In the present paper, we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics, viz., Riesz time-fractional Camassa–Holm (CH) equation by modified homotopy analysis method (MHAM). As a result, new types of solutions are obtained. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. The main aim of this paper is to employ a new approach, which enables us successful and efficient derivation of the analytical solutions for the Riesz time-fractional CH equation.

References

References
1.
Zhang
,
Y.
,
2013
, “
Time-Fractional Camassa–Holm Equation: Formulation and Solution Using Variational Methods
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
4
), p.
041020-1-8
.10.1115/1.4024970
2.
Boyd
,
J. P.
,
1997
, “
Peakons and Coshoidal Waves: Travelling Wave Solutions of the Camassa–Holm Equation
,”
Appl. Math. Comput.
,
81
(
2–3
), pp.
173
187
.10.1016/0096-3003(95)00326-6
3.
Camassa
,
R.
, and
Holm
,
D. D.
,
1993
, “
An Integrable Shallow Water Equation With Peaked Solitons
,”
Phys. Rev. Lett.
,
71
(
11
), pp.
1661
1664
.10.1103/PhysRevLett.71.1661
4.
Qian
,
T.
, and
Tang
,
M.
,
2001
, “
Peakons and Periodic Cusp Waves in a Generalized Camassa–Holm Equation
,”
Chaos, Solitons Fractals
,
12
(
7
), pp.
1347
1360
.10.1016/S0960-0779(00)00117-X
5.
Liu
,
Z.-R.
,
Wang
,
R.-Q.
, and
Jing
,
Z.-J.
,
2004
, “
Peaked Wave Solutions of Camassa–Holm Equation
,”
Chaos, Solitons Fractals
,
19
(
1
), pp.
77
92
.10.1016/S0960-0779(03)00082-1
6.
Liu
,
Z.
, and
Qian
,
T.
,
2001
, “
Peakons and Their Bifurcation in a Generalized Camassa–Holm Equation
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
,
11
(
3
), pp.
781
792
.10.1142/S0218127401002420
7.
Tian
,
L.
, and
Song
,
X.
,
2004
, “
New Peaked Solitary Wave Solutions of the Generalized Camassa–Holm Equation
,”
Chaos, Solitons Fractals
,
19
(
3
), pp.
621
637
.10.1016/S0960-0779(03)00192-9
8.
Kalisch
,
H.
,
2004
, “
Stability of Solitary Waves for a Nonlinearly Dispersive Equation
,”
Discrete Contin. Dyn. Syst. Ser. A
,
10
(
3
), pp.
709
717
.10.3934/dcds.2004.10.709
9.
Liu
,
Z.
, and
Ouyang
,
Z.
,
2007
, “
A Note on Solitary Waves for Modified Forms of Camassa–Holm and Degasperis–Procesi Equations
,”
Phys. Lett. A
,
366
(
4–5
), pp.
377
381
.10.1016/j.physleta.2007.01.074
10.
Camassa
,
R.
,
Holm
,
D. D.
, and
Hyman
,
J. M.
,
1994
, “
A New Integrable Shallow Water Equation
,”
Adv. Appl. Mech.
,
31
, pp.
1
33
.10.1016/S0065-2156(08)70254-0
11.
Cooper
,
F.
, and
Shepard
,
H.
,
1994
, “
Solitons in the Camassa–Holm Shallow Water Equation
,”
Phys. Lett. A
,
194
(
4
), pp.
246
250
.10.1016/0375-9601(94)91246-7
12.
He
,
B.
,
Rui
,
W.
,
Chen
,
C.
, and
Li
,
S.
,
2008
, “
Exact Travelling Wave Solutions of a Generalized Camassa–Holm Equation Using the Integral Bifurcation Method
,”
Appl. Math. Comput.
,
206
(
1
), pp.
141
149
.10.1016/j.amc.2008.08.043
13.
Wazwaz
,
A.
,
2006
, “
Solitary Wave Solutions for Modified Forms of Degasperis–Procesi and Camassa–Holm Equations
,”
Phys. Lett. A
,
352
(
6
), pp.
500
504
.10.1016/j.physleta.2005.12.036
14.
Wazwaz
,
A.
,
2007
, “
New Solitary Wave Solutions to the Modified Forms of Degasperis–Procesi and Camassa–Holm Equations
,”
Appl. Math. Comput.
,
186
(
1
), pp.
130
141
.10.1016/j.amc.2006.07.092
15.
Tian
,
L.
, and
Yin
,
J.
,
2004
, “
New Compact on Solutions and Solitary Wave Solutions of Fully Nonlinear Generalized Camassa–Holm Equations
,”
Chaos, Solitons and Fractals
,
20
, pp.
289
299
.10.1142/S0218127409024050
16.
Wang
,
Q.
, and
Tang
,
M.
,
2008
, “
New Exact Solutions for Two Nonlinear Equations
,”
Phys. Lett. A
,
372
(
17
), pp.
2995
3000
.10.1016/j.physleta.2008.01.012
17.
Yomba
,
E.
,
2008
, “
The Sub-ODE Method for Finding Exact Travelling Wave Solutions of Generalized Nonlinear Camassa–Holm, and Generalized Nonlinear Schrödinger Equations
,”
Phys. Lett. A
,
372
(
3
), pp.
215
222
.10.1016/j.physleta.2007.03.008
18.
Yomba
,
E.
,
2008
, “
A Generalized Auxiliary Equation Method and Its Application to Nonlinear Klein–Gordon and Generalized Nonlinear Camassa–Holm Equations
,”
Phys. Lett. A
,
372
(
7
), pp.
1048
1060
.10.1016/j.physleta.2007.09.003
19.
Liu
,
Z.
, and
Pan
,
J.
,
2009
, “
Coexistence of Multifarious Explicit Nonlinear Wave Solutions for Modified Forms of Camassa–Holm and Degasperis–Procesi Equations
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
,
19
(
7
), pp.
2267
2282
.10.1142/S0218127409024050
20.
Liu
,
Z.
, and
Liang
,
Y.
,
2011
, “
The Explicit Nonlinear Wave Solutions and Their Bifurcations of the Generalized Camassa–Holm Equation
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
,
21
(
11
), pp.
3119
3136
.10.1142/S0218127411030556
21.
Parkes
,
E. J.
, and
Vakhnenko
,
V. O.
,
2005
, “
Explicit Solutions of the Camassa–Holm Equation
,”
Chaos, Solitons Fractals
,
26
(
5
), pp.
1309
1316
.10.1016/j.chaos.2005.03.011
22.
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
.10.1115/1.4025770
23.
Saha Ray
,
S.
,
2008
, “
A New Approach for the Application of Adomian Decomposition Method for the Solution of Fractional Space Diffusion Equation With Insulated Ends
,”
Appl. Math. Comput.
,
202
(
2
), pp.
544
549
.10.1016/j.amc.2008.02.043
24.
Liao
,
S.
,
2003
,
Beyond Perturbation: Introduction to the Homotopy Analysis Method
,
Chapman and Hall/CRC Press
,
Boca Raton, FL
.
25.
Jafarian
,
A.
,
Ghaderi
,
P.
,
Golmankhaneh
,
A. K.
, and
Baleanu
,
D.
,
2014
, “
Analytical Treatment of System of Abel Integral Equations by Homotopy Analysis Method
,”
Rom. Rep. Phys.
,
66
(
3
), pp.
603
611
.
26.
Jafarian
,
A.
,
Ghaderi
,
P.
,
Golmankhaneh
,
A. K.
, and
Baleanu
,
D.
,
2014
, “
Analytical Approximate Solutions of the Zakharov–Kuznetsov Equations
,”
Rom. Rep. Phys.
,
66
(
2
), pp.
296
306
.
27.
Shen
,
S.
,
Liu
,
F.
,
Anh
,
V.
, and
Turner
,
I.
,
2008
, “
The Fundamental Solution and Numerical Solution of the Riesz Fractional Advection–Dispersion Equation
,”
IMA J. Appl. Math.
,
73
(
6
), pp.
850
872
.10.1093/imamat/hxn033
28.
Herrmann
,
R.
,
2011
,
Fractional Calculus: An Introduction for Physicists
,
World Scientific
,
Singapore
.
29.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
,
2002
,
Fractional Integrals and Derivatives: Theory and Applications
,
Taylor and Francis
,
London
.
30.
Podlubny
,
I.
,
1999
,
Fractional Differential Equation
,
Academic
,
New York
.
31.
Adomian
,
G.
,
1994
,
Solving Frontier Problems of Physics: The Decomposition Method
,
Kluwer Academic Publishers
,
Boston
.
You do not currently have access to this content.