In this paper, an analytical technique is proposed to determine the exact solution of fractional order modified Fornberg–Whitham equation. Since exact solution of fractional Fornberg–Whitham equation is unknown, first integral method has been applied to determine exact solutions. The solitary wave solution of fractional modified Fornberg–Whitham equation has been attained by using first integral method. The approximate solutions of fractional modified Fornberg–Whitham equation, obtained by optimal homotopy asymptotic method (OHAM), are compared with the exact solutions obtained by the first integral method. The obtained results are presented in tables to demonstrate the efficiency of these proposed methods. The proposed schemes are quite simple, effective, and expedient for obtaining solution of fractional modified Fornberg–Whitham equation.

References

References
1.
Tavazoei
,
M. S.
, and
Haeri
,
M.
,
2009
, “
Describing Function Based Methods for Predicting Chaos in a Class of Fractional Order Differential Equations
,”
Nonlinear Dyn.
,
57
(
3
), pp.
363
373
.
2.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
New York
.
3.
Bagley
,
R. L.
, and
Torvik
,
P. J.
,
1984
, “
On the Appearance of the Fractional Derivative in the Behavior of Real Materials
,”
ASME J. Appl. Mech.
,
51
(
2
), pp.
294
298
.
4.
Caputo
,
M.
,
1969
,
Elasticita` e Dissipazione
,
Zanichelli
,
Bologna, Italy
.
5.
Suarez
,
L. E.
, and
Shokooh
,
A.
,
1997
, “
An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives
,”
ASME J. Appl. Mech.
,
64
(
3
), pp.
629
635
.
6.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.
7.
Oldham
,
K. B.
, and
Spainer
,
J.
,
1974
,
The Fractional Calculus
,
Academic Press
,
New York
.
8.
Caputo
,
M.
,
1967
, “
Linear Model of Dissipation Whose Q is Almost Frequency Independent, Part II
,”
J. R. Astron. Soc.
,
13
(
5
), pp.
529
539
.
9.
Das
,
S.
,
2011
,
Functional Fractional Calculus
,
Springer-Verlag
,
Berlin
.
10.
Herrmann
,
R.
,
2011
,
Fractional Calculus an Introduction to Physicists
,
World Scientific
,
Singapore
.
11.
Gupta
,
A. K.
, and
Saha Ray
,
S.
,
2014
, “
On the Solutions of Fractional Burgers–Fisher and Generalized Fisher's Equations Using Two Reliable Methods
,”
Int. J. Math. Math. Sci.
,
2014
, p.
682910
.
12.
Saha Ray
,
S.
,
2012
, “
On Haar Wavelet Operational Matrix of General Order and Its Application for the Numerical Solution of Fractional Bagley Torvik Equation
,”
Appl. Math. Comput.
,
218
(
9
), pp.
5239
5248
.
13.
Gupta
,
A. K.
, and
Saha Ray
,
S.
,
2014
, “
Comparison Between Homotopy Perturbation Method and Optimal Homotopy Asymptotic Method for the Soliton Solution of Boussinesq–Burgers Equation
,”
Comput. Fluids
,
103
, pp.
34
41
.
14.
Saha Ray
,
S.
, and
Gupta
,
A. K.
,
2014
, “
A Two-Dimensional Haar Wavelet Approach for the Numerical Simulations of Time and Space Fractional Fokker–Planck Equations in Modelling of Anomalous Diffusion Systems
,”
J. Math. Chem.
,
52
(
8
), pp.
2277
2293
.
15.
Jiang
,
H.
,
Liu
,
F.
,
Turner
,
I.
, and
Burrage
,
I.
,
2012
, “
Analytical Solutions for the Multi-Term Time-Space Caputo–Riesz Fractional Advection-Diffusion Equations on a Finite Domain
,”
J. Math. Anal. Appl.
,
389
(
2
), pp.
1117
1127
.
16.
Yıldırım
,
A.
, and
Kocak
,
H.
,
2009
, “
Homotopy Perturbation Method for Solving the Space-Time Fractional Advection-Dispersion Equation
,”
Adv. Water Resour.
,
32
(
12
), pp.
1711
1716
.
17.
Saha Ray
,
S.
,
Chaudhuri
,
K. S.
, and
Bera
,
R. K.
,
2006
, “
Analytical Approximate Solution of Nonlinear Dynamic System Containing Fractional Derivative by Modified Decomposition Method
,”
Appl. Math. Comput.
,
182
(
1
), pp.
544
552
.
18.
Gorenflo
,
R.
,
Mainardi
,
F.
, and
Vivoli
,
A.
,
2007
, “
Continuous-Time Random Walk and Parametric Subordination in Fractional Diffusion
,”
Chaos, Solitons Fractals
,
34
(
1
), pp.
87
103
.
19.
He
,
J. H.
,
1998
, “
Nonlinear Oscillation With Fractional Derivative and Its Applications
,”
International Conference on Vibrating Engineering’98
, Dalian, China, pp.
288
291
.
20.
He
,
J. H.
,
1999
, “
Some Applications of Nonlinear Fractional Differential Equations and Their Approximations
,”
Bull. Sci. Technol.
,
15
(
2
), pp.
86
90
.
21.
Bagley
,
R. L.
, and
Torvik
,
P. J.
,
1983
, “
A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,”
J. Rheol.
,
27
(
3
), pp.
201
210
.
22.
Bagley
,
R. L.
, and
Torvik
,
P. J.
,
1983
, “
Fractional Calculus—A Different Approach to the Analysis of Viscoelastically Damped Structures
,”
AIAA J.
,
21
(
5
), pp.
741
748
.
23.
Bagley
,
R. L.
, and
Torvik
,
P. J.
,
1985
, “
Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures
,”
AIAA J.
,
23
(
6
), pp.
918
925
.
24.
Sun
,
H. H.
,
Onaral
,
B.
, and
Tsao
,
Y.
,
1984
, “
Application of Positive Reality Principle to Metal Electrode Linear Polarization Phenomena
,”
IEEE Trans. Biomed. Eng.
,
31
(
10
), pp.
664
674
.
25.
Sun
,
H. H.
,
Abdelvahab
,
A. A.
, and
Onaral
,
B.
,
1984
, “
Linear Approximation of Transfer Function With a Pole of Fractional Order
,”
IEEE Trans. Autom. Control
,
29
(
5
), pp.
441
444
.
26.
Mandelbrot
,
B.
,
1967
, “
Some Noises With 1/f Spectrum, a Bridge Between Direct Current and White Noise
,”
IEEE Trans. Inf. Theory
,
13
(
2
), pp.
289
298
.
27.
Hartley
,
T. T.
,
1995
, “
Chaos in a Fractional Order Chua System
,”
IEEE Trans. Circuits Syst.
,
42
(
8
), pp.
485
490
.
28.
Inc
,
M.
,
2008
, “
Approximate Analytical Solution of the Space-and Time-Fractional Burgers Equations With Initial Conditions by Variational Iteration Method
,”
J. Math. Anal. Appl.
,
345
(
1
), pp.
476
484
.
29.
Saha Ray
,
S.
, and
Bera
,
R. K.
,
2005
, “
An Approximate Solution of a Nonlinear Fractional Differential Equation by Adomian Decomposition Method
,”
Appl. Math. Comput.
,
167
(
1
), pp.
561
571
.
30.
Saha Ray
,
S.
, and
Bera
,
R. K.
,
2005
, “
Analytical Solution of the Bagley Torvik Equation by Adomian Decomposition Method
,”
Appl. Math. Comput.
,
168
(
1
), pp.
398
410
.
31.
Odibat
,
Z. M.
,
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
.
32.
Gupta
,
A. K.
, and
Saha Ray
,
S.
,
2015
, “
The Comparison of Two Reliable Methods for Accurate Solution of Time-Fractional Kaup–Kupershmidt Equation Arising in Capillary Gravity Waves
,”
Math. Methods Appl. Sci.
,
39
(
3
), pp.
583
592
.
33.
Saha Ray
,
S.
,
2013
, “
Soliton Solutions for Time Fractional Coupled Modified KdV Equations Using New Coupled Fractional Reduced Differential Transform Method
,”
J. Math. Chem.
,
51
(
8
), pp.
2214
2229
.
34.
Wazwaz
,
A. M.
,
2009
,
Partial Differential Equations and Solitary Waves Theory
,
Springer, Higher Education Press
,
Berlin
.
35.
Shang
,
N.
, and
Zheng
,
B.
,
2013
, “
Exact Solutions for Three Fractional Partial Differential Equations by the G/G Method
,”
Int. J. Appl. Math.
,
43
(
3
), pp.
1
6
.
36.
Saha Ray
,
S.
, and
Gupta
,
A. K.
,
2016
, “
Numerical Solution of Fractional Partial Differential Equation of Parabolic Type With Dirichlet Boundary Conditions Using Two-Dimensional Legendre Wavelets Method
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
1
), p.
011012
.
37.
Wang
,
Y.
, and
Fan
,
Q.
,
2012
, “
The Second Kind Chebyshev Wavelet Method for Solving Fractional Differential Equations
,”
Appl. Math. Comput.
,
218
(
17
), pp.
8592
8601
.
38.
Whitham
,
G. B.
,
1967
, “
Variational Methods and Applications to Water Wave
,”
Proc. R. Soc. London, Ser. A
,
299
(
1456
), pp.
6
25
.
39.
Fornberg
,
B.
, and
Whitham
,
G. B.
,
1978
, “
A Numerical and Theoretical Study of Certain Nonlinear Wave Phenomena
,”
Philos. Trans. R. Soc. London Ser. A
,
289
(
1361
), pp.
373
404
.
40.
He
,
B.
,
Meng
,
Q.
, and
Li
,
S.
,
2010
, “
Explicit Peakon and Solitary Wave Solutions for the Modified Fornberg–Whitham Equation
,”
Appl. Math. Comput.
,
217
(
5
), pp.
1976
1982
.
41.
Abidi
,
F.
, and
Omrani
,
K.
,
2010
, “
The Homotopy Analysis Method for Solving the Fornberg–Whitham Equation and Comparison With Adomian's Decomposition Method
,”
Comput. Math. Appl.
,
59
(
8
), pp.
2743
2750
.
42.
Zhou
,
J.
, and
Tian
,
L.
,
2008
, “
A Type of Bounded Traveling Wave Solutions for the Fornberg–Whitham Equation
,”
J. Math. Anal. Appl.
,
346
(
1
), pp.
255
261
.
43.
Gupta
,
P. K.
, and
Singh
,
M.
,
2011
, “
Homotopy Perturbation Method for Fractional Fornberg–Whitham Equation
,”
Comput. Math. Appl.
,
61
(
2
), pp.
250
254
.
44.
Sakar
,
M. G.
,
Erdogan
,
F.
, and
Yıldırım
,
A.
,
2012
, “
Variational Iteration Method for the Time-Fractional Fornberg–Whitham Equation
,”
Comput. Math. Appl.
,
63
(
9
), pp.
1382
1388
.
45.
Chen
,
A.
,
Li
,
J.
,
Deng
,
X.
, and
Huang
,
W.
,
2009
, “
Travelling Wave Solutions of the Fornberg–Whitham Equation
,”
Appl. Math. Comput.
,
215
(
8
), pp.
3068
3075
.
46.
Hesam
,
S.
,
Nazemi
,
A.
, and
Haghbin
,
A.
,
2012
, “
Reduced Differential Transform Method for Solving the Fornberg–Whitham Type Equation
,”
Int. J. Nonlinear Sci.
,
13
(
2
), pp.
158
162
.
47.
Lu
,
J.
,
2011
, “
An Analytical Approach to the Fornberg–Whitham Type Equations by Using the Variational Iteration Method
,”
Comput. Math. Appl.
,
61
(
8
), pp.
2010
2013
.
48.
Saha Ray
,
S.
, and
Gupta
,
A. K.
,
2015
, “
A Numerical Investigation of Time-Fractional Modified Fornberg–Whitham Equation for Analysing the Behaviour of Water Waves
,”
Appl. Math. Comput.
,
266
, pp.
135
148
.
49.
Raslan
,
K. R.
,
2008
, “
The First Integral Method for Solving Some Important Nonlinear Partial Differential Equations
,”
Nonlinear Dyn.
,
53
(
4
), pp.
281
286
.
50.
Abbasbandy
,
S.
, and
Shirzadi
,
A.
,
2010
, “
The First Integral Method for Modified Benjamin–Bona–Mahony Equation
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(
7
), pp.
1759
1764
.
51.
Lu
,
B.
,
2012
, “
The First Integral Method for Some Time Fractional Differential Equations
,”
J. Math. Anal. Appl.
,
395
(
2
), pp.
684
693
.
52.
Jafari
,
H.
,
Soltani
,
R.
,
Khalique
,
C. M.
, and
Baleanu
,
D.
,
2013
, “
Exact Solutions of Two Nonlinear Partial Differential Equations by Using the First Integral Method
,”
Boundary Value Probl.
,
2013
(
1
), p.
117
.
53.
Bekir
,
A.
,
Güner
,
Ö.
, and
Ünsal
,
Ö.
,
2015
, “
The First Integral Method for Exact Solutions of Nonlinear Fractional Differential Equations
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
2
), p.
021020
.
54.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
,
1993
,
Fractional Integrals and Derivatives: Theory and Applications
,
Taylor and Francis
,
London
.
55.
Yang
,
X. J.
,
2012
,
Advanced Local Fractional Calculus and Its Applications
,
World Science Publisher
,
New York
.
56.
Yang
,
X. J.
,
2012
, “
A Short Note on Local Fractional Calculus of Function of One Variable
,”
J. Appl. Libr. Inf. Sci.
,
1
(
1
), pp.
1
13
.
57.
Yang
,
X. J.
,
2012
, “
The Zero-Mass Renormalization Group Differential Equations and Limit Cycles in Non-Smooth Initial Value Problems
,”
Prespacetime J.
,
3
(
9
), pp.
913
923
.
58.
Hu
,
M. S.
,
Baleanu
,
D.
, and
Yang
,
X. J.
,
2013
, “
One-Phase Problems for Discontinuous Heat Transfer in Fractal Media
,”
Math. Probl. Eng.
,
2013
, p.
358473
.
59.
Agarwal
,
O. P.
,
2004
, “
A General Formulation and Solution Scheme for Fractional Optimal Control Problems
,”
Nonlinear Dyn.
,
38
, pp.
323
337
.
60.
Su
,
W. H.
,
Yang
,
X. J.
,
Jafari
,
H.
, and
Baleanu
,
D.
,
2013
, “
Fractional Complex Transform Method for Wave Equations on Cantor Sets Within Local Fractional Differential Operator
,”
Adv. Differ. Equations
,
2013
(
97
), pp.
1
8
.
61.
Yang
,
X. J.
,
Baleanu
,
D.
, and
Srivastava
,
H. M.
,
2015
,
Local Fractional Integral Transforms and Their Applications
,
Academic Press (Elsevier)
,
London
.
62.
Ding
,
T. R.
, and
Li
,
C. Z.
,
1996
,
Ordinary Differential Equations
,
Peking University Press
,
Peking, China
.
63.
Bourbaki
,
N.
,
1972
,
Commutative Algebra
,
Addison-Wesley
,
Paris, France
.
64.
Feng
,
Z.
, and
Wang
,
X.
,
2001
, “
Explicit Exact Solitary Wave Solutions for the Kundu Equation and the Derivative Schrödinger Equation
,”
Phys. Scr.
,
64
(
1
), pp.
7
14
.
65.
Feng
,
Z.
, and
Roger
,
K.
,
2007
, “
Traveling Waves to a Burgers–Korteweg–de Vries-Type Equation With Higher-Order Nonlinearities
,”
J. Math. Anal. Appl.
,
328
(
2
), pp.
1435
1450
.
66.
Marinca
,
V.
, and
Herisanu
,
N.
,
2011
,
Nonlinear Dynamical Systems in Engineering
,
Springer-Verlag
,
Berlin
.
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