, providing accurate, stable, and widely applicable solutions for complex differential models.
-
SUMUDU TRANSFORM DEFINITION
The Sumudu Transform is an integral transform, similar in purpose to the Laplace Transform, used to solve differential equations and other problems in engineering, physics, and applied mathematics. It converts a function of time () into a function of a new variable u, often simplifying calculations and preserving the dimensionality of the original problem.
Mathematically, for a function () defined for 0
{()}() = ().
0
where: t is the original time variable, u is the Sumudu transform variable, ensures convergence of the integral.
The Sumudu Transform, like the Laplace Transform, it helps turn differential equations into algebraic equations, making them easier to solve, and also preserves the unit of the original function. That is, if f(t) has units of seconds, u keeps compatible dimensions, unlike the Laplace transform variable s, which is reciprocal time.
-
METHODOLOGY OF SUMUDU ADOMIAN DECOMPOSITION METHOD (SADM)
Consider the general nonlinear ordinary differential equation of
the form:
3 = [[(2) + 2]]
Eventually, we have the general recursive relation as follows:
0 = ()
{
+1
= [[() +
]]
(()) + (()) + (()) = () (1)
The initial condition is given by ()(0) = ()
where L is an operator of the highest derivative,
(()) is the remainder of the differential operator.
(()) is the nonlinear term, () Can any function.
By taking the S-Transform on both sides of Eq. (1), we have:
[(())] + [(())] + [(())] = [()] (2)Then:
[(())] = [()] [(())] [(())]Hence, the exact or approximate solution is given by:
() = ()
=0
-
APPLICATIONS
In this section, we solve several example problems using the new method and present the corresponding numerical results. The results are illustrated with graphs to make the solutions easier to understand. We also compare the performance of the new method with the classical Adomian Decomposition Method (ADM), TADM, and MADM to highlight its accuracy and efficiency.
()
1
+()(0)
=0
= [()] [(())] [(())]
1
Application (1): Find the solution?
{ = +
() = , () =
Solution:
() = +()(0) + [()] [(())]
=0
[(())] (3)
Using ADM:
The approximate solution 0 + 1 + 2 + 3
Then, taking the inverse S-Transform:
() = () [[(())] +
() + + + + + + +
! ! ! ! ! ! !
Using MADM:
() = + + ( )
[(())]] (13);=0
!
=0
1
We have (, ) = + = +
1 2 !
() = ( +()(0) + [()])
We consider 1 = 1 , 2
=
!
=0
The methodology consists of approximating the solution of (1)
Then the Recursive Relations:
0 =
as an infinite Series given:
=
!
+ (0)
{ +2 = (+1)
() = ()
=0
1 =
!
+ (0) = () =
+
!
!
However, the nonlinear term (()) is decomposed as
(()) = ()
=0
2 = (1) = (
3 = (1) = (
!
!
+ ) =
!
+ ) =
!
!
!
+
!
+
1
The approximate solution 0 + 1 + 2 + 3
() = ! [(()]=0
() +
+
!
!
+
!
+
!
+
!
+
!
() = () [ [ () + ()]]
Using SADM:
=0
Then:
=0
=0
take the Sumudu transform:
() = () + ()
1 = [[(0) + 0]]
() (0) (0)
= () + ()
2 = [[(1) + 1]]
2
2
() 1
2 = 2 + () + ()
() = 1 + 2() + 2()
() = 1 + 3 + ()
(()) = (1 + 3) + (2())
3
Figure (1): Comparison of SADM/ADM Approximation with MADM and Exact Solution
() = 1 +
3!
+ [2()]
() = ()
=0
() = +
=0
!
+ [2 ( )]
=0
Then the Recursive Relations:
{ 0 = + !
+1 = [2()] 0
1 = [2(0)] = [2 ( +
= [2( + )]
)]
!
= [2 + ] =
!
+
!
2 = [2(1)] = [2 (
!
+ )]
!
= [2(2 + )] = [4 + ] =
!
+
!
3 = [2(2)] = [2 (
!
+ )]
!
Application (2): Find the solution?
= [2(4 + )] = [6 + ] =
+
{
()
=
The approximate solution ()
! !
0 + 1 + 2 + 3
()() = ; = , , ,
() = +
!
+
!
+
!
+
!
+
!
+
!
+
!
Solution:
Using ADM:
TABLE(1): Comparison of SADM/ADM Approximate Solution with MADM, Exact Solution, and Absolute Error
The approximate solution 0 + 1 + 2
X
SADM (ADM)
MADM
EX
error1
error2
0
1
1
1
0
0
0.25
1.034025
1.034025
1.03E+00
3.66E-10
3.76E-10
0.5
1.148721
1.148721
1.15E+00
8.46E-08
8.93E-08
1
1.718257
1.718254
1.72E+00
1.46E-05
1.62E-05
1.25
2.240192
2.240172
2.24E+00
6.73E-05
7.64E-05
1.5
2.981035
2.980929
2.981689
0.000219
0.000255
2
5.382363
5.380952
5.389056
0.001242
0.001504
2.25
7.220299
7.216226
7.237736
0.002409
0.002972
2.5
9.641271
9.630759
9.682494
0.004257
0.005343
3
16.90067
16.84643
17.08554
0.01082
0.013995
3.25
22.18086
22.06938
22.54034
0.015948
0.020894
3.5
28.94718
28.72998
29.61545
0.022565
0.029899
4
48.52875
47.80635
50.59815
0.040899
0.055176
4.25
62.37482
61.12819
65.85541
0.052852
0.071782
4.5
79.80807
77.72287
85.51713
0.066759
0.091143
5
129.0013
123.619
143.4132
0.100492
0.138022
= 2 (1 +
1!
Using TADM
2
+
2!
+ +
41
)
41!
= 2 1
1!
2
2!
3
3!
4
4!
5
5!
6
6!
7
7!
8
8!
9
9!
10
10!
11
11!
12
12!
13
13!
( )
We have a Taylor series for = 1 + + + 5 +
= 2 (1 +
2
+
3
+
4
+
5
+
1!
+ ) 1
5!
2
1! 2! 3! 4! 5! 1! 2!
3 4 5 6 7 8 9 10
3! 4! 5! 6! 7! 8! 9! 10!
11
11!
12
12!
13
13!
( )
2
3
4
5
6
7
8
9
10
() 1 1
1 1 1
1 1 1 1
= 1 + + + + + + + + + +
1! 2! 3! 4! 5! 6! 7! 8! 9! 10!
14 14 13 12 11 10 9 8 7 6
11
12
13
14
15
16
1 1 1 1 1
+ + + + 2 + 2 + 2
11! 12! 13! 14! 15! 16!
5
4
3
2
( )
2
=
1
(())
() = ()
=0
() = 1 + + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
214
2
3
4
5
6
7
8
+ 10 + 11 + 12 + 13 +
1
() = 1 + 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8!
=0
9 10 11 12 13 14
+ + + + + + 2
9! 10! 11! 12! 13! 14!
14(())
() = 1 2 3 4 5 6 7 8 9
10 11 12 13 + 2
15
+ 2
15!
16
+ 2
16!
+ ( () )
=0
14(())
1
We consider
= 1 , = , =
0 1 2
1!
2 , =
3
2!
3 , ,
3!
1[()] = 1 [1 2 3 4 5 6 7
Then the Recursive Relations:
8 9 10 11 12 13
{
0 = 1
=
(
) 0
2
+
1
] 1[14(())]
+1
+1
2
13
= ( ) = (1 )
() = 2 1
1(14(()))
1 1 0 1!
13 13
1! 2!
13!
= ( ) =
1! 13! 1! 13!
2
2 = 2 (1 ) =
(
13
)
() = ()
=0
Then the Recursive Relations:
2
=
{
2!
2!
14
14!
1!
26
+
26!
13!
0 = 2 1
1!
2
2!
13
;
13!
The approximate solution 0
+ 1
+ 2
+ 3
+1() = 1 (14. (()))
1 +
1!
Using SADM:
2
+
2!
13
13!
14
14!
26
+
26!
1 = 1[14. (0)]
= 1 [14. (2 1
1!
2
2!
13
)]
13!
Take the Sumudu transform: 214
(
(14)
) = 2
) (())
= 1 [
1
14 15 1617 18 1920
()
14
(0)
14
(0)
13
(2)(0)
12
(3)(0)
11
(4)(0)
10
(5)(0)
9
21 2223 24 252627]
(6)(0)
8
(7)(0)
7
(8)(0)
6
(9)(0)
5
= 1 [2 (1 2 3 4 5 6 7 8
(10)(0)
4
(11)(0)
3
(12)(0)
2
9
10
11
12
13
1
+ )
1
(13)(0)
=
2
1
(())
14 15 1617 18 1920
21 2223 24 252627]
= 1
[2.1 2 2223
24
25
26
27
= 2 2.1
2
1!
22
2!
23
3!
24
4!
25
5!
26
6!
27
7!
28 29 210 211 212
2
28
8!
29
9!
210
10!
211
11!
212
12!
213
13!
213 +
1
14 15 1617
214
215
216
217
218
18 1920 21 2223 24
14!
15!
16!
17!
18!
25
26
27]
219
19!
220
20!
221
21!
222
22!
223
23!
= [2 2
2
1!
22
2!
23
3!
24
4!
27
]
27!
224
24!
225
25!
226
26!
227
27!
28
28!
29
29!
2 22
= 2 + 2 + +
1! 2!
23 24
+ +
3! 4!
+ +
27
27!
30 31 32 33 34 35
30! 31! 32! 33! 34! 35!
2
= 1[14. (1)]
2
22
36
36!
37
37!
38
38!
39
39!
40
40!
41
41!
= 1 [14. (2 + 2 + +
1! 2!
The approximate solution () 0
+ 1
+ 2
23
+
24
+
27
+ + )]
= 2 (1 +
2
+
3
+
+ +
41
)
3! 4!
27!
1! 2! 3!
41!
= 1 [
214
1
+ 214 + 215 + 216+217 + 218
+ 219+220 + 221 + 222+223
+ 224 + 225+226+227 + 28
+ 29+30 + 31 + 32+33 + 34
+ 35+36 + 37 + 38+39+40+41]
TABLE(2): Comparison of SADM/ADM Approximate Solution with TADM, Exact Solution, and Absolute Error
x
SADM (ADM)
TADM
Exact
Error 1
Error 2
0
1
1
1
0
0
0.25
1.284025
1.28125
1.284025
0
0.002161
0.5
1.648721
1.625
1.648721
0
0.014388
0.75
2.117
2.03125
2.117
0
0.040505
1
2.718282
2.5
2.718282
0
0.080301
= 1 [2 (1 2 3 4 5 6 7 8
9 10 11 12 13 + 1 )
1
+ 214 + 215 + 216+217 + 218
+ 219+220 + 221 + 222+223
+ 224 + 225+226+227 + 28
+ 29+30 + 31 + 32+33 + 34
Figure (2): Comparison of SADM/ADM Approximation with TADM and Exact Solution
+ 35+36 + 37 + 38+39+40+41]
= 1 [2.1 2 22 23 24 25 26 27
28 29 210 211 212
213 + 2
1
214 215
216217 218 219220 221
222223 224 225226227
28 2930 31 3233 34
3536 37 38394041]
Application (3) : Find the solution?
{ 2 1 = 0
Remark that we can obtain the exact solution if we continue using Recursive Relations i.e
(0) = 0
Solution:
Using ADM:
The approximate solution () 0 + 1 + 2 + 3
3 25 177
() = 0 + 1 + 2 + 3 + +4 + = + 3 + 15 + 315 +
= tan ()
TABLE(3): Comparison of SADM/ADM Approximate Solution with Exact Solution and Absolute Error
Using SADM
() +
3
3
5
+ 2 15 +
177
x
SADM/ADM
Exact
Error
0
0
0
0
0.1
0.100335
1.00E-01
2.19E-10
0.2
0.20271
2.03E-01
5.61E-08
0.3
0.309336
3.09E-01
1.44E-06
0.4
0.422787
0.422793
1.45E-05
0.5
0.546255
5.46E-01
8.70E-05
0.6
0.683879
6.84E-01
3.77E-04
0.7
0.841187
8.42E-01
1.31E-03
0.8
1.025675
1.03E+00
3.85E-03
0.9
1.247545
1.26E+00
1.00E-02
1
1.520635
1.557408
0.023612
945
Take the Sumudu transform:
() (2) (1) = 0
() = (1) + (2)
()
(0)
= 1 + (2)
() = 1. + . (2)
1(()) = 1() + 1(. (2))
() = + 1(. (2))
Now suppose: () = ()
=0
, 2 = ()
=0
Figure(3) : Comparison of SADM/ADM Approximation with Exact Solution
() = + 1 [. ( ())]
=0 =0
Then the Recursive Relations:
0 =
{+1() = 1(. ()) 0
0() = (0)2
1 = 1[. (0())] = 1[. ()] = 1[. 2] = 1[3]
3
= 3
3
1() = 201 = 2. . 3 =
24
3
24
2 × 4! × 4
2 = 1[. (1())] = 1 [. (
3
)] = 1 [. ] 3
= 16. 1[5] = 16
5
5!
25
= 15
2() = 202 + (1)2
25
3 2
46
6
176
= 2. . 15 + ( 3 )
= 15 + 9 =
45
176
3 = 1[. (2())] = 1 [. ( )]
45
17 × 6! × 6
-
CONCLUSION
The Adomian Decomposition Method (ADM) and its variant,
= 1 [.
45
17 × 6!
]
17 × 6!
7
177
the S-Transform Adomian Decomposition Method (SADM), provide powerful tools for obtaining analytical or approximate
solutions to linear and nonlinear differential equations. While
= 45
1[7] =
45 × 7! =
315
both methods produce the same series solution, SADM offers
The approximate solution () 0 + 1 + 2 + 3 + 4
significant advantages: it simplifies calculations, eliminates theneed for repeated integration, and is particularly efficient
() +
3
3
25
+ 15
177
+ 315
for high-order or complex equations.
REFERENCES
-
T. A. Abbasi, Improved Adomian decomposition method for nonlinear initial value problems, Appl. Math. Comput., vol. 215, no. 5, pp. 2034 2045, 2010.
-
Z. A. Abdo and Y. Q. Hasan, Modified ADM for higher-order Emden Fowler equations, AJPAS, vol. 12, no. 1, pp. 4559, 2020.
-
G. Adomian, A review of the decomposition method and some recent applications, J. Math. Anal. Appl., vol. 135, no. 2, pp. 501545, 1988.
-
G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., vol. 135, no. 2, pp. 501544, 1988.
-
G. Adomian, A review of the decomposition method and its application to stochastic continuous-time dynamical systems, Math. Comput. Simul., vol. 32, pp. 1942, 1990.
-
E. U. Agom and F. O. Ogunfiditimi, Modified Adomian polynomial for nonlinear functional with integer exponent, J. Math., vol. 11, no. 6, pp. 4045, 2015.
-
E. U. Agom and F. O. Ogunfiditimi, Modified Adomian polynomials for nonlinear functions with integer exponents, Int. J. Math. Math. Sci., vol. 2015, pp. 112, 2015.
-
S. Al Baghdadi, A comparative study of Adomian decomposition method and variational iteration method in solving nonlinear differential equations, Open Access J., 2024.
-
S. Aljuhani et al., Application of Adomian decomposition method for solving nonlinear Boussinesq equations using Maple, Sci. Res. Publ., vol. 12, no. 5, pp. 115, 2023.
-
S. Aljuhani et al., Solving nonlinear Boussinesq equations using Adomian decomposition method with Maple, SCIRP J., 2023.
- [11] M. Al-Mazmumy, Enhanced ADM using inverse integral operators for ODEs, Mathematics, vol. 11, no. 12, p. 698, 2022.
-
M. Al-Mazmumy, ADM combined with Taylor series and orthogonal polynomials for fractional PDEs, AIMS Math., vol. 9, no. 10, pp. 1475 1490, 2024.
-
M. Al-Mazmumy and S. O. Almuhalbedi, Solution of nonlinear integro- differential equations by two-step Adomian decomposition method
(TSAM), Int. J. Mod. Nonlinear Theory Appl., vol. 5, no. 4, pp. 248255, 2016.
-
M. Almazmumy, S. O. Al-Muhalbedi, and M. Rashid, Two-step Adomian decomposition method for initial value problems in ordinary differential equations, Open Access J. Math., vol. 4, no. 3, pp. 120132, 2012.
-
M. Al-Mazmumy et al., Modified Adomian decomposition method for initial-boundary value problems of fractional PDEs, AIMS Math., vol. 10, no. 3, pp. 12501275, 2025.
-
S. Al-Mazmumy et al., Modified Adomian decomposition method for
initial-boundary value problems of fractional PDEs, AIMS Math., 2025.
-
F. Almeida et al., Semi-analytical solutions of fractional Riccati differential equations using Adomian decomposition, PPHMJ Open Access J., 2023.
-
F. Almeida et al., Semi-analytical solutions of fractional Riccati differential equations using Adomian decomposition method, J. Math. Res., vol. 15, no. 2, pp. 4560, 2023.
-
J. Bazar, E. Babolian, G. Kember, A. Nouri, and R. Islam, An alternate algorithm for computing Adomian polynomials in special cases, Appl. Math. Comput., vol. 138, nos. 23, pp. 523529, 2003.
-
J. Bazar, E. Babolian, A. Nouri, and S. Islam, Alternative algorithm for computing Adomian polynomials, Appl. Math. Comput., vol. 140, nos. 23, pp. 303309, 2003.
-
F. B. M. Belgacem and A. A. Karaballi, Sumudu transform: Fundamental properties, investigations and applications, J. Appl. Math. Stoch. Anal., vol. 2006, pp. 110, 2006.
-
J. Biazar and S. M. Shafiof, A simple algorithm for calculating Adomian polynomials, Int. J. Contemp. Math. Sci., vol. 2, no. 20, pp. 975982, 2007.
-
J. Biazar and S. M. Shafiof, Simplified algorithm for multivariable Adomian polynomials, Comput. Math. Appl., vol. 54, no. 4, pp. 451457, 2007.
-
E. Centre and D. Uk, On the Adomian decomposition method for the solution of differential equations, Appl. Math., vol. 1, no. 2, pp. 2029, 2013.
-
B. Daoud, Modified Adomian decomposition method for boundary layer convective heat transfer problems, J. Comput. Phys., vol. 356, pp. 203218, 2018.
