Travelling Wave Solutions to Riesz TimeFractional Camassa–Holm Equation in Modelling for ShallowWater Waves
 By : Admin
 Category : Free Essays
Traveling Wave Solutions to Riesz TimeFractional Camassa–Holm Equation in Modeling for ShallowWater Waves
Abstraction
There's a specialist from your university waiting to help you with that essay.
Tell us what you need to have done now!
order now
In the present paper, we construct the analytical exact solutions of a nonlinear development equation in mathematical natural philosophies, viz. Riesz timefractional Camassa–Holm equation by modified homotopy analysis method. As a consequence, new types of solutions are obtained. Then, we analyse the consequences by numerical simulations, which demonstrate the simpleness and effectivity of the present method.
Cardinal words:Riesz timefractional Camassa–Holm equation ; Riesz fractional derived function ; Riemann–Liouville fractional builtin ; RiemannLiouville fractional derived function ; Modified Homotopy Analysis Method ; Adomian multinomial.
 Introduction
Let us see the Camassa–Holm equation [ 1 ] with Rieszfractional clip derivative
( 1.1 )
whereandis arbitrary invariable.is the Riesz fractional derived function.
Recently the CamassaHolm ( CH ) equation has been of great research involvement due to its shallow H2O wave nature and multiple solitary wave nature as pointed by Boyd [ 2 ] . Previously, Camassa and Holm [ 3 ] derived a wholly integrable diffusing shallow H2O equation, i.e. CamassaHolm equation and obtained the lone moving ridge solution of the signifier. The lone moving ridge obtained by Camassa and Holm is called peakons wave due to the discontinuity of the first derived function at the moving ridge extremum and have been discussed in [ 29 ] . Camassa et Al. [ 10 ] presented numerical solutions of the clip dependent signifier and discussed the CH equation as a Hamiltonian system. For general, Cooper and Shepard [ 11 ] derived the variational estimates to the lone moving ridge solution of CH equation. He et Al. [ 12 ] obtained some exact going wave solutions by utilizing the builtin bifurcation method. The exact going wave solutions for CamassaHolm equation are studied by Wazwaz [ 13, 14 ] and others [ 15, 16 ] . Yomba [ 17, 18 ] applied the subODE method and the generalised subsidiary equation method for obtaining the exact solution of CamassaHolm equation. The coexistence of manysided solution studied by Liu and Pan [ 19 ] . The expressed nonlinear moving ridge solutions given by Liu et Al. [ 20 ] and Parkes et Al. [ 21 ] .
In the past few old ages, the many research workers investigated the analytical solutions of fractional development equations [ 22 ] . In this paper, the derived analytical solutions are based on homotopy analysis method [ 2325 ] with some alteration. In this present analysis, we employ a new attack such as using the homotopy analysis method along with Adomian’s multinomials. It enables successful derivation of the analytical solutions for the Riesz timefractional Camassa–Holm equation. By taking the 3rd order modified homotopy analysis method ( MHAM ) solution 2dimensional and 3 dimensional graphs have been plotted. Then we compare the MHAM consequences with the solution obtained by VIM [ 1 ] .
This paper is organized as follows. In Section 2 some basic definitions of Riesz fractional derived function and builtin are given. In Sections 3 and 4, the solution process and consequences of the modified homotopy analysis method ( MHAM ) are given severally. We present the corresponding numerical simulations of the proposed method with mistake analysis in Sections 5 and 6 severally. The decisions are drawn in Section 7.
 Preliminaries
In this portion we give some definition for the fractional concretion, which is further used throughout the staying subdivision of the paper.
Definition 2.1A existent multivariable map,is said to be in the infinite,with regard toif there exists a existent figure, such that, where. Obviously,if.
Definition 2.2The lefthand side of the Riemann–Liouville fractional integral of a map,is defined by
,,
Definition 2.3The RiemannLiouville fractional derivative [ 2629 ] of the orderof a map,are defined as
,
,
Definition 2.4The Riesz fractional builtin [ 2629 ] of the orderof a map,are defined as
Where,
and,are the left and righthand side RiemannLiouville fractional integral operator severally.
Definition 2.5The Riesz fractional derived function of the orderof a map,are defined as [ 2629 ]
where,
and,are the left and righthand side RiemannLiouville fractional derived function operator severally.
Lemma 2.6Letandbe such that,and, so we have following index regulation:
Remark 2.7The Riesz fractional operatorof the ordercan be express as Riesz fractional integral operatorby following individuality, define in [ 2629 ]
,
 Basic thought of modified homotopy analysis method ( MHAM )
In this paper, we apply the HAM [ 2325 ] to the discussed job. To demo the basic thought, allow us see the undermentioned Riesz Fractional differential equation
( 3.1 )
whereis a Riesz Fractional differential equation and Riesz Fractional derived function operator defined in,tenandTdenote independent variables andis an unknown map. For simpleness, we ignore all boundary or initial conditions, which can be treated in the similar manner.
Then by usingin the combining weight. ( 3.1 ) and by lemma 2.7, we cut down the Riesz differential equation as
. ( 3.2 )
whereNitrogenis a nonlinear derived function operator.
By agencies of the HAM, one first constructs the nothing^{Thursday}order distortion equation of combining weight. ( 3.2 ) as
( 3.3 )
whereLiteris an subsidiary linear operator,is an unknown map,is an initial conjecture of,is an subsidiary parametric quantity andis the implanting parametric quantity. For the interest of convenience, the look in nonlinear operator signifier has been modified in HAM. In this modified homotopy analysis method, nonlinear term appeared in look for nonlinear operator signifier has been expanded utilizing Adomian type of multinomials as[ 30 ] .
Obviously, whenP= 0 andP= 1, we have
,( 3.4 )
severally. Therefore asPadditions from 0 to 1, the solutionvaries from the initial conjecture to thesolution. Expandingin Taylor series with regard to the embedding parametric quantityP, outputs
( 3.5 )
where
The convergence of the series ( 3.5 ) depends upon the subsidiary parametric quantity. If it is convergent atP= 1, we have
which must be one of the solutions of the original nonlinear equation.
Distinguishing the zerothorder distortion combining weight. ( 3.3 )mtimes with regard toPand so putingP= 0 and eventually spliting them bym! , we obtain the followersm^{Thursday}order distortion equation
( 3.6 )
where
and
( 3.7 )
It should be noted thatforis governed by the additive equation ( 3.6 ) which can be solved by symbolic computational package.
 Execution of the MHAM method for approximative solution ofRiesz timefractional CamassaHolm equation
In this subdivision, we foremost see the application of MHAM for the solution of Riesz timefractional CamassaHolm equation of combining weight. ( 1.1 ) with given initial status [ 1 ]
( 4.1 )
By usingand by lemma 2.7, the combining weight. ( 1.1 ) can be written in the undermentioned signifier as
( 4.2 )
Expandingin Taylor series with regard toP, we have
( 4.3 )
where
To obtain the approximative solution of the combining weight. ( 1.1 ) , we choose the linear operators
( 4.4 )
From combining weight. ( 3.2 ) , we define nonlinear term as
( 4.5 )
Using combining weight. ( 3.3 ) in the above subdivision, we construct the alleged nothing^{Thursday}order distortion equations
( 4.6 )
Obviously, whenP=0 andP=1, combining weight. ( 4.3 ) outputs
;
Therefore, as the embedding parametric quantityPadditions from 0 to 1,varies from the initial conjecture to the exact solution.
If the subsidiary linear operator, the initial conjecture, and the subsidiary parametric quantitiesare so decently chosen, the above series in combining weight. ( 4.3 ) converges atP= 1 and we obtain
( 4.7 )
Harmonizing to combining weight. ( 3.6 ) , we have them^{Thursday}order distortion equation
,( 4.8 )
where
( 4.9 )
Now, the solutions of them^{Thursday}order distortion equations ( 4.8 ) forbecomes
( 4.10 )
Note:In position of the righthand side Riemann–Liouville fractional derived function is interpreted as a future province of the procedure in natural philosophies. For this ground, the rightderivative is normally neglected in applications, when the present province of the procedure does non depend on the consequences of the hereafter development, and so the rightderivative is used equal to zero in the undermentioned computations.
By seting the initial conditions in combining weight. ( 4.1 ) into combining weight. ( 4.10 ) and work outing them, we now in turn obtain
and so on.
By the homotopy 3rd order series, the solution of combining weight. ( 1.1 ) is approximated as
( 4.11 )
5.Thegraph and numerical simulations for MHAM method
As pointed out by Liao [ 23 ] in general, by agencies of the allegedcurve, it is consecutive frontward to take a proper value ofwhich ensures that the solution series is convergent.
Fig. 1.Thecurve for partial derived functions ofatfor the MHAM solution.
To look into the influence ofon the solution series, we plot the so calledcurve of partial derived functions ofatobtained from the MHAM solutions as shown in Fig. 1. In this manner, it is found that our series converge when.
In this present numerical experiment, combining weight. ( 4.11 ) obtained by MHAM has been used to pull the graphs as shown in Fig. 2 and Fig. 3 for different values of. The numerical solutions of combining weight. ( 1.1 ) has been shown in Fig.2 and Fig.3 with the aid of the homotopy series solutions of, whenand.
Case 1:For
( a ) ( B )
Fig. 2.( a ) The MHAM method going wave solution for, ( B ) matching 2D solution forwhen.
Case 2:For
( a ) ( B )
Fig. 3.( a ) The MHAM method going wave solution for, ( B ) matching 2D solution forwhen.
In Figs. 2 and 3, the MHAM approximative solutions graph 3dimensional and 2dimensional are plotted for the intervals?1?ten?1 and?1?T?1 utilizing different fractional orders that is forandseverally.
 Comparison of present MHAM solution with respect to VIM solution
In this present analysis, we examine the comparing for the solutions of MHAM with Variational Iteration Method ( VIM ) [ 1 ] . Here we tabulate the solutions for combining weight. ( 1.1 ) utilizing different values ofandT.
Table 1.Comparison of the solutions between 3rd order MHAM and VIM solutions for different values ofandTwhen.
Comparison of the solutions between 3rd order MHAM and VIM solutions 

0.1 
0.3 
0.5 
0.7 
0.9 

MHAM 
Energy 
MHAM 
Energy 
MHAM 
Energy 
MHAM 
Energy 
MHAM 
Energy 

0.1 
0.00901 
0090186 
0.009028 
0.00903 
0.009036 
0.009044 
0.00904 
0.009053 
0.009050 
0.009061 
0.3 
0.00695 
0.006957 
0.006958 
0.00695 
0.006959 
0.006960 
0.00696 
0.006961 
0.006961 
0.006963 
0.5 
0.00477 
0.004770 
0.004762 
0.00475 
0.004754 
0.004746 
0.00474 
0.004738 
0.004742 
0.004731 
0.7 
0.00237 
0.002367 
0.002346 
0.00233 
0.002325 
0.002307 
0.00230 
0.002286 
0.002295 
0.002267 
0.9 
0.000338 
0.0003522 
0.0003922 
0.000419 
0.0004299 
0.0004666 
0.000460 
0.0005063 
0.0004877 
0.000541 
Table 2.Theandmistakes for 3rd order MHAM solutions with respect to VIM solutions for different values ofwhen.
ten 
Comparison of MHAM Solution with respect to VIM solution 

0.1 
8.02236E6 
9.57534E6 
0.3 
1.06901E6 
7.74385E7 
0.5 
7.40666E6 
8.84726E6 
0.7 
1.94153E5 
7.44980E6 
0.9 
3.79915E5 
5.35954E5 
In order to compare the solutions obtained by present method with respect to those obtained by VIM method [ 1 ] ,andmistake norms have been besides presented in Table2. It may be observed that there is a good understanding between the present MHAM solution and VIM solution.
 Decision
In this paper, we have proposed a new analytical technique MHAM method to obtain the approximative solution of the Riesz timefractional Camassa–Holm equation. The Riesz timefractional Camassa–Holm equation has been first clip solved by MHAM method in order to warrant pertinence of the present method. MHAM provides us with a convenient manner to command the convergence of approximative series solution and solves the job without any demand for discretization of the variables. To command the convergence of the solution, we can take the proper values of, in this paper we choose. Besides here we presented a comparing between MHAM solutions and VIM solution. The proposed MHAM method is really simple and efficient for work outing Riesz timefractional Camassa–Holm equation.
Mentions
[ 1 ] Zhang Y. , 2013, “TimeFractional Camassa–Holm Equation: Formulation and Solution Using Variational Methods”Journal of Computational and Nonlinear Dynamics,8, pp. 0410207.
[ 2 ] Boyd J. P. , 1997, “Peakons and coshoidal moving ridges: going wave solutions of the Camassa–Holm equation,Appl. Math. Comput. ,81 ( 2–3 ) , pp. 173–87.
[ 3 ] Camassa R. , Holm D. D. , 1993, “An integrable shallow H2O equation with ailing solitons” ,Physical Review Letters, 71 ( 11 ) , pp. 1661–1664.
[ 4 ] Qian T. , Tang M. , 2001, “Peakons and periodic cusp moving ridges in a generalised CamassaHolm equation, ”Chaos, Solitons andFractals, 12 ( 7 ) , pp. 1347–1360.
[ 5 ] Liu Z.R. , Wang R.Q. , Jing Z.J. , 2004, “Peaked wave solutions of CamassaHolm equation, ”Chaos, Solitons and Fractals, 19 ( 1 ) , pp. 77–92.
[ 6 ] Liu Z. , Qian T. , 2001, “Peakons and their bifurcation in a generalised CamassaHolm equation” ,International Journal ofBifurcation and Chaos in Applied Sciences and Engineering, 11 ( 3 ) , pp. 781–792.
[ 7 ] Tian L. , Song X. , 2004, “New peaked lone wave solutions of the generalised CamassaHolm equation” ,Chaos, Solitons andFractals, 19 ( 3 ) , pp. 621–637.
[ 8 ] Kalisch H. , 2004, “Stability of lone moving ridges for a nonlinearly diffusing equation” ,Discrete and Continuous Dynamical Systems. Series A, 10 ( 3 ) , pp. 709–717.
[ 9 ] Liu Z. , Ouyang Z. , 2007 “A note on lone moving ridges for modified signifiers of CamassaHolm and DegasperisProcesi equations” ,Physicss Letters A, 366 ( 45 ) , pp. 377–381.
[ 10 ] Camassa R. , Holm D. D. , Hyman J. M. , 1994, “A new integrable Shallow H2O equation” ,Progresss in Applied Mechanics 31, Academic, pp. 134.
[ 11 ] Cooper F. , Shepard H. , 1994, “Solitons in the Camassa–Holm shallow H2O equation” ,Phys. Lett. Angstrom, 194 ( 4 ) , pp. 246–250.
[ 12 ] He B. , Rui W. , Chen C. , Li S. , 2008, “Exact going wave solutions of a generalised CamassaHolm equation utilizing the builtin bifurcation method” ,Applied Mathematics and Computation, 206 ( 1 ) , pp. 141–149.
[ 13 ] Wazwaz A. , 2006, “Solitary wave solutions for modified signifiers of DegasperisProcesi and CamassaHolm equations” ,Physicss LetterssA, 352 ( 6 ) , pp. 500–504.
[ 14 ] Wazwaz A. , 2007, “New lone wave solutions to the modified signifiers of DegasperisProcesi and CamassaHolm equations, ”AppliedMathematicss and Calculation, 186 ( 1 ) , pp. 130–141.
[ 15 ] Liu Z. , Pan J. , 2009 “Coexistence of manysided expressed nonlinear wave solutions for modified signifiers of CamassaHolm and DegaperisProcesi equations” ,International Journal of Bifurcationand Chaos in Applied Sciences and Engineering, 19 ( 7 ) , pp. 2267–2282.
[ 16 ] Wang Q. , Tang M. , 2008, “New exact solutions for two nonlinear equations” ,Physicss Letters A, 372 ( 17 ) , pp. 2995–3000.
[ 17 ] Yomba E. , 2008, “The subODE method for happening exact going wave solutions of generalised nonlinear CamassaHolm, and generalized nonlinear Schrodinger equations” ,Physicss Letters A, 372 ( 3 ) , pp. 215–222.
[ 18 ] Yomba E. , 2008, “A generalized subsidiary equation method and its application to nonlinear KleinGordon and generalized nonlinear CamassaHolm equations” ,Physicss Letters A, 372 ( 7 ) , pp. 1048–1060.
[ 19 ] Liu Z. , Pan J. , 2009 “Coexistence of manysided expressed nonlinear wave solutions for modified signifiers of CamassaHolm and DegaperisProcesi equations” ,International Journal of Bifurcationand Chaos in Applied Sciences and Engineering, 19 ( 7 ) , pp. 2267–2282.
[ 20 ] Liu Z. Liang Y. , 2011, “The explicit nonlinear wave solutions and their bifurcations of the generalised CamassaHolm equation” ,International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 21 ( 11 ) , pp. 3119–3136.
[ 21 ] Parkes E.J. , Vakhnenko V.O. , 2005, “Explicit solutions of the Camassa–Holm equation” ,Chaos, Solitons andFractals, 26, pp. 1309–1316.
[ 22 ] Jafari H. , Tajadodi H. , Baleanu D. , 2014, “Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Fractional Evolution Equations” ,Journal of computational and nonlinear kineticss, 9 ( 2 ) , pp. 02101914.
[ 23 ] Liao S. , 2003,Beyond Perturbation: Introduction to the to the homotopy analysis method, Chapman and Hall/CRC Press, Boca Raton.
[ 24 ] Jafarian A. , Ghaderi P. , A. K. Golmankhaneh, Baleanu D. , 2014, “Analytical intervention of system of abel builtin equations by homotopy analysis method” ,Rumanian studies in natural philosophies,66 ( 3 ) , pp. 603611
[ 25 ] Jafarian A. , Ghaderi P. , A. K. Golmankhaneh, Baleanu D. , 2014, “Analytical approximative solutions of the zakharovkuznetsov equations” ,Rumanian studies in natural philosophies,66 ( 2 ) , pp. 296306.
[ 26 ] Shen S. , Liu F. , Anh V. , Turner I. , 2008, “The cardinal solution and numerical solution of the Riesz fractional advection–dispersion equation” ,IMA J. Appl. Math., 73 ( 6 ) , pp.850–872.
[ 27 ] Herrmann R. , 2011,Fractional Calculus: An Introduction for Physicists, World Scientific Publishing Co. Pte. Ltd. , Singapore
[ 28 ] Samko S. G. , Kilbas A. A. , Marichev O. I. , 2002,Fractional Integrals and derived functions: Theory and Applications, Taylor and Francis, London.
[ 29 ] Podlubny I. , 1999,Fractional differential Equation, Academic Press, New York.
[ 30 ] Adomian G. , 1994,SolvingFrontier Problems of Physics: The Decomposition method. Kluwer Academic Pubishers: Boston.
No Comments