This paper presents the formulation of the time-fractional Camassa–Holm equation using the Euler–Lagrange variational technique in the Riemann–Liouville derivative sense and derives an approximate solitary wave solution. Our results witness that He's variational iteration method was a very efficient and powerful technique in finding the solution of the proposed equation.

References

References
1.
He
,
J.
,
1997
, “
A New Approach to Nonlinear Partial Differential Equations
,”
Commun. Nonlinear Sci. Numer. Simul.
,
2
(
4
), pp.
230
235
.10.1016/S1007-5704(97)90007-1
2.
He
,
J.
,
1998
, “
Approximate Analytical Solution for Seepage Flow With Fractional Derivatives in Porous Media
,”
Comput. Meth. Appl. Mech. Eng.
,
167
, pp.
57
68
.10.1016/S0045-7825(98)00108-X
3.
Molliq
,
R. Y.
,
Noorani
,
M. S. M.
, and
Hashim
,
I.
,
2009
, “
Variational Iteration Method for Fractional Heat- and Wave-Like Equations
,”
Nonlinear Anal. Real World Appl.
,
10
, pp.
1854
1869
.10.1016/j.nonrwa.2008.02.026
4.
Momani
,
S.
,
Odibat
,
Z.
, and
Alawnah
,
A.
,
2008
, “
Variational Iteration Method for Solving the Space-and Time-Fractional KdV Equation
,”
Numer. Meth. Part. Differ. Equ.
,
24
(
1
), pp.
261
271
.10.1002/num.20247
5.
Camassa
,
R.
,
Holm
,
D.
, and
Hyman
,
J.
,
1994
, “
A New Integrable Shallow Water Equation
,”
Adv. Appl. Mech.
,
31
, pp.
1
33
.10.1016/S0065-2156(08)70254-0
6.
Johnson
,
R. S.
,
2002
, “
Camassa–-Holm, Korteweg-deVries and Related Models for Water Waves
,”
J. Fluid Mech.
,
455
, pp.
63
82
.10.1017/S0022112001007224
7.
Fokas
,
A.
, and
Fuchssteiner
,
B.
,
1981
, “
Symplectic Structures, Their Bäcklund Transformation and Hereditary Symmetries
,”
Phys. D
,
4
, pp.
47
66
.10.1016/0167-2789(81)90004-X
8.
Lenells
,
J.
,
2005
, “
Conservation Laws of the Camassa–-Holm Equation
,”
J. Phys. A
,
38
, pp.
869
880
.10.1088/0305-4470/38/4/007
9.
Camassa
,
R.
, and
Holm
,
D.
,
1993
, “
An Integrable Shallow Water Equation With Peaked Solutions
,”
Phys. Rev. Lett.
,
71
, pp.
1661
1664
.10.1103/PhysRevLett.71.1661
10.
El-Wakil
,
S.
,
Abulwafa
,
E.
,
Zahran
,
M.
, and
Mahmoud
,
A.
,
2011
, “
Time-Fractional KdV Equation: Formulation and Solution Using Variational Methods
,”
Nonlinear Dyn.
,
65
, pp.
55
63
.10.1007/s11071-010-9873-5
11.
Gorenflo
,
R.
,
Mainardi
,
F.
,
Scalas
,
E.
, and
Raberto
,
M.
,
2001
, “
Fractional Calculus and Continuous-Time Finance III. The Diffusion Limit
,”
Mathematical Finance (Konstanz, 2000), (Trends in Mathematics)
,
Birkhäuser
,
Basel
, pp.
171
180
.
12.
Hilfer
,
R.
,
2000
,
Applications of Fractional Calculus in Physics
,
World Scientific
,
Singapore
.
13.
Lundstrom
,
B.
,
Higgs
,
M.
,
Spain
,
W.
, and
Fairhall
A.
,
2008
, “
Fractional Differentiation by Neocortical Pyramidal Neurons
,”
Nature Neurosci.
,
11
, pp.
1335
1342
.10.1038/nn.2212
14.
Malinowska
,
A. B.
, and
Torres
,
D. F. M.
,
2012
,
Introduction to the Fractional Calculus of Variations
,
Imperial College Press
,
London
.
15.
Metzler
,
R.
, and
Klafter
,
J.
,
2004
, “
The Restaurant at the End of the Random Walk: Recent Developments in the Description of Anomalous Transport by Fractional Dynamics
,”
J. Phys. A
,
37
, pp.
R161
R208
.10.1088/0305-4470/37/31/R01
16.
Rossikhin
,
Y. A.
, and
Shitikova
,
M. V.
,
1997
, “
Application of Fractional Derivatives to the Analysis of Damped Vibrations of Viscoelastic Single Mass Systems
,
Acta Mech.
,
120
, pp.
109
125
.10.1007/BF01174319
17.
Sabatelli
,
L.
,
Keating
,
S.
,
Dudley
,
J.
, and
Richmond
,
P.
,
2002
, “
Waiting Time Distributions in Financial Markets
,”
Eur. Phys. J. B.
,
27
, pp.
273
275
.
18.
Schumer
,
R.
,
Benson
,
D. A.
,
Meerschaert
,
M. M.
, and
Baeumer
,
B.
,
2003
, “
Multiscaling Fractional Advection-Dispersion Equations and Their Solutions
,”
Water Resour. Res.
,
39
, pp.
1022
1032
.
19.
Schumer
,
R.
,
Benson
,
D. A.
,
Meerschaert
,
M. M.
, and
Wheatcraft
,
S.W.
,
2001
, “
Eulerian Derivation of the Fractional Advection-Dispersion Equation
,”
J. Contamin. Hydrol.
,
48
, pp.
69
88
.10.1016/S0169-7722(00)00170-4
20.
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
.10.1007/s11071-008-9447-y
21.
Cresson
,
J.
,
2007
, “
Fractional Embedding of Differential Operators and Lagrangian Systems
,”
J. Math. Phys.
,
48
, p.
033504
.10.1063/1.2483292
22.
Herzallah
,
M. A. E.
, and
Baleanu
,
D.
,
2012
, “
Fractional Euler–Lagrange Equations Revisited
,”
Nonlinear Dyn.
,
69
(
3
), pp.
977
982
.10.1007/s11071-011-0319-5
23.
Malinowska
,
A. B.
,
2012
, “
A Formulation of the Fractional Noether-Type Theorem for Multidimensional Lagrangians
,”
Appl. Math. Lett.
,
25
, pp.
1941
1946
.10.1016/j.aml.2012.03.006
24.
Riewe
,
F.
,
1996
, “
Nonconservative Lagrangian and Hamiltonian Mechanics
,”
Phys. Rev. E
,
53
(
2
), pp.
1890
1899
.10.1103/PhysRevE.53.1890
25.
Riewe
,
F.
,
1997
, “
Mechanics With Fractional Derivatives
,”
Phys. Rev. E
,
55
(
3
), pp.
3581
3592
.10.1103/PhysRevE.55.3581
26.
Wu
,
G. C.
, and
Baleanu
,
D.
,
2013
, “
Variational Iteration Method for the Burgers' Flow With Fractional Derivatives—New Lagrange Multipliers
,”
Appl. Math. Model.
,
37
(
9
), pp.
6183
6190
.10.1016/j.apm.2012.12.018
27.
Odzijewicz
,
T.
,
Malinowska
,
A. B.
, and
Torres
,
D. F M.
,
2012
, “
Generalized Fractional Calculus With Applications to the Calculus of Variations
,”
Comput. Math. Appl.
,
64
(
10
), pp.
3351
3366
.10.1016/j.camwa.2012.01.073
28.
Odzijewicz
,
T.
,
Malinowska
,
A. B.
, and
Torres
,
D. F. M.
,
2012
, “
Fractional Variational Calculus With Classical and Combined Caputo Derivatives
,”
Nonlinear Anal. TMA
,
75
(
3
), pp.
1507
1515
.10.1016/j.na.2011.01.010
29.
Agrawal
,
O. P.
,
2002
, “
Formulation of Euler–Lagrange Equations for Fractional Variational Problems
,”
J. Math. Anal. Appl.
,
272
(
1
), pp.
368
379
.10.1016/S0022-247X(02)00180-4
30.
Agrawal
,
O. P.
,
2004
, “
A General Formulation and Solution Scheme for Fractional Optimal Control Problems
,”
Nonlinear Dyn.
,
38
(
4
), pp.
323
337
.10.1007/s11071-004-3764-6
31.
Agrawal
,
O. P.
,
2007
, “
Fractional Variational Calculus in Terms of Riesz Fractional Derivatives
,”
J. Phys. A Math. Theor.
,
40
, pp.
62
87
.
32.
Baleanu
,
D.
, and
Muslih
,
S. I.
,
2005
, “
Lagrangian Formulation of Classical Fields Within Riemann–Liouville Fractional Derivatives
,
Phys Scr.
,
72
, pp.
119
123
.10.1238/Physica.Regular.072a00119
33.
Inokuti
,
M.
,
Sekine
,
H.
, and
Mura
,
T.
,
1978
, “
General Use of the Lagrange Multiplier in Non-Linear Mathematical Physics
,”
Variational Method in the Mechanics of Solids
,
S.
Nemat-Nasser
(ed.),
Pergamon Press
,
Oxford
.
34.
Saha Ray
,
S.
, and
Bera
,
R.
,
2005
, “
An Approximate Solution of a Nonlinear Fractional Differential Equation by Adomian Decomposition Method
,”
Appl. Math. Comput.
,
167
, pp.
561
571
.10.1016/j.amc.2004.07.020
35.
Cang
,
J.
,
Tan
,
Y.
,
Xu
,
H.
, and
Liao
S.
,
2009
, “
Series Solutions of Nonlinear Fractional Riccati Differential Equations
,”
Chaos Solitons Fractals
,
40
(
1
), pp.
1
9
.10.1016/j.chaos.2007.04.018
36.
Liao
,
S.
,
1992
, “
The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems
,” Ph.D. thesis, Shanghai Jiao Tong University, Shanghai, China.
37.
Sweilam
,
N. H.
,
Khader
,
M. M.
, and
Al-Bar
,
R. F.
,
2007
, “
Numerical Studies for a Multi-Order Fractional Differential Equation
,”
Phys. Lett. A
,
371
, pp.
26
33
.10.1016/j.physleta.2007.06.016
38.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
, Vol.
204
(North-Holland Mathematics Studies),
Elsevier
,
Amsterdam, The Netherlands
.
39.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego
.
40.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
,
1993
.
Fractional Integrals and Derivatives: Theory and Applications
,
Gordon and Breach
,
New York
.
41.
He
,
J.
,
1997
, “
Semi-Inverse Method of Establishing Generalized Variational Principles for Fluid Mechanics With Emphasis on Turbo-Machinery Aerodynamics
,”
Int. J. Turbo Jet-Engines
,
14
(
1
), pp.
23
28
.
42.
He
J.
,
2004
,
Variational Principles for Some Nonlinear Partial Differential Equations With Variable Coefficients
,”
Chaos Solitons Fractals
,
19
, pp.
847
851
.10.1016/S0960-0779(03)00265-0
You do not currently have access to this content.