Travelling Wave Solutions to Riesz Time-Fractional Camassa–Holm Equation in Modelling for Shallow-Water Waves

Traveling Wave Solutions to Riesz Time-Fractional Camassa–Holm Equation in Modeling for Shallow-Water Waves

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In the present paper, we construct the analytical exact solutions of a nonlinear development equation in mathematical natural philosophies, viz. Riesz time-fractional 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 time-fractional Camassa–Holm equation ; Riesz fractional derived function ; Riemann–Liouville fractional built-in ; Riemann-Liouville fractional derived function ; Modified Homotopy Analysis Method ; Adomian multinomial.

  1. Introduction

Let us see the Camassa–Holm equation [ 1 ] with Riesz-fractional clip derivative

( 1.1 )

whereandis arbitrary invariable.is the Riesz fractional derived function.

Recently the Camassa-Holm ( 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. Camassa-Holm 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 [ 2-9 ] . 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 built-in bifurcation method. The exact going wave solutions for Camassa-Holm equation are studied by Wazwaz [ 13, 14 ] and others [ 15, 16 ] . Yomba [ 17, 18 ] applied the sub-ODE method and the generalised subsidiary equation method for obtaining the exact solution of Camassa-Holm equation. The coexistence of many-sided 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 [ 23-25 ] 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 time-fractional 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 built-in 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.

  1. 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 left-hand side of the Riemann–Liouville fractional integral of a map,is defined by

,,

Definition 2.3The Riemann-Liouville fractional derivative [ 26-29 ] of the orderof a map,are defined as

,

,

Definition 2.4The Riesz fractional built-in [ 26-29 ] of the orderof a map,are defined as

Where,

and,are the left- and right-hand side Riemann-Liouville fractional integral operator severally.

Definition 2.5The Riesz fractional derived function of the orderof a map,are defined as [ 26-29 ]

where,

and,are the left- and right-hand side Riemann-Liouville 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 [ 26-29 ]

,

  1. Basic thought of modified homotopy analysis method ( MHAM )

In this paper, we apply the HAM [ 23-25 ] 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 nothingThursday-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 zeroth-order distortion combining weight. ( 3.3 )m-times with regard toPand so putingP= 0 and eventually spliting them bym! , we obtain the followersmThursday-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.

  1. Execution of the MHAM method for approximative solution ofRiesz time-fractional Camassa-Holm equation

In this subdivision, we foremost see the application of MHAM for the solution of Riesz time-fractional Camassa-Holm 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 nothingThursday-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 themThursday-order distortion equation

,( 4.8 )

where

( 4.9 )

Now, the solutions of themThursday-order distortion equations ( 4.8 ) forbecomes

( 4.10 )

Note:In position of the right-hand side Riemann–Liouville fractional derived function is interpreted as a future province of the procedure in natural philosophies. For this ground, the right-derivative 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 right-derivative 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 alleged-curve, it is consecutive frontward to take a proper value ofwhich ensures that the solution series is convergent.

Fig. 1.The-curve for partial derived functions ofatfor the MHAM solution.

To look into the influence ofon the solution series, we plot the so called-curve 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 2-D solution forwhen.

Case 2:For

( a ) ( B )

Fig. 3.( a ) The MHAM method going wave solution for, ( B ) matching 2-D 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.

  1. 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

8.02236E-6

9.57534E-6

0.3

1.06901E-6

7.74385E-7

0.5

7.40666E-6

8.84726E-6

0.7

1.94153E-5

7.44980E-6

0.9

3.79915E-5

5.35954E-5

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.

  1. Decision

In this paper, we have proposed a new analytical technique MHAM method to obtain the approximative solution of the Riesz time-fractional Camassa–Holm equation. The Riesz time-fractional 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 time-fractional Camassa–Holm equation.

Mentions

[ 1 ] Zhang Y. , 2013, “Time-Fractional Camassa–Holm Equation: Formulation and Solution Using Variational Methods”Journal of Computational and Nonlinear Dynamics,8, pp. 041020-7.

[ 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 Camassa-Holm equation, ”Chaos, Solitons andFractals, 12 ( 7 ) , pp. 1347–1360.

[ 5 ] Liu Z.-R. , Wang R.-Q. , Jing Z.-J. , 2004, “Peaked wave solutions of Camassa-Holm equation, ”Chaos, Solitons and Fractals, 19 ( 1 ) , pp. 77–92.

[ 6 ] Liu Z. , Qian T. , 2001, “Peakons and their bifurcation in a generalised Camassa-Holm 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 Camassa-Holm 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 Camassa-Holm and Degasperis-Procesi equations” ,Physicss Letters A, 366 ( 4-5 ) , 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. 1-34.

[ 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 Camassa-Holm equation utilizing the built-in bifurcation method” ,Applied Mathematics and Computation, 206 ( 1 ) , pp. 141–149.

[ 13 ] Wazwaz A. , 2006, “Solitary wave solutions for modified signifiers of Degasperis-Procesi and Camassa-Holm equations” ,Physicss LetterssA, 352 ( 6 ) , pp. 500–504.

[ 14 ] Wazwaz A. , 2007, “New lone wave solutions to the modified signifiers of Degasperis-Procesi and Camassa-Holm equations, ”AppliedMathematicss and Calculation, 186 ( 1 ) , pp. 130–141.

[ 15 ] Liu Z. , Pan J. , 2009 “Coexistence of many-sided expressed nonlinear wave solutions for modified signifiers of Camassa-Holm and Degaperis-Procesi 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 sub-ODE method for happening exact going wave solutions of generalised nonlinear Camassa-Holm, 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 Klein-Gordon and generalized nonlinear Camassa-Holm equations” ,Physicss Letters A, 372 ( 7 ) , pp. 1048–1060.

[ 19 ] Liu Z. , Pan J. , 2009 “Coexistence of many-sided expressed nonlinear wave solutions for modified signifiers of Camassa-Holm and Degaperis-Procesi 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 Camassa-Holm 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. 021019-1-4.

[ 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 built-in equations by homotopy analysis method” ,Rumanian studies in natural philosophies,66 ( 3 ) , pp. 603-611

[ 25 ] Jafarian A. , Ghaderi P. , A. K. Golmankhaneh, Baleanu D. , 2014, “Analytical approximative solutions of the zakharov-kuznetsov equations” ,Rumanian studies in natural philosophies,66 ( 2 ) , pp. 296-306.

[ 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.

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