Abstract

In this paper, we find the solution for fractional coupled system arisen in magnetothermoelasticity with rotation using q-homotopy analysis transform method (q-HATM). The proposed technique is graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme, and fractional derivative defined with Mittag–Leffler kernel. The fixed point hypothesis is considered to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. To illustrate the efficiency of the future technique, we analyzed the projected model in terms of fractional order. Moreover, the physical behavior of q-HATM solutions has been captured in terms of plots for different arbitrary order. The attained consequences confirm that the considered algorithm is highly methodical, accurate, very effective, and easy to implement while examining the nature of fractional nonlinear differential equations arisen in the connected areas of science and engineering.

References

References
1.
Liouville
,
J.
,
1832
, “
Memoire Surquelques Questions de Geometrieet de Mecanique, et Sur un Nouveau Genre de Calcul Pour Resoudreces Questions
,”
J. Ecole. Polytech.
,
13
, pp.
1
69
.
2.
Riemann
,
G. F. B.
,
1896
,
Versuch Einer Allgemeinen Auffassung Der Integration Und Differentiation, Gesammelte Mathematische Werke
,
Cambridge University Press
,
Cambridge, UK
.
3.
Caputo
,
M.
,
1969
, “
Elasticita e Dissipazione
,”
Zanichelli, Bologna
.
4.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to Fractional Calculus and Fractional Differential Equations
,
A Wiley
,
New York
.
5.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
New York
.
6.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
Amsterdam, The Netherlands
.
7.
Baleanu
,
D.
,
Guvenc
,
Z. B.
, and
Tenreiro Machado
,
J. A.
,
2010
,
New Trends in Nanotechnology and Fractional Calculus Applications
,
Springer
,
Berlin
.
8.
Gao
,
W.
,
Veeresha
,
P.
,
Prakasha
,
D. G.
, and
Baskonus
,
H. M.
,
2020
, “
Novel Dynamic Structures of 2019-nCoV With Nonlocal Operator Via Powerful Computational Technique
,”
Biology
,
9
(
5
), p.
107
.10.3390/biology9050107
9.
Veeresha
,
P.
, and
Prakasha
,
D. G.
,
2019
, “
A Novel Technique for (2 + 1)-Dimensional Time-Fractional Coupled Burgers Equations
,”
Math. Comput. Simul.
,
166
, pp.
324
345
.10.1016/j.matcom.2019.06.005
10.
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
11.
Veeresha
,
P.
, and
Prakasha
,
D. G.
,
2020
, “
An Efficient Technique for Two-Dimensional Fractional Order Biological Population Model
,”
Int. J. Model. Simul. Sci. Comput.
,
11
(
01
), p.
2050005
.10.1142/S1793962320500051
12.
Veeresha
,
P.
, and
Prakasha
,
D. G.
,
2020
, “
A Reliable Analytical Technique for Fractional Caudrey-Dodd-Gibbon Equation With Mittag-Leffler Kernel
,”
Nonlinear Eng.
,
9
(
1
), pp.
319
328
.10.1515/nleng-2020-0018
13.
Liang
,
Y.
,
Wang
,
S.
,
Chen
,
W.
,
Zhou
,
Z.
, and
Magin
,
R. L.
,
2019
, “
A Survey of Models of Ultraslow Diffusion in Heterogeneous Materials
,”
ASME Appl. Mech. Rev.
,
71
(
4
), p. 040802.10.1115/1.4044055
14.
Boit
,
M.
,
1956
, “
Thermoelasticity and Irreversible Thermodynamics
,”
J. Appl. Phys
,
27
, pp.
240
253
.10.1063/1.1722351
15.
Gepreel
,
K. A.
,
Abo-Dahab
,
S. M.
, and
Nofal
,
T. A.
,
2012
, “
Homotopy Perturbation Method and Variational Iteration Method for Harmonic Waves Propagation in Nonlinear Magneto-Thermoelasticity With Rotation
,”
Math. Probl. Eng.
,
2012
, pp.
1
30
.10.1155/2012/827901
16.
Jafarian
,
A.
,
Ghaderi
,
P.
,
Golmankhaneh
,
A. K.
, and
Baleanu
,
D.
,
2013
, “
Analytic Solution for a Nonlinear Problem of Magneto-Thermoelasticity
,”
Rep. Math. Phys.
,
71
(
3
), pp.
399
410
.10.1016/S0034-4877(13)60039-7
17.
Caputo
,
M.
, and
Fabrizio
,
M.
,
2015
, “
A New Definition of Fractional Derivative Without Singular Kernel
,”
Prog. Fract. Differ. Appl.
,
1
(
2
), pp.
73
85
.10.12785/pfda/010201
18.
Atangana
,
A.
, and
Baleanu
,
D.
,
2016
, “
New Fractional Derivatives With Non-Local and Non-Singular Kernel Theory and Application to Heat Transfer Model
,”
Therm. Sci.
,
20
(
2
), pp.
763
769
.10.2298/TSCI160111018A
19.
Yingjie
,
L.
, and
Wen
,
C.
,
2018
, “
A Non-Local Structural Derivative Model for Characterization of Ultraslow Diffusion in Dense Colloids
,”
Commun. Nonlinear Sci. Numer. Simul.
,
56
, pp.
131
137
.10.1016/j.cnsns.2017.07.027
20.
Chen
,
W.
,
Liang
,
Y.
, and
Hei
,
X.
,
2016
, “
Structural Derivative Based on Inverse Mittag-Leffler Function for Modeling Ultraslow Diffusion
,”
Fract. Calc. Appl. Anal.
,
19
(
5
), pp.
1250
1261
.10.1515/fca-2016-0064
21.
Liao
,
S. J.
,
1997
, “
Homotopy Analysis Method and Its Applications in Mathematics
,”
J. Basic Sci. Eng.
,
5
(
2
), pp.
111
125
.
22.
Singh
,
J.
,
Kumar
,
D.
, and
Swroop
,
R.
,
2016
, “
Numerical Solution of Time- and Space-Fractional Coupled Burgers' Equations Via Homotopy Algorithm
,”
Alexandria Eng. J.
,
55
(
2
), pp.
1753
1763
.10.1016/j.aej.2016.03.028
23.
Srivastava
,
H. M.
,
Kumar
,
D.
, and
Singh
,
J.
,
2017
, “
An Efficient Analytical Technique for Fractional Model of Vibration Equation
,”
Appl. Math. Model.
,
45
, pp.
192
204
.10.1016/j.apm.2016.12.008
24.
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
25.
Gao
,
W.
,
Veeresha
,
P.
,
Prakasha
,
D. G.
,
Senel
,
B.
, and
Baskonus
,
H. M.
,
2020
, “
Iterative Method Applied to the Fractional Nonlinear Systems Arising in Thermoelasticity With Mittag-Leffler Kernel
,”
Fractals
, 28(8), pp. 1–16.10.1142/S0218348X2040040X
26.
Veeresha
,
P.
,
Prakasha
,
D. G.
,
Singh
,
J.
,
Kumar
,
D.
, and
Baleanu
,
D.
,
2020
, “
Fractional Klein-Gordon-Schrödinger Equations With Mittag-Leffler Memory
,”
Chin. J. Phys.
,
68
, pp.
65
78
.10.1016/j.cjph.2020.08.023
27.
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
28.
Veeresha
,
P.
,
Prakasha
,
D. G.
, and
Singh
,
J.
,
2020
, “
A Novel Approach for Nonlinear Equations Occurs in Ion Acoustic Waves in Plasma With Mittag-Leffler Law
,”
Eng. Comput.
,
37
(
6
), pp.
1865
1897
.10.1108/EC-09-2019-0438
29.
Singh
,
J.
,
2019
, “
A New Analysis for Fractional Rumor Spreading Dynamical Model in a Social Network With Mittag-Leffler Law
,”
Chaos
,
29
(
1
), p.
013137
.10.1063/1.5080691
30.
Kiran
,
M. S.
,
Betageri
,
V. S.
,
Prakasha
,
D. G.
,
Veeresha
,
P.
, and
Kumar
,
S.
,
2020
, “
Evolution and Analysis of COVID-2019 Through a Fractional Mathematical Model
,”
Authorea
, Epub.10.22541/au.158879146.68774508
31.
Prakasha
,
D. G.
, and
Veeresha
,
P.
,
2020
, “
Analysis of Lakes Pollution Model With Mittag-Leffler Kernel
,”
J. Ocean Eng. Sci.
, Epub.10.1016/j.joes.2020.01.004
32.
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
33.
Singh
,
J.
,
Kumar
,
D.
,
Hammouch
,
Z.
, and
Atangana
,
A.
,
2018
, “
A Fractional Epidemiological Model for Computer Viruses Pertaining to a New Fractional Derivative
,”
Appl. Math. Comput.
,
316
, pp.
504
515
.10.1016/j.amc.2017.08.048
34.
Prakasha
,
D. G.
,
Malagi
,
N. S.
, and
Veeresha
,
P.
,
2020
, “
New Approach for Fractional Schrödinger–Boussinesq Equations With Mittag-Leffler Kernel
,”
Math. Meth. Appl. Sci.
, Epub.10.1002/mma.6635
35.
Atangana
,
A.
, and
Alkahtani
,
B. T.
,
2015
, “
Analysis of the Keller-Segel Model With a Fractional Derivative Without Singular Kernel
,”
Entropy
,
17
(
12
), pp.
4439
4453
.10.3390/e17064439
36.
Atangana
,
A.
, and
Alkahtani
,
B. T.
,
2016
, “
Analysis of Non- Homogenous Heat Model With New Trend of Derivative With Fractional Order
,”
Chaos Solitons Fractals
,
89
, pp.
566
571
.10.1016/j.chaos.2016.03.027
37.
Veeresha
,
P.
,
Prakasha
,
D. G.
, and
Kumar
,
D.
,
2020
,
Fractional SIR Epidemic Model of Childhood Disease With Mittag-Leffler Memory, Fractional Calculus in Medical and Health Science
,
CRC Press
,
Boca Raton, FL
, pp.
229
248
.
38.
Veeresha
,
P.
,
Prakasha
,
D. G.
,
Singh
,
J.
,
Khan
,
I.
, and
Kumar
,
D.
,
2020
, “
Analytical Approach for Fractional Extended Fisher–Kolmogorov Equation With Mittag-Leffler Kernel
,”
Adv. Differ. Equ.
,
174
.10.1186/s13662-020-02617-w
39.
Kumar
,
D.
,
Agarwal
,
R. P.
, and
Singh
,
J.
,
2018
, “
A Modified Numerical Scheme and Convergence Analysis for Fractional Model of Lienard's Equation
,”
J. Comput. Appl. Math
,
339
, pp.
405
413
.10.1016/j.cam.2017.03.011
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