Abstract
Dynamical behaviors of the time-fractional nonlinear model of the coupled spring-mass system with damping have been explored here. Fractional derivatives with singular and nonsingular kernels are used to assess the suggested model. The fractional Adams–Bashforth numerical method based on Lagrange polynomial interpolation is applied to solve the system with nonlocal operators. Existence, Ulam–Hyers stability, and uniqueness of the solution are established by using fixed-point theory and nonlinear analysis. Further, the error analysis of the present method has also been included. Finally, the behavior of the solution is explained by graphical representations through numerical simulations.
Issue Section:
Research Papers
References
1.
Jena
,
R. M.
,
Chakraverty
,
S.
, and
Jena
,
S. K.
, 2020
, “
Analysis of the Dynamics of Phytoplankton Nutrient and Whooping Cough Models With Nonsingular Kernel Arising in the Biological System
,” Chaos, Solitons Fractals
,
141
, pp. 1
–10
.10.1016/j.chaos.2020.1103732.
Jena
,
R. M.
,
Chakraverty
,
S.
, and
Baleanu
,
D.
, 2020
, “
SIR Epidemic Model of Childhood Diseases Through Fractional Operators With Mittag-Leffler and Exponential Kernels
,” Math. Comput. Simul.
,
182
, pp. 514
–534
.10.1016/j.matcom.2020.11.0173.
Jena
,
R. M.
,
Chakraverty
,
S.
,
Baleanu
,
D.
, and
Jena
,
S. K.
, 2020
, “
Analysis of Time-Fractional Dynamical Model of Romantic and Interpersonal Relationships With Non-Singular Kernels: A Comparative Study
,” Math. Methods Appl. Sci.
,
44
, pp. 2183
–2199
.10.1002/mma.69294.
Srivastava
,
H. M.
,
Jena
,
R. M.
,
Chakraverty
,
S.
, and
Jena
,
S. K.
, 2020
, “
Dynamic Response Analysis of Fractionally-Damped Generalized Bagley-Torvik Equation Subject to External Loads
,” Russ. J. Math. Phys.
,
27
(2
), pp. 254
–268
.10.1134/S10619208200201205.
Jena
,
R. M.
,
Chakraverty
,
S.
, and
Baleanu
,
D.
, 2020
, “
A Novel Analytical Technique for the Solution of Time-Fractional Ivancevic Option Pricing Model
,” Phys. A
,
550
, p. 124380
.10.1016/j.physa.2020.1243806.
Wang
,
Z.
,
Wang
,
X. H.
,
Xia
,
J. W.
,
Shen
,
H.
, and
Meng
,
B.
, 2020
, “
Adaptive Sliding Mode Output Tracking Control based-FODOB for a Class of Uncertain Fractional-Order Nonlinear Time-Delayed Systems
,” Sci. China Technol. Sci.
,
63
, pp. 1854
–1862
.10.1007/s11431-019-1476-47.
Jia
,
J.
,
Huang
,
X.
,
Li
,
Y.
,
Cao
,
J.
, and
Alsaedi
,
A.
, 2020
, “
Global Stabilization of Fractional-Order Memristor-Based Neural Networks With Time Delay
,” IEEE Trans. Neural Networks Learn. Syst.
,
31
(3
), pp. 997
–1009
.10.1109/TNNLS.2019.29153538.
Machado
,
J. T.
,
Kiryakova
,
V.
,
Kochubei
,
A.
, and
Luchko
,
Y.
, 2019
, “
Recent History of the Fractional Calculus: Data and Statistics
,” Handbook of Fractional Calculus with Applications
, Vol.
1
, De Gryuter, Berlin, Germany, pp. 1
–21
.10.1515/9783110571622-0019.
Abdo
,
M. S.
,
Panchal
,
S. K.
,
Shah
,
K.
, and
Abdeljawad
,
T.
, (2020
, “
2020) Existence Theory and Numerical Analysis of Three Species Prey-Predator Model Under Mittag-Leffler Power Law
,” Adv. Differ. Equ.
,
249
, pp. 1–16.10.1186/s13662-020-02709-710.
Owolabi
,
K. M.
, 2019
, “
Mathematical Modelling and Analysis of Love Dynamics: A Fractional Approach
,” Phys. A
,
525
, pp. 849
–865
.10.1016/j.physa.2019.04.02411.
Toufik
,
M.
, and
Atangana
,
A.
, 2017
, “
New Numerical Approximation of Fractional Derivative With Non-Local and Non-Singular Kernel: Application to Chaotic Models
,” Eur. Phys. J. Plus
,
132
, p. 444
.10.1140/epjp/i2017-11717-012.
Werner
,
W.
, 1984
, “
Polynomial Interpolation: Lagrange Versus Newton
,” Math. Comput.
,
43
, pp. 205
–217
.10.1090/S0025-5718-1984-0744931-013.
Krogh
,
T. F.
, 1970
, “
Efficient Algorithms for Polynomial Interpolation and Numerical Differentiation
,” Math. Comput.
,
24
, pp. 185
–190
.10.1090/S0025-5718-1970-0258240-X14.
Yang
,
Y.
, and
Gordon
,
S. P.
, 2015
, “
Visualizing and Understanding the Components of Lagrange and Newton Interpolation
,” Probl., Resour., Issues Math. Undergrad. Stud.
,
26
(1
), pp. 39
–52
.10.1080/10511970.2015.104505315.
Srivastava
,
R. B.
, and
Shukla
,
S.
, 2012
, Numerical Accuracies of Lagrange's and Newton Polynomial Interpolation: Numerical Accuracies of Interpolation Formulas
,
LAP LAMBERT Academic Publishing
, Wallingford, UK.16.
Rosenfeld
,
J. A.
, and
Dixon
,
W. E.
, 2017
, “
Approximating the Caputo Fractional Derivative Through the Mittag-Leffler Reproducing Kernel Hilbert Space and the Kernelized Adams-Bashforth-Moulton Method
,” SIAM J. Numer. Anal.
,
55
(3
), pp. 1201
–1217
.10.1137/16M105689417.
Diethelm
,
K.
,
Ford
,
N. J.
, and
Freed
,
A. D.
, 2002
, “
A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
,” Nonlinear Dyn.
,
29
, pp. 3
–22
.10.1023/A:101659221934118.
Diethelm
,
K.
,
Ford
,
N. J.
, and
Freed
,
A. D.
, 2004
, “
Detailed Error Analysis for a Fractional Adams Method
,” Numer. Algorithms
,
36
(1
), pp. 31
–52
.10.1023/B:NUMA.0000027736.85078.be19.
Gnitchogna
,
R.
, and
Atangana
,
A.
, 2017
, “
New Two Step Laplace Adam- Bashforth Method for Integer a Noninteger Order Partial Differential Equations
,” Numer. Methods Partial Differ. Equ.
,
34
(5
), pp. 1739
–1758
.10.1002/num.2221620.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
, 1993
, Fractional Integrals and Derivatives: Theory and Applications
,
Gordon and Breach
,
Yverdon
.21.
Caputo
,
M.
, and
Fabrizio
,
M.
, 2015
, “
A New Definition of Fractional Derivative Without Singular Kernel
,” Prog. Fract. Differ. Appl.
,
1
(2
), pp. 73
–85
.http://www.naturalspublishing.com/files/published/0gb83k287mo759.pdf22.
Atangana
,
A.
, and
Baleanu
,
D.
, 2016
, “
New Fractional Derivatives With Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model
,” Therm. Sci.
,
20
, pp. 1
–16
.10.2298/TSCI160111018A23.
Hristov
,
J.
, 2016
, “
Transient Heat Diffusion With a Non-Singular Fading Memory from the Cattaneo Constitutive Equation With Jeffrey's Kernel to the Caputo-Fabrizio Time-Fractional Derivative
,” Therm. Sci.
,
20
(2
), pp. 757
–762
.10.2298/TSCI160112019H24.
Hristov
,
J.
, 2017
, “
Derivation of the Fractional Dodson Equation and Beyond: Transient Diffusion With a Non-Singular Memory and Exponentially Fading-Out Diffusivity
,” Prog. Fract. Differ. Appl.
,
3
(4
), pp. 255
–270
.10.18576/pfda/03040225.
Hristov
,
J.
, 2019
, “
Linear Viscoelastic Responses and Constitutive Equations in Terms of Fractional Operators With Non-Singular Kernels: Pragmatic Approach, Memory Kernel Correspondence Requirement and Analyses
,” Eur. Phys. J. Plus
,
134
(283
), pp. 1
–32
.10.1140/epjp/i2019-12697-726.
Atangana
,
A.
, 2017
, “
Fractal-Fractional Differentiation and Integration: Connecting Fractal Calculus and Fractional Calculus to Predict Complex System
,” Chaos, Solitons Fractals
,
102
, pp. 396
–406
.10.1016/j.chaos.2017.04.02727.
Atangana
,
A.
,
Akgül
,
A.
, and
Owolabi
,
K. M.
, 2020
, “
Analysis of Fractal Fractional Differential Equations
,” Alexandria Eng J.
,
59
(3
), pp. 1117–1134.10.1016/j.aej.2020.01.00528.
Atangana
,
A.
, and
Qureshi
,
S.
, 2019
, “
Modeling Attractors of Chaotic Dynamical Systems With Fractal-Fractional Operators
,” Chaos, Solitons Fractals
,
123
, pp. 320
–337
.10.1016/j.chaos.2019.04.02029.
Fay
,
T. H.
, and
Graham
,
S. D.
, 2003
, “
Coupled Spring Equations
,” Int. J. Math. Educ. Sci. Technol.
,
34
(1
), pp. 65
–79
.10.1080/002073902100002925830.
Zafar
,
Z. U. A.
,
Younas
,
S.
,
Hussain
,
M. T.
, and
Tunçd
,
C.
, 2021
, “
Fractional Aspects of Coupled Mass-Spring System
,” Chaos, Solitons Fractals
,
144
, p. 110677
.10.1016/j.chaos.2021.11067731.
Losada
,
J.
, and
Nieto
,
J. J.
, 2015
, “
Properties of the New Fractional Derivative Without Singular Kernel
,” Progr. Fract. Differ. Appl.
,
1
(2
), pp. 87
–92
.10.12785/pfda/01020232.
Ali
,
Z.
,
Kumam
,
P.
,
Shah
,
K.
, and
Zada
,
A.
, 2019
, “
Investigation of Ulam Stability Results of a Coupled System of Nonlinear Implicit Fractional Differential Equations
,” Mathematics
,
7
(4
), p. 341
.10.3390/math704034133.
Ali
,
Z.
,
Zada
,
A.
, and
Shah
,
K.
, 2019
, “
On Ulam's Stability for a Coupled Systems of Nonlinear Implicit Fractional Differential Equations
,” Bull. Malays. Math. Sci. Soc.
,
42
(5
), pp. 2681
–2699
.10.1007/s40840-018-0625-xCopyright © 2022 by ASME
You do not currently have access to this content.
