Abstract

The aim of the present investigation to find the solution for fractional generalized Hirota–Satsuma coupled Korteweg–de-Vries (KdV) and coupled modified KdV (mKdV) equations with the aid of an efficient computational scheme, namely, fractional natural decomposition method (FNDM). The considered fractional models play an important role in studying the propagation of shallow-water waves. Two distinct initial conditions are choosing for each equation to validate and demonstrate the effectiveness of the suggested technique. The simulation in terms of numeric has been demonstrated to assure the proficiency and reliability of the future method. Further, the nature of the solution is captured for different value of the fractional order. The comparison study has been performed to verify the accuracy of the future algorithm. The achieved results illuminate that, the suggested computational method is very effective to investigate the considered fractional-order model.

References

1.
Liouville
,
J.
,
1832
, “
Memoire Surquelques Questions de Geometrieet de Mecanique, Etsur un Nouveau Genre de Calcul Pour Resoudreces Questions
,”
J. Ec. Polytech.
,
13
, pp.
1
69
.
2.
Caputo
,
M.
,
1969
,
Elasticita e Dissipazione
,
Zanichelli
,
Bologna, Italy
.
3.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.
4.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
New York
.
5.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
Amsterdam, The Netherlands
.
6.
Baleanu
,
D.
,
Wu
,
G. C.
, and
Zeng
,
S. D.
,
2017
, “
Chaos Analysis and Asymptotic Stability of Generalized Caputo Fractional Differential Equations
,”
Chaos Solitons Fractals
,
102
, pp.
99
105
.10.1016/j.chaos.2017.02.007
7.
Baleanu
,
D.
,
Guvenc
,
Z. B.
, and
Machado
,
J. A. T.
,
2010
,
New Trends in Nanotechnology and Fractional Calculus Applications
,
Springer
,
New York
.
8.
Prakasha
,
D. G.
, and
Veeresha
,
P.
,
2020
, “
Analysis of Lakes Pollution Model With Mittag-Leffler Kernel
,”
J. Ocean Eng. Sci.
, pp.
1
13
.10.1016/j.joes.2020.01.004
9.
Veeresha
,
P.
,
Prakasha
,
D. G.
, and
Baskonus
,
H. M.
,
2019
, “
Novel Simulations to the Time-Fractional Fisher's Equation
,”
Math. Sci.
,
13
(
1
), pp.
33
42
.10.1007/s40096-019-0276-6
10.
Gao
,
W.
,
Veeresha
,
P.
,
Prakasha
,
D. G.
,
Baskonus
,
H. M.
, and
Yel
,
G.
,
2020
, “
New Numerical Results for the Time-Fractional Phi-Four Equation Using a Novel Analytical Approach
,”
Symmetry
,
12
(
3
), p.
478
.10.3390/sym12030478
11.
Veeresha
,
P.
,
Prakasha
,
D. G.
, and
Baskonus
,
H. M.
,
2019
, “
New Numerical Surfaces to the Mathematical Model of Cancer Chemotherapy Effect in Caputo Fractional Derivatives
,”
Chaos
,
29
(
1
), p.
013119
.10.1063/1.5074099
12.
Gao
,
W.
,
Veeresha
,
P.
,
Prakasha
,
D. G.
,
Baskonus
,
H. M.
, and
Yel
,
G.
,
2019
, “
A Powerful Approach for Fractional Drinfeld–Sokolov–Wilson Equation With Mittag-Leffler Law
,”
Alexandria Eng. J.
,
58
(
4
), pp.
1301
1311
.10.1016/j.aej.2019.11.002
13.
Gao
,
W.
,
Veeresha
,
P.
,
Prakasha
,
D. G.
,
Baskonus
,
H. M.
, and
Yel
,
G.
,
2020
, “
New Approach for the Model Describing the Deathly Disease in Pregnant Women Using Mittag-Leffler Function
,”
Chaos Solitons Fractals
,
134
, p.
109696
.10.1016/j.chaos.2020.109696
14.
Prakash
,
A.
,
Veeresha
,
P.
,
Prakasha
,
D. G.
, and
Goyal
,
M.
,
2019
, “
A Homotopy Technique for Fractional Order Multi-Dimensional Telegraph Equation Via Laplace Transform
,”
Eur. Phys. J. Plus
,
134
(
19
), pp.
1
18
.10.1140/epjp/i2019-12411-y
15.
Yin
,
C.
,
Dadras
,
S.
,
Huang
,
X.
,
Malek
,
H.
, and
Cheng
,
Y.
,
2018
, “
Optimal Lighting Control Strategy for Lighting System Based on Multivariable Fractional-Order Extremum Seeking Method
,” Annual American Control Conference (
ACC
), Milwaukee, WI, June 27–29, pp.
3098
3103
.10.23919/ACC.2018.8431004
16.
Veeresha
,
P.
, and
Prakasha
,
D. G.
,
2020
, “
Solution for Fractional Generalized Zakharov Equations With Mittag-Leffler Function
,”
Results Eng.
,
5
, p.
100085
.10.1016/j.rineng.2019.100085
17.
Yin
,
C.
,
Dadras
,
S.
,
Cheng
,
Y. H.
,
Huang
,
X.
,
Cao
,
J.
, and
Malek
,
H.
,
2019
, “
Multidimensional Fractional-Order Newton-Based Extremum Seeking for Online Light-Energy Saving Technique of Lighting System
,”
IEEE Trans. Ind. Electron.
, epub.10.1109/TIE.2019.2950867
18.
Prakasha
,
D. G.
,
Veeresha
,
P.
, and
Baskonus
,
H. M.
,
2019
, “
Analysis of the Dynamics of Hepatitis E Virus Using the Atangana-Baleanu Fractional Derivative
,”
Eur. Phys. J. Plus
,
134
, p. 214.10.1140/epjp/i2019-12590-5
19.
Wu
,
Y.
,
Geng
,
X.
,
Hu
,
X.
, and
Zhu
,
S.
,
1999
, “
A Generalized Hirota–Satsuma Coupled Korteweg–de Vries Equation and Miura Transformations
,”
Phys. Lett. A
,
255
(
4–6
), pp.
259
264
.10.1016/S0375-9601(99)00163-2
20.
Hirota
,
R.
, and
Satsuma
,
J.
,
1981
, “
Soliton Solutions of a Coupled Korteweg-de Vries Equation
,”
Phys. Lett. A
,
85
(
8–9
), pp.
407
408
.10.1016/0375-9601(81)90423-0
21.
Adomian
,
G.
,
1984
, “
A New Approach to Nonlinear Partial Differential Equations
,”
J. Math. Anal. Appl.
,
102
(
2
), pp.
420
434
.10.1016/0022-247X(84)90182-3
22.
Rawashdeh
,
M. S.
, and
Al-Jammal
,
H.
,
2016
, “
New Approximate Solutions to Fractional Nonlinear Systems of Partial Differential Equations Using the FNDM
,”
Adv. Differ. Equations
,
235
, pp.
1
19
.10.1186/s13662-016-0960-x
23.
Rawashdeh
,
M. S.
, and
Al-Jammal
,
H.
,
2016
, “
Numerical Solutions for System of Nonlinear Fractional Ordinary Differential Equations Using the FNDM
,”
Mediterr. J. Math.
,
13
(
6
), pp.
4661
4677
.10.1007/s00009-016-0768-7
24.
Rawashdeh
,
M. S.
,
2017
, “
The Fractional Natural Decomposition Method: Theories and Applications
,”
Math. Methods Appl. Sci.
,
40
(
7
), pp.
2362
2376
.10.1002/mma.4144
25.
Prakasha
,
D. G.
,
Veeresha
,
P.
, and
Rawashdeh
,
M. S.
,
2019
, “
Numerical Solution for ( 2 + 1 )-Dimensional Time-Fractional Coupled Burger Equations Using Fractional Natural Decomposition Method
,”
Math. Methods Appl. Sci.
,
42
(
10
), pp.
3409
3427
.10.1002/mma.5533
26.
Prakasha
,
D. G.
,
Veeresha
,
P.
, and
Baskonus
,
H. M.
,
2019
, “
Two Novel Computational Techniques for Fractional Gardner and Cahn-Hilliard Equations
,”
Comp. Math. Methods
,
1
(
2
), pp.
1
19
.10.1002/cmm4.1021
27.
Veeresha
,
P.
,
Prakasha
,
D. G.
, and
Singh
,
J.
,
2020
, “
Solution for Fractional Forced KdV Equation Using Fractional Natural Decomposition Method
,”
AIMS Math.
,
5
(
2
), pp.
798
810
.10.3934/math.2020054
28.
Ganji
,
D. D.
, and
Rafei
,
M.
,
2006
, “
Solitary Wave Solutions for a Generalized Hirota-Satsuma Coupled KdV Equation by Homotopy Perturbation Method
,”
Phys. Lett. A
,
356
(
2
), pp.
131
137
.10.1016/j.physleta.2006.03.039
29.
Ganji
,
Z. Z.
,
Ganji
,
D. D.
, and
Rostamiyan
,
Y.
,
2009
, “
Solitary Wave Solutions for a Time-Fraction Generalized Hirota-Satsuma Coupled KdV Equation by an Analytical Technique
,”
Appl. Math. Model.
,
33
(
7
), pp.
3107
3113
.10.1016/j.apm.2008.10.034
30.
Abazari
,
R.
, and
Abazari
,
M.
,
2012
, “
Numerical Simulation of Generalized Hirota-Satsuma Coupled KdV Equation by RDTM and Comparison With DTM
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
(
2
), pp.
619
629
.10.1016/j.cnsns.2011.05.022
31.
Liu
,
J.
, and
Li
,
H.
,
2013
, “
Approximate Analytic Solutions of Time-Fractional Hirota-Satsuma Coupled KdV Equation and Coupled mKdV Equation
,”
Abstr. Appl. Anal.
,
2013
, p.
561980
.10.1155/2013/561980
32.
Martínez
,
H. Y.
, and
Aguilar
,
J. F. G.
,
2018
, “
Fractional Sub-Equation Method for Hirota–Satsuma-Coupled KdV Equation and Coupled mKdV Equation Using the Atangana's Conformable Derivative
,”
Waves Random Complex Media
, 29(4), pp.
678
693
.10.1080/17455030.2018.1464233
33.
Rady
,
A. S. A.
,
Osman
,
E. S.
, and
Khalfallah
,
M.
,
2010
, “
Onsoliton Solutions for a Generalized Hirota-Satsuma Coupled KdV Equation
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(
2
), pp.
264
274
.10.1016/j.cnsns.2009.03.011
34.
Feng
,
D.
, and
Li
,
K.
,
2011
, “
Exact Traveling Wave Solutions for a Generalized Hirota-Satsuma Coupled KdV Equation by Fan Sub-Equation Method
,”
Phys. Lett. A
,
375
(
23
), pp.
2201
2210
.10.1016/j.physleta.2011.04.039
35.
Saberi
,
E.
, and
Hejazi
,
S. R.
,
2018
, “
Lie Symmetry Analysis, Conservation Laws and Exact Solutions of the Time-Fractional Generalized Hirota-Satsuma Coupled KdV System
,”
Phys. A
,
492
, pp.
296
307
.10.1016/j.physa.2017.09.092
36.
Gupta
,
S.
,
Kumar
,
D.
, and
Singh
,
J.
,
2017
, “
An Efficient Computational Approach for Generalized Hirota-Satsuma Coupled KdV Equations Arising in Shallow Water Waves
,”
Waves Wavelets Fractals Adv. Anal.
,
3
(
1
), pp.
14
30
.10.1515/wwfaa-2017-0002
37.
Zhao
,
G. Z.
,
Yu
,
X. J.
,
Xu
,
Y.
,
Zhu
,
J.
, and
Wu
,
D.
,
2013
, “
Approximate Analytic Solutions for a Generalized Hirota-Satsuma Coupled KdV Equation and a Coupled mKdV Equation
,”
Chin. Phys. B
,
2013
(
8
), pp.
1
10
.10.1088/1674-1056/19/8/080204
38.
Kumar
,
D.
,
Singh
,
J.
,
Tanwar
,
K.
, and
Baleanu
,
D.
,
2019
, “
A New Fractional Exothermic Reactions Model Having Constant Heat Source in Porous Media With Power, Exponential and Mittag-Leffler Laws
,”
Int. J. Heat Mass Transfer
,
138
, pp.
1222
1227
.10.1016/j.ijheatmasstransfer.2019.04.094
39.
Kumar
,
D.
,
Singh
,
J.
, and
Baleanu
,
D.
,
2020
, “
On the Analysis of Vibration Equation Involving a Fractional Derivative With Mittag-Leffler Law
,”
Math. Meth. Appl. Sci.
,
43
(
1
), pp.
443
457
.10.1002/mma.5903
40.
Kumar
,
D.
,
Singh
,
J.
,
Qurashi
,
M. A.
, and
Baleanu
,
D.
,
2019
, “
A New Fractional SIRS-SI Malaria Disease Model With Application of Vaccines, Anti-Malarial Drugs, and Spraying
,”
Adv. Differ. Equations
, p.
278
.10.1186/s13662-019-2199-9
41.
Singh
,
J.
,
Kumar
,
D.
, and
Baleanu
,
D.
,
2019
, “
New Aspects of Fractional Biswas-Milovic Model With Mittag-Leffler Law
,”
Math. Model. Nat. Phenom.
,
14
(
3
), p.
303
.10.1051/mmnp/2018068
42.
Prakasha
,
D. G.
,
Veeresha
,
P.
, and
Singh
,
J.
,
2019
, “
Fractional Approach for Equation Describing the Water Transport in Unsaturated Porous Media With Mittag-Leffler Kernel
,”
Front. Phys.
,
7
, p.
193
.10.3389/fphy.2019.00193
43.
Singh
,
J.
,
Kumar
,
D.
,
Baleanu
,
D.
, and
Rathore
,
S.
,
2019
, “
On the Local Fractional Wave Equation in Fractal Strings
,”
Math. Methods Appl. Sci.
,
42
(
5
), pp.
1588
1595
.10.1002/mma.5458
44.
Kumar
,
D.
,
Singh
,
J.
, and
Baleanu
,
D.
,
2018
, “
A New Numerical Algorithm for Fractional Fitzhugh-Nagumo Equation Arising in Transmission of Nerve Impulses
,”
Nonlinear Dyn.
,
91
(
1
), pp.
307
317
.10.1007/s11071-017-3870-x
45.
Veeresha
,
P.
,
Prakasha
,
D. G.
, and
Baskonus
,
H. M.
,
2019
, “
An Efficient Technique for a Fractional-Order System of Equations Describing the Unsteady Flow of a Polytropic Gas
,”
Pramana
,
93
, p. 75.10.1007/s12043-019-1829-9
46.
Kumar
,
D.
,
Singh
,
J.
, and
Baleanu
,
D.
,
2018
, “
Analysis of Regularized Long-Wave Equation Associated With a New Fractional Operator With Mittag-Leffler Type Kernel
,”
Phys. A
,
492
, pp.
155
167
.10.1016/j.physa.2017.10.002
47.
Mittag-Leffler
,
G. M.
,
1903
, “Sur la Nouvelle Function E α ( x ),”
C. R. Acad. Sci. Paris
,
137
, pp.
554
558
.
48.
Saif
,
M.
,
Khan
,
F.
,
Sooppy Nisar
,
K.
, and
Araci
,
S.
,
2020
, “
N-Transform—Properties and Applications
,”
NUST J. Eng. Sci.
,
21
(
2
), pp.
127
133
.10.22436/jmcs.021.02.04
49.
Loonker
,
D.
, and
Banerji
,
P. K.
,
2013
, “
Solution of Fractional Ordinary Differential Equations by Natural Transform
,”
Int. J. Math. Eng. Sci.
,
12
(
2
), pp.
1
7
.
50.
Rezazadeh
,
H.
,
Seadawy
,
A. R.
,
Eslami
,
M.
, and
Mirzazadeh
,
M.
,
2019
, “
Generalized Solitary Wave Solutions to the Time Fractional Generalized Hirota-Satsuma Coupled KdV Via New Definition for Wave Transformation
,”
J. Ocean Eng. Sci.
,
4
(
2
), pp.
77
84
.10.1016/j.joes.2019.01.002
51.
Kurt
,
A.
,
Rezazadeh
,
H.
,
Senol
,
M.
,
Neirameh
,
A.
,
Tasbozan
,
O.
,
Eslami
,
M.
, and
Mirzazadeh
,
M.
,
2019
, “
Two Effective Approaches for Solving Fractional Generalized Hirota-Satsuma Coupled KdV System Arising in Interaction of Long Waves
,”
J. Ocean Eng. Sci.
,
4
(
1
), pp.
24
32
.10.1016/j.joes.2018.12.004
You do not currently have access to this content.