Abstract
This study expands and modifies the homotopy analysis method to handle differential equations with generalized Caputo-type fractional derivatives. Analytical approximate solutions for such models were successfully provided using the proposed modification. The determination of the valid region of convergence for the proposed method, with respect to the auxiliary control parameter, was discussed when using fractional operators. Then, mainly, the accuracy and effectiveness of the proposed method was verified through illustrative examples and comparisons with the predictor corrector method and RK4 method. Finally, it is expected that the studied generalized operators and the suggested method can be widely applied in the field of fractional calculus.
References
1.
Oldham
,
K. B.
, and
Spanier
,
J.
, 1974
, The Fractional Calculus
,
Academic Press
,
New York
.2.
Bagley
,
R. L.
, and
Torvik
,
P. L.
, 1986
, “
On the Fractional Calculus Model of Viscoelastic Behavior
,” J. Rheol.
,
30
(1
), pp. 133
–155
.10.1122/1.5498873.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
, 1993
, Fractional Integrals and Derivatives: Theory and Applications
,
Gordon and Breach
,
Amsterdam
.4.
Mainardi
,
F.
, 1996
, “
Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena
,” Chaos Soliton Fract.
,
7
(9
), pp. 1461
–1477
.10.1016/0960-0779(95)00125-55.
Podlubny
,
I.
, 1999
, Fractional Differential Equations
,
Academic Press
,
New York
.6.
Hilfer
,
R.
, 2000
, Applications of Fractional Calculus in Physics
,
World Scientific
,
Singapore
.7.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
, 2006
, Theory and Applications of Fractional Differential Equations
,
Elsevier
,
Amsterdam, The Netherlands
.8.
Stojanović
,
M.
, 2011
, “
Numerical Method for Solving Diffusion-Wave Phenomena
,” Comput. Appl. Math.
,
235
(10
), pp. 3121
–3137
.10.1016/j.cam.2010.12.0109.
Herrmann
,
R.
, 2014
, Fractional Calculus: An Introduction for Physicists
,
World Scientific
,
Singapore
.10.
West
,
B. J.
, 2015
, Fractional Calculus View of Complexity: Tomorrow's Science
,
Taylor and Francis
,
UK
.11.
Sun
,
H. G.
,
Zhang
,
Y.
,
Baleanu
,
D.
,
Chen
,
W.
, and
Chen
,
Y. Q.
, 2018
, “
A New Collection of Real World Applications of Fractional Calculus in Science and Engineering
,” Commun. Nonlinear Sci. Numer. Simul.
,
64
, pp. 213
–231
.10.1016/j.cnsns.2018.04.01912.
Odibat
,
Z.
, and
Baleanu
,
D.
, 2019
, “
A Linearization-Based Approach of Homotopy Analysis Method for Non-Linear Time-Fractional Parabolic PDEs
,” Math. Methods Appl. Sci.
,
42
(18
), pp. 7222
–7232
.10.1002/mma.582913.
Odibat
,
Z.
, and
Kumar
,
S.
, 2019
, “
A Robust Computational Algorithm of Homotopy Asymptotic Method for Solving Systems of Fractional Differential Equations
,” ASME J. Comput. Nonlinear Dyn.
,
14
(8
), p. 81004
.10.1115/1.404361714.
Al-Qudah
,
A.
,
Odibat
,
Z.
, and
Shawagfeh
,
N.
, 2022
, “
A Linearization-Based Computational Algorithm of Homotopy Analysis Method for Nonlinear Reaction-Diffusion Systems
,” Math. Comput. Simulat.
,
194
, pp. 505
–522
.10.1016/j.matcom.2021.11.02715.
Al-Qudah
,
A.
,
Odibat
,
Z.
, and
Shawagfeh
,
N.
, 2022
, “
An Optimal Homotopy Analysis Transform Method for Handling Nonlinear PDEs
,” Int. J. Appl. Comput. Math.
,
8
(5
), p. 260
.10.1007/s40819-022-01401-616.
Yadav
,
S.
,
Kumar
,
D.
, and
Nisar
,
K. S.
, 2021
, “
A Reliable Numerical Method for Solving Fractional Reaction-Diffusion Equations
,” J. King Saud Univ. Sci.
,
33
(2
), p. 101320
.10.1016/j.jksus.2020.10132017.
Shawagfeh
,
N.
, 2002
, “
Analytical Approximate Solutions for Nonlinear Fractional Differential Equations
,” Appl. Math. Comput.
,
131
(2–3
), pp. 517
–529
.10.1016/S0096-3003(01)00167-918.
Duan
,
J. S.
,
Chaolu
,
T.
,
Rach
,
R.
, and
Lu
,
L.
, 2013
, “
The Adomian Decomposition Method With Convergence Acceleration Techniques for Nonlinear Fractional Differential Equations
,” Comput. Math. Appl.
,
66
(5
), pp. 728
–736
.10.1016/j.camwa.2013.01.01919.
Laoubi
,
M.
,
Odibat
,
Z.
, and
Maayah
,
B.
, 2022
, “
A Legendre-Based Approach of the Optimized Decomposition Method for Solving Nonlinear Caputo-Type Fractional Differential Equations
,” Math. Methods Appl. Sci.
,
45
(12
), pp. 7307
–7321
.10.1002/mma.823720.
Odibat
,
Z.
, and
Momani
,
S.
, 2006
, “
Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order
,” Int. J. Nonlinear Sci. Numer. Simul.
,
7
(1
), pp. 27
–34
.21.
Odibat
,
Z.
, and
Momani
,
S.
, 2009
, “
The Variational Iteration Method: An Effcient Scheme for Handling Fractional Partial Differential Equations in Fluid Mechanics
,” Comput. Math. Appl.
,
58
(11–12
), pp. 2199
–2208
.10.1016/j.camwa.2009.03.00922.
Anwar
,
A. M. O.
,
Jarad
,
F.
,
Baleanu
,
D.
, and
Ayaz
,
F.
, 2013
, “
Fractional Caputo Heat Equation Within the Double Laplace Transform
,” Rom. J. Phys.
,
58
(1
), pp. 15
–22
.23.
Jarad
,
F.
, and
Abdeljawad
,
T.
, 2020
, “
Generalized Fractional Derivatives and Laplace Transform
,” Discrete Contin. Dyn. Syst. - S
,
13
(3
), pp. 709
–722
.10.3934/dcdss.202003924.
Prakash
,
P.
,
Kumar
,
M.
, and
Baleanu
,
D.
, 2018
, “
A New Iterative Technique for a Fractional Model of Nonlinear Zakharov-Kuznetsov Equations Via Sumudu Transform
,” Appl. Math. Comput.
,
334
, pp. 30
–40
.25.
Singh
,
J.
, 2020
, “
Analysis of Fractional Blood Alcohol Model With Composite Fractional Derivative
,” Chaos Solitons Fractals
,
140
, p. 110127
.10.1016/j.chaos.2020.11012726.
Singh
,
J.
,
Gupta
,
A.
, and
Baleanu
,
D.
, 2022
, “
On the Analysis of an Analytical Approach for Fractional Caudrey-Dodd-Gibbon Equations
,” Alex. Eng. J.
,
61
(7
), pp. 5073
–5082
.10.1016/j.aej.2021.09.05327.
Odibat
,
Z.
,
Momani
,
S.
, and
Erturk
,
V. S.
, 2008
, “
Generalized Differential Transform Method: Application to Differential Equations of Fractional Order
,” Appl. Math. Comput.
,
197
(2
), pp. 467
–477
.28.
Odibat
,
Z.
,
Kumar
,
S.
,
Shawagfeh
,
N.
,
Alsaedi
,
A.
, and
Hayat
,
T.
, 2017
, “
A Study on the Convergence Conditions of Generalized Differential Transform Method
,” Math. Methods Appl. Sci.
,
40
(1
), pp. 40
–48
.10.1002/mma.396129.
Diethelm
,
K.
,
Ford
,
N.
, and
Freed
,
A.
, 2002
, “
A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
,” Nonlin. Dyn.
,
29
(1/4
), pp. 3
–22
.10.1023/A:101659221934130.
Cen
,
Z.
,
Le
,
A.
, and
Xu
,
A.
, 2017
, “
A Robust Numerical Method for a Fractional Differential Equation
,” Appl. Math. Comput.
,
315
, pp. 445
–452
.10.1016/j.amc.2017.08.01131.
Dabiri
,
A.
, and
Butcher
,
E. A.
, 2018
, “
Numerical Solution of Multi-Order Fractional Differential Equations With Multiple Delays Via Spectral Collocation Methods
,” Appl. Math. Modell.
,
56
, pp. 424
–448
.10.1016/j.apm.2017.12.01232.
Odibat
,
Z.
, and
Baleanu
,
D.
, 2020
, “
Numerical Simulation of Initial Value Problems With Generalized Caputo-Type Fractional Derivatives
,” Appl. Numer. Math.
,
156
, pp. 94
–105
.10.1016/j.apnum.2020.04.01533.
Odibat
,
Z.
, 2021
, “
A Universal Predictor-Corrector Algorithm for Numerical Simulation of Generalized Fractional Differential Equations
,” Nonlinear Dyn.
,
105
(3
), pp. 2363
–2374
.10.1007/s11071-021-06670-234.
Singh
,
J.
,
Ganbari
,
B.
,
Kumar
,
D.
, and
Baleanu
,
D.
, 2021
, “
Analysis of Fractional Model of Guava for Biological Pest Control With Memory Effect
,” J. Adv. Res.
,
32
, pp. 99
–108
.10.1016/j.jare.2020.12.00435.
Zheng
,
X.
, 2022
, “
Numerical Approximation for a Nonlinear Variable-Order Fractional Differential Equation Via a Collocation Method
,” Math. Comput. Simul.
,
195
, pp. 107
–118
.10.1016/j.matcom.2022.01.00536.
Heydari
,
M. H.
,
Razzaghi
,
M.
, and
Baleanu
,
D.
, 2023
, “
A Numerical Method Based on the Piecewise Jacobi Functions for Distributed-Order Fractional Schrödinger Equation
,” Commun. Nonlinear Sci. Numer. Simul.
,
116
, p. 106873
.10.1016/j.cnsns.2022.10687337.
Liao
,
S.
, 2003
, Beyond Perturbation: Introduction to the Homotopy Analysis Method
,
Chapman and Hall/CRC Press
,
Florida
.38.
Liao
,
S.
, 2011
, Homotopy Analysis Method in Nonlinear Differential Equations
,
Springer-Verlag
,
Berlin
.39.
Sun
,
Q.
, 2005
, “
Solving the klein-gordon Equation by Means of the Homotopy Analysis Method
,” Appl. Math. Comput.
,
169
(1
), pp. 355
–365
.10.1016/j.amc.2004.09.05640.
Wu
,
W.
, and
Liao
,
S.
, 2005
, “
Solving Solitary Waves With discontinuity by Means of the Homotopy Analysis Method
,” Chaos Soliton Fractals
,
26
(1
), pp. 177
–185
.10.1016/j.chaos.2004.12.01641.
Abbasbandy
,
S.
, 2007
, “
The Application of Homotopy Analysis Method to Solve a Generalized Hirota-Satsuma Coupled KdV Equation
,” Phys. Lett. A
,
361
(6
), pp. 478
–483
.10.1016/j.physleta.2006.09.10542.
Abbasbandy
,
S.
, 2007
, “
Homotopy Analysis Method for Heat Radiation Equations
,” Int. Commun. Heat Mass Transfer
,
34
(3
), pp. 380
–387
.10.1016/j.icheatmasstransfer.2006.12.00143.
Abbasbandy
,
S.
, 2008
, “
Approximate Solution for the Nonlinear model of diffusion and Reaction in Porous Catalysts by Means of the Homotopy Analysis Method
,” Chem. Eng. J.
,
136
(2–3
), pp. 144
–150
.10.1016/j.cej.2007.03.02244.
Song
,
L.
, and
Zhang
,
H.
, 2007
, “
Application of Homotopy Analysis Method to Fractional KdV-Burgers-Kuramoto Equation
,” Phys. Lett. A
,
367
(1–2
), pp. 88
–94
.10.1016/j.physleta.2007.02.08345.
Odibat
,
Z.
,
Momani
,
S.
, and
Xu
,
H.
, 2010
, “
A Reliable Algorithm of Homotopy Analysis Method for Solving Nonlinear Fractional Differential Equations
,” Appl. Math. Modell.
,
34
(3
), pp. 593
–600
.10.1016/j.apm.2009.06.02546.
Ganjiani
,
M.
, 2010
, “
Solution of Nonlinear Fractional Differential Equations Using Homotopy Analysis Method
,” Appl. Math. Model.
,
34
(6
), pp. 1634
–1641
.10.1016/j.apm.2009.09.01147.
Ghazanfari
,
B.
, and
Veisi
,
F.
, 2011
, “
Homotopy Analysis Method for the Fractional Nonlinear Equations
,” J. King Saud Univ. Sci.
,
23
(4
), pp. 389
–393
.10.1016/j.jksus.2010.07.01948.
Hashim
,
I.
,
Abdulaziz
,
O.
, and
Momani
,
S.
, 2009
, “
Homotopy Analysis Method for Fractional IVPs
,” Commun. Nonlinear Sci. Numer. Simul.
,
14
(3
), pp. 674
–684
.10.1016/j.cnsns.2007.09.01449.
Miller
,
K. S.
, and
Ross
,
B.
, 1993
, An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.50.
Caputo
,
M.
, 1967
, “
Linear Models of Dissipation Whose Q is Almost Frequency independent-II
,” Geophys. J. Int.
,
13
(5
), pp. 529
–539
.10.1111/j.1365-246X.1967.tb02303.x51.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
, 2017
, The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science
,
Wiley
,
New York
.52.
Osler
,
T. J.
, 1970
, “
The Fractional Derivative of a Composite Function
,” SIAM J. Math. Anal.
,
1
(2
), pp. 288
–293
.10.1137/050102653.
Fafa
,
W.
,
Odibat
,
Z.
, and
Shawagfeh
,
N.
, 2022
, “
Analytical Approximate Solutions for Differential Equations With Generalized Caputo-Type Fractional Derivatives
,” Int. J. Appl. Comput. Math.
,
8
(5
), p. 231
.10.1007/s40819-022-01448-554.
Almeida
,
R.
, 2017
, “
A Caputo Fractional Derivative of a Function With Respect to Another Function
,” Commun. Nonlinear Sci. Numer. Simul.
,
44
, pp. 460
–481
.10.1016/j.cnsns.2016.09.00655.
Odibat
,
Z.
, 2019
, “
On the Optimal Selection of the Linear Operator and the Initial Approximation in the Application of the Homotopy Analysis Method to Nonlinear Fractional Differential Equations
,” Appl. Numer. Math.
,
137
, pp. 203
–212
.10.1016/j.apnum.2018.11.003Copyright © 2023 by ASME
You do not currently have access to this content.
