In this paper, we present new ideas for the implementation of homotopy asymptotic method (HAM) to solve systems of nonlinear fractional differential equations (FDEs). An effective computational algorithm, which is based on Taylor series approximations of the nonlinear equations, is introduced to accelerate the convergence of series solutions. The proposed algorithm suggests a new optimal construction of the homotopy that reduces the computational complexity and improves the performance of the method. Some numerical examples are tested to validate and illustrate the efficiency of the proposed algorithm. The obtained results demonstrate the improvement of the accuracy by the new algorithm.

References

References
1.
Liao
,
S.
,
1997
, “
Homotopy Analysis Method: A New Analytical Technique for Nonlinear Problems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
2
(
2
), pp.
95
100
.
2.
Liao
,
S.
,
2003
,
Beyond Perturbation: Introduction to the Homotopy Analysis Method
,
Chapman & Hall/CRC Press
,
Boca Raton, FL
.
3.
Liao
,
S.
,
2004
, “
On the Homotopy Analysis Method for Nonlinear Problems
,”
Appl. Math. Comput.
,
147
, pp.
499
513
.
4.
Liao
,
S.
, and
Tan
,
Y.
,
2007
, “
A General Approach to Obtain Series Solutions of Nonlinear Differential Equations
,”
Stud. Appl. Math.
,
119
(
4
), pp.
297
354
.
5.
Liao
,
S.
,
2009
, “
Notes on the Homotopy Analysis Method: Some Definitions and Theorems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
14
(
4
), pp.
983
997
.
6.
Wang
,
Z.
,
Zou
,
L.
, and
Zhang
,
H.
,
2007
, “
Applying Homotopy Analysis Method for Solving Differential-Difference Equation
,”
Phys. Lett. A
,
369
(
1–2
), pp.
77
84
.
7.
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
.
8.
Molabahrami
,
A.
, and
Khani
,
F.
,
2009
, “
The Homotopy Analysis Method to Solve the Burgers-Huxley Equation
,”
Nonlinear Anal.: Real World Appl.
,
10
(
2
), pp.
589
600
.
9.
Rashidi
,
M.
, and
Dinarvand
,
S.
,
2009
, “
Purely Analytic Approximate Solutions for Steady Three-Dimensional Problem of Condensation Film on Inclined Rotating Disk by Homotopy Analysis Method
,”
Nonlinear Anal.: Real World Appl.
,
10
(
4
), pp.
2346
2356
.
10.
Chen
,
Y.
, and
Liu
,
J.
,
2009
, “
A Study of Homotopy Analysis Method for Limit Cycle of Van Der Pol Equation
,”
Comm. Nonlinear Sci. Numer. Simul.
,
14
(
5
), pp.
1816
1821
.
11.
Odibat
,
Z.
,
2010
, “
A Study on the Convergence of Homotopy Analysis Method
,”
Appl. Math. Comput.
,
217
(
2
), pp.
782
789
.
12.
Martin
,
O.
,
2013
, “
On the Homotopy Analysis Method for Solving a Particle Transport Equation
,”
Appl. Math. Model.
,
37
(
6
), pp.
3959
3967
.
13.
Shivanian
,
E.
, and
Abbasbandy
,
S.
,
2014
, “
Predictor Homotopy Analysis Method: Two Points Second Order Boundary Value Problems
,”
Nonlinear Anal.: Real World Appl.
,
15
, pp.
89
99
.
14.
Kumar
,
S.
,
Singh
,
J.
,
Kumar
,
D.
, and
Kapoor
,
S.
,
2014
, “
New Homotopy Analysis Transform Algorithm to Solve Volterra Integral Equation
,”
Ain Shams Eng. J.
,
5
(
1
), pp.
243
246
.
15.
Massa
,
F.
,
Lallemand
,
B.
, and
Tison
,
T.
,
2015
, “
Multi-Level Homotopy Perturbation and Projection Techniques for the Reanalysis of Quadratic Eigenvalue Problems: The Application of Stability Analysis
,”
Mech. Syst. Signal Process.
,
52
, pp.
88
104
.
16.
Sardanys
,
J.
,
Rodrigues
,
C.
,
Janurio
,
C.
,
Martins
,
N.
,
Gil-Gmez
,
G.
, and
Duarte
,
J.
,
2015
, “
Activation of Effector Immune Cells Promotes Tumor Stochastic Extinction: A Homotopy Analysis Approach
,”
Appl. Math. Comput.
,
252
(
1
), pp.
484
495
.
17.
Hetmaniok
,
E.
,
Słota
,
D.
,
Wituła
,
R.
, and
Zielonka
,
A.
,
2015
, “
Solution of the One-Phase Inverse Stefan Problem by Using the Homotopy Analysis Method
,”
Appl. Math. Model.
,
39
(
22
), pp.
6793
6805
.
18.
Odibat
,
Z.
, and
Bataineh
,
A.
,
2015
, “
An Adaptation of HAM for Reliable Treatment of Strongly Nonlinear Problems: Construction of Homotopy Polynomials
,”
Math. Methods Appl. Sci.
,
38
(
5
), pp.
991
1000
.
19.
Liu
,
Q. X.
,
Liu
,
J. K.
, and
Chen
,
Y. M.
,
2016
, “
Asymptotic Limit Cycle of Fractional Van Der Pol Oscillator by Homotopy Analysis Method and Memory-Free Principle
,”
Appl. Math. Model.
,
40
(
4
), pp.
3211
3220
.
20.
Yang
,
Z.
, and
Liao
,
S.
,
2017
, “
A HAM-Based Wavelet Approach for Nonlinear Partial Differential Equations: Two Dimensional Bratu Problem as an Application
,”
Comm. Nonlinear Sci. Numer. Simul.
,
53
, pp.
249
262
.
21.
Gorder
,
R.
, and
Vajravelu
,
K.
,
2009
, “
On the Selection of Auxiliary Functions, Operators, and Convergence Control Parameters in the Application of the Homotopy Analysis Method to Nonlinear Differential Equations: A General Approach
,”
Comm. Nonlinear Sci. Numer. Simul.
,
14
(
12
), pp.
4078
4089
.
22.
Liao
,
S.
,
2010
, “
An Optimal Homotopy-Analysis Approach for Strongly Nonlinear Differential Equations
,”
Comm. Nonlinear Sci. Numer. Simul.
,
15
(
8
), pp.
2003
2016
.
23.
Pandey
,
R.
,
Singh
,
O.
,
Baranwal
,
V.
, and
Tripathi
,
M.
,
2012
, “
An Analytic Solution for the Space-Time Fractional Advection-Dispersion Equation Using the Optimal Homotopy Asymptotic Method
,”
Comput. Phys. Commun.
,
183
(
10
), pp.
2089
2106
.
24.
Golbabai
,
A.
,
Fardi
,
M.
, and
Sayevand
,
K.
,
2013
, “
Application of the Optimal Homotopy Asymptotic Method for Solving a Strongly Nonlinear Oscillatory System
,”
Math. Comput. Model.
,
58
(
11–12
), pp.
1837
1843
.
25.
Marinca
,
V.
, and
Herisanu
,
N.
,
2014
, “
The Optimal Homotopy Asymptotic Method for Solving Blasius Equation
,”
Appl. Math. Comput.
,
231
, pp.
134
139
.
26.
Mallory
,
K.
, and
Gorder
,
R.
,
2014
, “
Optimal Homotopy Analysis and Control of Error for Solutions to the Non-Local Whitham Equation
,”
Numer. Algorithms
,
66
(
4
), pp.
843
863
.
27.
Sarwar
,
S.
,
Alkhalaf
,
S.
,
Iqbal
,
S.
, and
Zahid
,
M. A.
,
2015
, “
A Note on Optimal Homotopy Asymptotic Method for the Solutions of Fractional Order Heat- and Wave-Like Partial Differential Equations
,”
Comput. Math. Appl.
,
70
(
5
), pp.
942
953
.
28.
Hamarsheh
,
M.
,
Ismail
,
A.
, and
Odibat
,
Z.
,
2015
, “
Optimal Homotopy Asymptotic Method for Solving Fractional Relaxation-Oscillation Equation
,”
J. Interpolat. Approx. Sci. Comput.
,
2015
(
2
), pp.
98
111
.
29.
Jia
,
W.
,
He
,
X.
, and
Guo
,
L.
,
2017
, “
The Optimal Homotopy Analysis Method for Solving Linear Optimal Control Problems
,”
Appl. Math. Model.
,
45
, pp.
865
880
.
30.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus
,
Academic Press
,
New York
.
31.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.
32.
Gorenflo
,
R.
, and
Mainardi
,
F.
,
1997
, “
Fractional Calculus: Integral and Differential Equations of Fractional Order
,”
Fractals and Fractional Calculus in Continuum Mechanics
,
A.
Carpinteri
and
F.
Mainardi
, eds.,
Springer Verlag
,
New York
, pp.
277
290
.
33.
Hilfer
,
R.
,
2000
,
Applications of Fractional Calculus in Physics
,
World Scientific Publishing Company
,
Singapore
.
34.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
Amsterdam, The Netherlands
.
35.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
,
2016
,
The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science
,
Wiley
,
New York
.
36.
Zaky
,
M. A.
,
Doha
,
E. H.
, and
Tenreiro Machado
,
J. A.
,
2018
, “
A Spectral Numerical Method for Solving Distributed-Order Fractional Initial Value Problems
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(
10
), p.
101007
.
37.
David
,
S. A.
,
Quintino
,
D. D.
,
Inacio
,
C. M. C.
, and
Machado
,
J. A. T.
,
2018
, “
Fractional Dynamic Behavior in Ethanol Prices Series
,”
J. Comput. Appl. Math.
,
339
, pp.
85
93
.
38.
Jajarmi
,
A.
, and
Baleanu
,
D.
,
2018
, “
A New Fractional Analysis on the Interaction of HIV With CD4+ T-Cells
,”
Chaos Solitons Fractals
,
113
, pp.
221
229
.
39.
Baleanu
,
D.
,
Jajarmi
,
A.
,
Bonyah
,
E.
, and
Hajipour
,
M.
,
2018
, “
New Aspects of Poor Nutrition in the Life Cycle Within the Fractional Calculus
,”
Adv. Differ. Equations
,
2018
, p.
230
.
40.
Jajarmi
,
A.
, and
Baleanu
,
D.
,
2018
, “
Suboptimal Control of Fractional-Order Dynamic Systems With Delay Argument
,”
J. Vib. Control
,
24
(
12
), pp.
2430
2446
.
41.
Momani
,
S.
, and
Al-Khaled
,
K.
,
2005
, “
Numerical Solutions for Systems of Fractional Differential Equations by the Decomposition Method
,”
Appl. Math. Comput.
,
162
(
3
), pp.
1351
1365
.
42.
Jafari
,
H.
, and
Daftardar-Gejji
,
V.
,
2006
, “
Revised Adomian Decomposition Method for Solving Systems of Ordinary and Fractional Differential Equations
,”
Appl. Math. Comput.
,
181
(
1
), pp.
598
608
.
43.
Momani
,
S.
, and
Odibat
,
Z.
,
2007
, “
Numerical Approach to Differential Equations of Fractional Order
,”
J. Comput. Appl. Math.
,
207
(
1
), pp.
96
110
.
44.
Jafari
,
H.
, and
Seifi
,
S.
,
2009
, “
Solving a System of Nonlinear Fractional Partial Differential Equations Using Homotopy Analysis Method
,”
Comm. Nonlinear Sci. Numer. Simul.
,
14
(
5
), pp.
1962
1969
.
45.
Odibat
,
Z. M.
,
Corson
,
N.
,
Aziz-Alaoui
,
M. A.
, and
Bertelle
,
C.
,
2010
, “
Synchronization of Chaotic Fractional-Order Systems Via Linear Control
,”
Int. J. Bifurcation Chaos
,
20
(
1
), pp.
81
97
.
46.
Odibat
,
Z.
,
2010
, “
Analytic Study on Linear Systems of Fractional Differential Equations
,”
Comput. Math. Appl.
,
59
(
3
), pp.
1171
1183
.
47.
Erturk
,
V.
,
Odibat
,
Z.
, and
Momani
,
S.
,
2011
, “
An Approximate Solution of a Fractional Order Differential Equation Model of Human T-Cell Lymphotropic Virus I (HTLV-I) Infection of CD4+ T-Cells
,”
Comput. Math. Appl.
,
62
(
3
), pp.
996
1002
.
48.
Odibat
,
Z.
,
2012
, “
A Note on Phase Synchronization in Coupled Chaotic Fractional Order Systems
,”
Nonlinear Anal.: Real World Appl.
,
13
(
2
), pp.
779
789
.
49.
Bhrawy
,
A. H.
, and
Zaky
,
M. A.
,
2016
, “
Shifted Fractional-Order Jacobi Orthogonal Functions: Application to a System of Fractional Differential Equations
,”
Appl. Math. Model.
,
40
(
2
), pp.
832
845
.
50.
Ouannas
,
A.
,
Odibat
,
Z.
, and
Hayat
,
T.
,
2017
, “
Fractional Analysis of Co-Existence of Some Types of Chaos Synchronization
,”
Chaos Solitons Fractals
,
105
, pp.
215
223
.
51.
Ouannas
,
A.
,
Grassi
,
G.
,
Ziar
,
T.
, and
Odibat
,
Z.
,
2017
, “
On a Function Projective Synchronization Scheme for Non-Identical Fractional-Order Chaotic (Hyperchaotic) Systems With Different Dimensions and Orders
,”
Optik
,
136
, pp.
513
523
.
52.
Du
,
W.
,
Miao
,
Q.
,
Tong
,
L.
, and
Tang
,
Y.
,
2017
, “
Identification of Fractional-Order Systems With Unknown Initial Values and Structure
,”
Phys. Let. A
,
381
(
23
), pp.
1943
1949
.
53.
Shukla
,
M. K.
, and
Sharma
,
B. B.
,
2017
, “
Backstepping Based Stabilization and Synchronization of a Class of Fractional Order Chaotic Systems
,”
Chaos Solitons Fractals
,
102
, pp.
274
284
.
54.
Odibat
,
Z.
,
Corson
,
N.
,
Aziz-Alaoui
,
M. A.
, and
Alsaedi
,
A.
,
2017
, “
Chaos in Fractional Order Cubic Chua System and Synchronization
,”
Int. J. Bifurcation Chaos
,
27
(
10
), p.
1750161
.
55.
Wang
,
J.
,
Xu
,
T. Z.
,
Wei
,
Y. Q.
, and
Xie
,
J. Q.
,
2018
, “
Numerical Simulation for Coupled Systems of Nonlinear Fractional Order Integro-Differential Equations Via Wavelets Method
,”
Appl. Math. Comput.
,
324
, pp.
36
50
.
56.
Lenka
,
B. K.
, and
Banerjee
,
S.
,
2018
, “
Sufficient Conditions for Asymptotic Stability and Stabilization of Autonomous Fractional Order Systems
,”
Comm. Nonlinear Sci. Numer. Simul.
,
56
, pp.
365
379
.
57.
Cang
,
J.
,
Tan
,
Y.
,
Xu
,
H.
, and
Liao
,
S.
,
2009
, “
Series Solutions of Non-Linear Riccati Differential Equations With Fractional Order
,”
Chaos Solitons Fractals
,
40
(
1
), pp.
1
9
.
58.
Odibat
,
Z.
,
Momani
,
S.
, and
Xu
,
H.
,
2010
, “
A Reliable Algorithm of Homotopy Analysis Method for Solving Nonlinear Fractional Differential Equations
,”
Appl. Math. Model.
,
34
(
3
), pp.
593
600
.
59.
Zurigat
,
M.
,
Momani
,
S.
,
Odibat
,
Z.
, and
Alawneh
,
A.
,
2010
, “
The Homotopy Analysis Method for Handling Systems of Fractional Differential Equations
,”
Appl. Math. Model.
,
34
(
1
), pp.
24
35
.
60.
Odibat
,
Z.
,
2011
, “
On Legendre Polynomial Approximation With the VIM or HAM for Numerical Treatment of Nonlinear Fractional Differential Equations
,”
J. Comput. Appl. Math.
,
235
(
9
), pp.
2956
2968
.
61.
Kumar
,
S.
,
Kumar
,
A.
, and
Odibat
,
Z.
,
2017
, “
A Nonlinear Fractional Model to Describe the Population Dynamics of Two Interacting Species
,”
Math. Methods Appl. Sci.
,
40
(
11
), pp.
4134
4148
.
62.
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
.
You do not currently have access to this content.