In this paper, we apply two decomposition methods, the Adomian decomposition method (ADM) and a well-established iterative method, to solve time-fractional Klein–Gordon type equation. We compare these methods and discuss the convergence of them. The obtained results reveal that these methods are very accurate and effective.

References

References
1.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus
,
Academic Press
,
New York
.
2.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations, Mathematics in Science and Engineering
, Vol.
198
,
Academic Press
,
San Diego, CA
.
3.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.
4.
Baleanu
,
D.
,
Diethelm
,
K.
,
Scalas
,
E.
, and
Trujillo
,
J.
,
2012
,
Fractional Calculus Models and Numerical Methods
(Series on Complexity, Nonlinearity and Chaos),
World Scientific Publishing Company
, Singapore.
5.
Adomian
,
G.
, and
Rach
,
R.
,
1996
, “
Modified Adomian Polynomials
,”
Math. Comput. Modell.
,
24
(
11
), pp.
39
46
.
6.
Adomian
,
G.
,
1988
, “
A Review of the Decomposition Method in Applied Mathematics
,”
J. Math. Anal. Appl.
,
135
(
2
), pp.
501
544
.
7.
Daftardar-Gejji
,
V.
, and
Jafari
,
H.
,
2005
, “
Adomian Decomposition: A Tool for Solving a System of Fractional Differential Equations
,”
J. Math. Anal. Appl.
,
301
(
2
), pp.
508
518
.
8.
Wazwaz
,
A.-M.
,
2006
, “
The Modified Decomposition Method for Analytic Treatment of Differential Equations
,”
Appl. Math. Comput.
,
173
(
1
), pp.
165
176
.
9.
Deeba
,
E.
, and
Khuri
,
S.
,
1996
, “
A Decomposition Method for Solving the Nonlinear Klein-Gordon Equation
,”
J. Comput. Phys.
,
124
(
2
), pp.
442
448
.
10.
Yusufoğlu
,
E.
,
2008
, “
The Variational Iteration Method for Studying the Klein-Gordon Equation
,”
Appl. Math. Lett.
,
21
(
7
), pp.
669
674
.
11.
Jafari
,
H.
,
Tajadodi
,
H.
, and
Baleanu
,
D.
,
2013
, “
A Modified Variational Iteration Method for Solving Fractional Riccati Differential Equation by Adomian Polynomials
,”
Fractional Calculus Appl. Anal.
,
16
(
1
), pp.
109
122
.
12.
Kurulay
,
M.
,
2012
, “
Solving the Fractional Nonlinear Klein–Gordon Equation by Means of the Homotopy Analysis Method
,”
Adv. Differ. Equations
,
2012
(
1
), p.
187
.
13.
Golbabai
,
A.
, and
Sayevand
,
K.
,
2011
, “
Analytical Modelling of Fractional Advection-Dispersion Equation Defined in a Bounded Space Domain
,”
Math. Comput. Modell.
,
53
(
9–10
), pp.
1708
1718
.
14.
Jafari
,
H.
,
Tajadodi
,
H.
, and
Baleanu
,
D.
,
2014
, “
Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Fractional Evolution Equations
,”
ASME J. Comput. Nonlinear Dyn.
,
9
(
2
), p.
021019
.
15.
Firoozjaee
,
M.
,
Yousefi
,
S.
,
Jafari
,
H.
, and
Baleanu
,
D.
,
2015
, “
On a Numerical Approach to Solve Multi-Order Fractional Differential Equations With Initial/Boundary Conditions
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
6
), p.
061025
.
16.
Golbabai
,
A.
, and
Nikan
,
O.
,
2015
, “
Application of the RBF Meshless Approach for Solving Fractional Order Differential Equations
,”
J. Comput. Complex Appl.
,
1
(
2
), pp.
64
78
.
17.
Evirgen
,
F.
, and
Ozdemir
,
N.
,
2011
, “
Multistage Adomian Decomposition Method for Solving NLP Problems Over a Nonlinear Fractional Dynamical System
,”
ASME J. Comput. Nonlinear Dyn.
,
6
(
2
), p.
021003
.
18.
Bekir
,
A.
,
2009
, “
New Exact Travelling Wave Solutions of Some Complex Nonlinear Equations
,”
Commun. Nonlinear Sci. Numer. Simul.
,
14
(
4
), pp.
1069
1077
.
19.
Kumar
,
S.
,
Yildirim
,
A.
,
Khan
,
Y.
,
Jafari
,
H.
,
Sayevand
,
K.
, and
Wei
,
L.
,
2012
, “
Analytical Solution of Fractional Black-Scholes European Option Pricing Equation by Using Laplace Transform
,”
J. Fractional Calculus Appl.
,
2
(
8
), pp.
1
9
.
20.
Sejdic
,
E.
,
Djurovic
,
I.
, and
Stankovic
,
L.
,
2011
, “
Fractional Fourier Transform as a Signal Processing Tool: An Overview of Recent Developments
,”
Signal Process.
,
91
(
6
), pp.
1351
1369
.
21.
Miurs
,
M.
,
1978
,
Backlund Transformation
,
Springer
,
Berlin
.
22.
Daftardar-Gejji
,
V.
, and
Jafari
,
H.
,
2006
, “
An Iterative Method for Solving Nonlinear Functional Equations
,”
J. Math. Anal. Appl.
,
316
(
2
), pp.
753
763
.
23.
Golmankhaneh
,
A. K.
,
Khatuni
,
T.
,
Porghoveh
,
N. A.
, and
Baleanu
,
D.
,
2012
, “
Comparison of Iterative Methods by Solving Nonlinear Sturm–Liouville, Burgers and Navier–Stokes Equations
,”
Cent. Eur. J. Phys.
,
10
(
4
), pp.
966
976
.
24.
Golmankhaneh
,
A. K.
,
Golmankhaneh
,
A. K.
, and
Baleanu
,
D.
,
2011
, “
On Nonlinear Fractional Klein–Gordon Equation
,”
Signal Process.
,
91
(
3
), pp.
446
451
.
25.
Hariharan
,
G.
,
2013
, “
Wavelet Method for a Class of Fractional Klein–Gordon Equations
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
2
), p.
021008
.
26.
Duan
,
J.
,
2015
, “
The Adomian Polynomials and the New Modified Decomposition Method for BVPs of Nonlinear ODEs
,”
Math. Comput.
,
4
(
1
), pp.
1
6
.
27.
Duan
,
J.-S.
,
2011
, “
New Recurrence Algorithms for the Nonclassic Adomian Polynomials
,”
Comput. Math. Appl.
,
62
(
8
), pp.
2961
2977
.
28.
Duan
,
J.-S.
,
2011
, “
New Ideas for Decomposing Nonlinearities in Differential Equations
,”
Appl. Math. Comput.
,
218
(
5
), pp.
1774
1784
.
29.
Zeng
,
Y.
,
2016
, “
Approximate Solutions of Three Integral Equations by the New Adomian Decomposition Method
,”
J. Comput. Complex Appl.
,
2
(
1
), pp.
38
43
.
30.
Abdelrazec
,
A.
, and
Pelinovsky
,
D.
,
2011
, “
Convergence of the Adomian Decomposition Method for Initial-Value Problems
,”
Numer. Methods Partial Differ. Equations
,
27
(
4
), pp.
749
766
.
31.
Bhalekar
,
S.
, and
Daftardar-Gejji
,
V.
,
2011
, “
Convergence of the New Iterative Method
,”
Int. J. Differ. Equations
,
2011
, p.
989065
.
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