Research ArticleA Numerical Solution for Hirota-SatsumaCoupled KdV Equation

M S Ismail and H A Ashi

Department of Mathematics College of Science King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia

Correspondence should be addressed to M S Ismail msismailkauedusa

Received 6 February 2014 Accepted 16 July 2014 Published 17 August 2014

Academic Editor Fuding Xie

Copyright copy 2014 M S Ismail and H A Ashi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A Petrov-Galerkin method and product approximation technique are used to solve numerically the Hirota-Satsuma coupledKorteweg-de Vries equation using cubic 119861-splines as test functions and a linear 119861-spline as trial functions The implicit midpointrule is used to advance the solution in time Newtonrsquos method is used to solve the block nonlinear pentadiagonal system we haveobtainedThe resulting schemes are of second order accuracy in both directions space and timeThe vonNeumann stability analysisof the schemes shows that the two schemes are unconditionally stable The single soliton solution and the conserved quantities areused to assess the accuracy and to show the robustness of the schemes The interaction of two solitons three solitons and birth ofsolitons is also discussed

1 Introduction

In 1981 Hirota and Satsuma [1] introduced the coupledKorteweg-de Vries equation (CKdV) as follows

120597119906

120597119905= 119886(

1205973119906

1205971199093+ 6119906

120597119906

120597119909) + 2119887V

120597V120597119909

120597V120597119905

= minus1205973V1205971199093

minus 3119906120597V120597119909

(1)

where 119886 and 119887 are arbitrary constants The CKdV equationdescribes interactions of two longwaveswith different disper-sion relations Namely it is connected withmost types of longwaves with weak dispersion internal acoustic and planetarywaves in geophysical hydrodynamics [2 3] By using Hirotamethod [1 4] the single solitary wave solution of this systemis

119906 (119909 119905) = 21205822sech2 (120585) V (119909 119905) =

1

2radic119908sech (120585)

120585 = 120582 (119909 minus 1205822119905) +

1

2 log (119908) 119908 =

minus119887

8 (4119886 + 1) 1205824

(2)

where 120582 is an arbitrary constant The two and three solitonssolutions for 119886 = 12 can be found in [1] The CKdV systemhas three conserved quantities

1198681= int

infin

minusinfin

119906119889119909

1198682= int

infin

minusinfin

(1199062+2

3119887V2)119889119909

1198683= int

infin

minusinfin

[(1 + 119886) (1199063minus1

21199062

119909) + 119887 (119906V2 minus V2

119909)] 119889119909

(3)

The proof can be found in [1 5 6]The CKdV equation has been discussed analytically by

many authors Kaya and Inan [7] used Adomian decompo-sition method to solve this system Fan used tanh methodto find some traveling wave solution [8] Assas [9] solvedthis system using variational iteration method Abbasbandy[10] used homotopy analysis method to solve the generalizedCKdV system

The numerical solutions of coupled nonlinear systemsare very interesting and important in applied science forexample the coupled nonlinear Schrodinger equation whichadmits soliton solution and it has many applications in

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 819367 9 pageshttpdxdoiorg1011552014819367

2 Abstract and Applied Analysis

communication and optical fibers this system has beensolved numerically by Ismail using finite difference and finiteelementmethods [11ndash13]TheCKdVhas been also considerednumerically by some researchers Halim et al [2 3] haveintroduced a numerical scheme for general CKdV systemsThe scheme is valid for solving an arbitrary number ofequations with arbitrary constant coefficients the methodis applied to the Hirota system and compared with itsknown explicit solution investigating the influence of initialconditions and grid sizes on accuracyWazwaz [14] produceda finite difference scheme for the periodic initial boundaryproblem of the CKdV system Ismail [6] solved this systemusing collocation method and quintic splines Kutluay andUcar [15] solved this system using a quadratic B-splineGalerkin approach

In this work we are going to solve the CKdV equationusing Petrov-Galerkin method [4] We choose the cubic119861-spline as test functions and the linear 119861-spline as trial(basis) functions Implicit midpoint rule is used in the timedirection Newtonrsquos method is used to solve the nonlinearblock pentadigonal system obtained from the schemes wehave derived The von Neumann stability analysis shows thatthe scheme is unconditionally stable Regarding the accuracythe scheme is of second order in space and time

The paper is organized as follows In Section 2 Petrov-Galerkin method is used to derive a numerical method fortheCKdVequation a coupled nonlinear pentadigonal systemis obtained Analysis of the method is given in Section 3 InSection 4 product approximation technique is used to derivea second method for solving the CKdV equation Numericalresults of various tests are contained in Section 5 We recapand sum up our conclusions in Section 6

2 Numerical Method

To derive numerical method for the CKdV system weconsider the initial boundary value problem

120597119906

120597119905minus 119886(

1205973119906

1205971199093+ 6119906

120597119906

120597119909) minus 2119887

120597V120597119909

= 0

(119909 119905) isin [119909119871 119909119877] times (0 119879]

120597V120597119905

+1205973V1205971199093

+ 3119906120597V120597119909

= 0 (119909 119905) isin [119909119871 119909119877] times (0 119879]

(4)

subject to the initial conditions119906 (119909 0) = 119892 (119909) V (119909 0) = 0 (5)

and the boundary conditions119906 (119909119871 119905) = 119906 (119909

119877 119905) = 0

119906119909(119909119871 119905) = 119906

119909(119909119877 119905) = 0

(6)

A standard weak formulation is obtained by multiplying (4)by a twice differentiable test function120595(119909) and integrating byparts to obtain

(119906119905 120595) minus 6119886 (119906119906

119909 120595) minus 119886 (119906

119909 120595119909119909) minus 2119887 (VV

119909 120595) = 0

(V119905 120595) + 3 (119906V

119909 120595) + (V

119909 120595119909119909) = 0

(7)

where ( ) denotes the usual 1198712inner product

(119891 119892) = int

119909119877

119909119871

119891 (119909) 119892 (119909) 119889119909 (8)

The space interval [119909119871 119909119877] is discretized by uniform (119873+

1) grid points Consider

119909119898= 119909119871+ 119898ℎ 119898 = 0 1 119873 (9)

where the grid spacing ℎ is given by ℎ = (119909119877minus 119909119871)119873 We

introduce finite elements in space in (7) Approximate theexact solutions of 119906(119909 119905) and V(119909 119905) by

119906 (119909 119905) =

119873

sum

119898=0

119880119898(119905) 120601119898(119909)

V (119909 119905) =119873

sum

119898=0

119881119898(119905) 120601119898(119909)

(10)

where the trial functions 120601119898(119909) = 120601((119909 minus 119909

119898)ℎ) 119898 =

0 1 119873 are the piecewise linear functions Consider

120601 (119909) =

1 + 119909 if minus 1 lt 119909 le 0

1 minus 119909 if 0 lt 119909 le 1

0 otherwise(11)

The unknown functions 119880119898(119905) 119881119898(119905) 119898 = 0 1 2 119873

are determined from the ordinary differential system Con-sider

(119880119905 120595119895) minus 6119886 (119880119880

119909 120595119895) minus 119886 (119880

119909 (120595119895)119909119909)

minus 2119887 (119881119881119909 120595119895) = 0

(119881119905 120595119895) + 3 (119880119881

119909 120595119895) + (119881

119909 (120595119895)119909119909) = 0

119895 = 0 1 2 119873

(12)

where the test functions120595119895(119909) = 120595((119909minus119909

119894)ℎ) 119895 = 1 2 119873

are the cubic 119861-spline with compact support Consider

120595 (119909) =1

4

(119909 + 2)3 if minus 2 le 119909 le minus1

[(2 + 119909)3minus 4(1 + 119909)

3] if minus 1 lt 119909 le 0

[(2 minus 119909)3minus 4(1 minus 119909)

3] if 0 lt 119909 le 1

(2 minus 119909)3 if 1 lt 119909 le 2

0 otherwise

(13)

Abstract and Applied Analysis 3

Direct calculation of the inner products given in (12)will lead us to the following system of first order ordinarydifferential system (ODES)

1

80(119898minus2

+ 26119898minus1

+ 66119898+ 26

119898+1+ 119898+2

)

minus6119886

80ℎ119865 (119880119898 119880119898) minus

2119887

80ℎ119865 (119881119898 119881119898)

minus3119886

4ℎ3(119880119898+2

minus 2119880119898+1

+ 2119880119898minus1

minus 119880119898minus2

) = 0

1

80(119898minus2

+ 26119898minus1

+ 66119898+ 26

119898+1+ 119898+2

)

+3

4ℎ3(119881119898+2

minus 2119881119898+1

+ 2119881119898minus1

minus 119881119898minus2

)

+3

80ℎ119865 (119880119898 119881119898) = 0

(14)

where119865 (119880119898 119881119898) = (minus119881

119898minus2+ 119881119898minus1

) (119880119898minus2

+ 4119880119898minus1

)

+ (minus119881119898minus1

+ 119881119898) (22119880

119898minus1+ 33119880

119898)

+ (minus119881119898+ 119881119898+1

) (33119880119898+ 22119880

119898+1)

+ (minus119881119898+1

+ 119881119898+2

) (4119880119898+1

+ 119880119898+2

)

(15)

Now to the ODES we assume 119880119899119898and 119881119899

119898to be the fully

discrete approximation to the exact solutions 119906(119909119898 119905119899) and

V(119909119898 119905119899) where 119905

119899= 119899119896 and 119896 is the time step size By using

the finite difference approximation

119898=119880119899+1

119898minus 119880119899

119898

119896 119880

119898=119880119899+1

119898+ 119880119899

119898

2= 119880lowast

119898=119881119899+1

119898minus 119881119899

119898

119896 119881

119898=119881119899+1

119898+ 119881119899

119898

2= 119881lowast

(16)

the ODES (14) will be reduced to the implicit nonlinear finitedifference scheme

(119880119899+1

119898minus2+ 26119880

119899+1

119898minus1+ 66119880

119899+1

119898+ 26119880

119899+1

119898+1+ 119880119899+1

119898+2)

minus (119880119899

119898minus2+ 26119880

119899

119898minus1+ 66119880

119899

119898+ 26119880

119899

119898+1+ 119880119899

119898+2)

minus 1199011(119880lowast

119898+2minus 2119880lowast

119898+1+ 2119880lowast

119898minus1minus 119880lowast

119898minus2)

+ 1199012119865 (119880lowast 119880lowast) + 1199013119865 (119881lowast 119881lowast) = 0

(17)

(119881119899+1

119898minus2+ 26119881

119899+1

119898minus1+ 66119881

119899+1

119898+ 26119881

119899+1

119898+1+ 119881119899+1

119898+2)

minus (119881119899

119898minus2+ 26119881

119899

119898minus1+ 66119881

119899

119898+ 26119881

119899

119898+1+ 119881119899

119898+2)

+ 1199014(119881lowast

119898+2minus 2119881lowast

119898+1+ 2119881lowast

119898minus1minus 119881lowast

119898minus2)

+ 1199015119865 (119880lowast 119881lowast) = 0 119898 = 1 2 119873

(18)

1199011=minus60119886119896

ℎ3 119901

2=minus6119886119896

ℎ 119901

3=minus2119887119896

ℎ

1199014=60119896

ℎ3 119901

5=3119896

ℎ

(19)

The system (17)-(18) is a nonlinear block penta diagonalsystem in the unknown vectors U119899+1 and V119899+1 and is solvedusing Newtonrsquos methodWe denote this method by Scheme 1

3 Analysis of the Method

In this section we will discuss the stability and the accuracyof Scheme 1

31 Stability of the Scheme To study the stability of Scheme 1von Neumann stability analysis will be used and this can beonly applied for linear finite difference schemes accordinglywe consider the linear version of the proposed method

(119880119899+1

119898minus2+ 26119880

119899+1

119898minus1+ 66119880

119899+1

119898+ 26119880

119899+1

119898+1+ 119880119899+1

119898+2)

minus (119880119899

119898minus2+ 26119880

119899

119898minus1+ 66119880

119899

119898+ 26119880

119899

119898+1+ 119880119899

119898+2)

+ 1199011(119880lowast

119898+2minus 2119880lowast

119898+1+ 2119880lowast

119898minus1minus 119880lowast

119898minus2)

+ 1199012119865 (119880119880

lowast) + 1199013119865 (119881119881

lowast) = 0

(119881119899+1

119898minus2+ 26119881

119899+1

119898minus1+ 66119881

119899+1

119898+ 26119881

119899+1

119898+1+ 119881119899+1

119898+2)

minus (119881119899

119898minus2+ 26119881

119899

119898minus1+ 66119881

119899

119898+ 26119881

119899

119898+1+ 119881119899

119898+2)

+ 1199014(119881lowast

119898+2minus 2119881lowast

119898+1+ 2119881lowast

119898minus1minus 119881lowast

119898minus2)

+ 1199015119865 (119880119881

lowast) = 0

(20)

where

119865 (119880119881) = 119880 [119881119898+2

+ 10119881119898+1

minus 10119881119898minus1

minus 119881119898minus2

] (21)

1199011015840

2= 51199012 11990110158403= 51199013 and 1199011015840

5= 51199015 119880 and 119881 are assumed

to be constant on the whole range We assume the solution ofthe linearized scheme (20) to be of the form

119880119899

119898= 119880119899119890119894120573119898ℎ

119881119899

119898= 119881119899119890119894120573119898ℎ

(22)

where120573 is a real constant Direct substitution of (22) into (20)will lead us to the following system

[1205741+ 11989411990111205742+ 1198941199011015840

21198801205743]119880119899+1

+ 1198941199011015840

31198811205743119881119899+1

= [1205741minus 11989411990111205742minus 1198941199011015840

21198801205743]119880119899minus 1198941199011015840

31198811205743119881119899

[1205741+ 11989411990141205742+ 1198941199011015840

51198801205743]119881119899+1

= [1205741minus 11989411990141205742minus 1198941199011015840

51198801205743]119881119899

(23)

where

1205741= 66 + 2 cos (2120573ℎ) + 52 cos (120573ℎ)

1205742= 2 sin (2120573ℎ) minus 4 sin (120573ℎ)

1205743= 2 sin (2120573ℎ) + 20 sin (120573ℎ)

(24)

The system (23) can be written in a matrix vector form as

Ψ119899+1

= 119861Ψ119899 (25)

4 Abstract and Applied Analysis

where Ψ = [119880 119881]119905 and 119861 is the (2 times 2)matrix Consider

119861 = [1205741+ 1198941198881

1198882

0 1205741+ 1198941198883

]

minus1

[1205741minus 1198941198881

minus1198882

0 1205741minus 1198941198883

] (26)

where

1198881= 11990111205742+ 1199011015840

21198801205743 119888

2= 1199011015840

31198811205743

1198883= 11990141205742+ 1199011015840

51198801205743

(27)

von Neumann stability condition for the system (25) statesthat for the scheme to be stable the maximum modulus ofthe eigenvalues of the amplification matrix 119861 should be lessthan or equal to one The eigenvalues of the matrix 119861 are

1205821=1205741minus 1198941198881

1205741+ 1198941198881

1205822=1205741minus 1198941198883

1205741+ 1198941198883

(28)

which hasmodulus equal to one and hence the scheme underconsideration is unconditionally stable in the linearizedsense

32 Accuracy of the Scheme In order to study the accuracy ofthe scheme we replace the numerical solution 119880119899

119898and 119881119899

119898in

(17) and (18) by the exact solutions 119906119899119898and V119899119898 By making use

of the following Taylor series expansions at the point (119909119898 119905119899)

1

119896[(119906119899+1

119898minus2+ 26119906

119899+1

119898minus1+ 66119906

119899+1

119898+ 26119906

119899+1

119898+1+ 119906119899+1

119898+2)

minus (119906119899

119898minus2+ 26119906

119899

119898minus1+ 66119906

119899

119898+ 26119906

119899

119898+1+ 119906119899

119898+2) ]

=3

2119906119905+3119896

4119906119905119905+3ℎ2

8119906119909119909119905

+1198962

4119906119905119905119905+ sdot sdot sdot

(29)

3

8ℎ3[119906119899+1

119898+2minus 2119906119899+1

119898+1+ 2119906119899+1

119898minus1minus 119906119899+1

119898minus2

+ 119906119899

119898+2minus 2119906119899

119898+1+ 2119906119899

119898minus1minus 119906119899

119898minus2]

=3

2119906119909119909119909

+3119896

4119906119909119909119909119905

+3ℎ2

8119906119909119909119909119909119909

+31198962

8119906119909119909119909119905119905

+ sdot sdot sdot

(30)

1

80ℎ119865 (119906lowast 119906lowast) =

3

2119906119906119909+3119896

4(119906119906119909)119905

+ ℎ2[3

8119906119906119909119909119909

+7

8119906119909119906119909119909]

+31198962

8[119906119909119906119905119905+ 119906119905119906119909119905+ 119906119906119909119905119905] + sdot sdot sdot

(31)

1

80ℎ119865 (119906lowast Vlowast) =

3

2119906V119909+3119896

4(119906V119909)119905

+ ℎ2[3

8119906V119909119909119909

+3

8V119909119906119909119909+1

2119906119909V119909119909]

+31198962

8[V119909119906119905119905+ 119906119905V119909119905+ 119906V119909119905119905] + sdot sdot sdot

(32)

and the CKdV system (1) we obtain the local truncation error(LTE) for the proposed scheme

LTE (17) = minus3ℎ2

2119906119909119906119909119909minus119887ℎ2

2V119906119909119909minus1198962

8119906119905119905119905

+91198962119886

4119906119905119906119909119905minus31198871198962

4V119905V119909119905+ sdot sdot sdot

LTE (18) = minus3ℎ2

2119906119909V119909119909minus1198962

8119906119905119905119905minus31198962

8119906119905V119909119905+ sdot sdot sdot

(33)

and hence the scheme is of second order in both directionsspace and time

4 Product Approximation Technique

A modified Petrov-Galerkin method for solving the CKdVsystem (1) can be achieved by using the product approxi-mation technique where we used special approximation tothe nonlinear terms in the differential system In order toapply this technique we rewrite the CKdV in the followingform

120597119906

120597119905= 119886(

1205973119906

1205971199093+ 3

120597 (1199062)

120597119909) + 119887

120597 (V2)120597119909

(34)

120597V120597119905

= minus1205973V1205971199093

minus 3119906120597V120597119909

(35)

The product approximation technique is used to approx-imate the nonlinear terms in (34) in the following man-ner

1199062(119909 119905) =

119873

sum

119898=1

1198802

119898(119905) 120601119898(119909)

V2 (119909 119905) =119873

sum

119898=1

1198812

119898(119905) 120601119898(119909)

(36)

By using the same procedure in deriving Scheme 1 and theapproximation (37) we can obtain after some manipulationsthe following scheme

1199011[(119880119899+1

119898minus2+ 26119880

119899+1

119898minus1+ 66119880

119899+1

119898+ 26119880

119899+1

119898+1+ 119880119899+1

119898+2)

minus (119880119899

119898minus2+ 26119880

119899

119898minus1+ 66119880

119899

119898+ 26119880

119899

119898+1+ 119880119899

119898+2) ]

minus 1199012[(119880lowast

119898+2)2+ 10(119880

lowast

119898+1)2minus 10(119880

lowast

119898minus1)2minus (119880lowast

119898minus2)2]

Abstract and Applied Analysis 5

Table 1 Petrov-Galerkin method

119879 119871infin(119880) 119871

infin(119881) 119868

11198682

1198683

0 0000000 0000000 2000000 05000000 01125292 0000015 0000004 2000000 05000000 01125294 0000021 0000005 2000000 05000000 01125296 0000023 0000006 2000000 05000000 01125298 0000024 0000007 2000000 05000000 011252910 0000025 0000008 2000000 05000000 0112529

minus 1199013(119880lowast

119898+2minus 2119880lowast

119898+1+ 2119880lowast

119898minus1minus 119880lowast

119898minus2)

minus 1199014[(119881lowast

119898+2)2+ 10(119881

lowast

119898+1)2minus 10(119881

lowast

119898minus1)2minus (119881lowast

119898minus2)2] = 0

(37)

1199011[(119881119899+1

119898minus2+ 26119881

119899+1

119898minus1+ 66119881

119899+1

119898+ 26119881

119899+1

119898+1+ 119881119899+1

119898+2)

minus (119881119899

119898minus2+ 26119881

119899

119898minus1+ 66119881

119899

119898+ 26119881

119899

119898+1+ 119881119899

119898+2) ]

+ 1199015(119881lowast

119898+2minus 2119881lowast

119898+1+ 2119881lowast

119898minus1minus 119881lowast

119898minus2)

+ 1199016[minus (119880lowast

119898minus2+ 4119880lowast

119898minus1) 119881lowast

119898minus2

+ (119880lowast

119898minus2minus 18119880

lowast

119898minus1minus 33119880

lowast

119898) 119881lowast

119898minus1

+ 22 (119880lowast

119898minus1minus 119880lowast

119898+1) 119881lowast

119898

+ (33119880lowast

119898+ 18119880

lowast

119898+1minus 119880lowast

119898+2) 119881lowast

119898+1

+ (4119880lowast

119898+1+ 119880lowast

119898+2) 119881lowast

119898+2] = 0

(38)

where

119880lowast

119898= 119880119899+1

119898+ 119880119899

119898 119881

lowast

119898= 119881119899+1

119898+ 119881119899

119898

1199011=

1

80 119901

2=3119886119896

64ℎ 119901

3=3119886119896

8ℎ3

1199014=

119887119896

64ℎ 119901

5=

3119896

8ℎ3 119901

6=

3119896

320ℎ

(39)

Again the method is unconditionally stable accordingto von Neumann stability analysis and it is of secondorder in both directions The scheme produced a nonlinearblock pentadiagonal system and its solution obtained usingNewtonrsquosmethodWehave noticed that the accuracy has beenimproved in the first equation (38) as we will see in the nextsection We will denote the scheme obtained by using theproduct approximation technique by Scheme 2

5 Numerical Results

To gain insight into the performance of the proposedschemes we perform different numerical tests like sin-gle soliton two and three solitons interaction and birthof solitons The conservation properties of the proposedschemes are examined by calculating 119868

1 1198682 and 119868

3using

the trapezoidal rule The 119871infin(119906) and 119871

infin(V) error norms are

defined as

119871infin(119906) =

1003817100381710038171003817U119899minus u1198991003817100381710038171003817infin = max

119898

1003816100381610038161003816119880119899

119898minus 119906119899

119898

1003816100381610038161003816

119871infin (V) =

1003817100381710038171003817V119899minus k1198991003817100381710038171003817infin = max

119898

1003816100381610038161003816119881119899

119898minus V119899119898

1003816100381610038161003816

(40)

are used to examine the accuracy of the proposed schemes

51 Single Soliton In the first test we choose the initialconditions

119906 (119909 0) = 21205822sech2 (120585) V (119909 0) =

1

2radic119908sech (120585)

120585 = 120582119909 +1

2 log (119908) 119908 =

minus119887

8 (4119886 + 1) 1205824

(41)

which represents a single soliton solution at 119905 = 0 To studythe behavior of numerical solution using Scheme 1 andScheme 2 we choose the set of parameters as ℎ = 005119896 = 001 tol = 10

minus7 119909119871= minus25 119909

119877= 25 119886 = minus0125

119887 = minus3 and 120582 = 05 The conserved quantities and theerror norms 119871

infin(119880) 119871

infin(119881) are displayed in Tables 1 and 2

for Scheme 1 and Scheme 2 respectively It is clear from thesetables that our schemes are highly accurate In addition theschemes preserve the conserved quantities exactly during theevolution of the numerical solution from 119905 = 0 to 119905 = 10 Theexecution time required to produce Table 1 is 2328 secondand 2171 second to produce Table 2 We have noticed thatScheme 2 has an upper hand over Scheme 1 with respect toaccuracy and CPU time In Figures 1 and 2 we display thenumerical solution of 119880119899

119898and 119881119899

119898for 119905 = 0 1 2 20

By choosing the set of values 119896 = 001 119886 = 05 119887 = minus30120582 = 05 and 119905 = 1 we perform a comparison of Scheme 1 andScheme 2 with Ismail [6] and we display this in Table 3 wecan easily see that the three methods produce highly accurateresults with some credits for collocation method

52 Two Solitons Interaction To study the interaction of twosolitons we choose the initial conditions as

119906 (119909 0) =

2

sum

119895=1

119906119895(119909 0) V (119909 0) =

2

sum

119895=1

V119895(119909 0) (42)

6 Abstract and Applied Analysis

Table 2 Product approximation technique

119879 119871infin(119880) 119871

infin(119881) 119868

11198682

1198683

0 0000000 0000000 2000000 05000000 01125292 0000004 0000005 2000000 05000000 01125294 0000008 0000007 2000000 05000000 01125296 0000010 0000009 2000000 05000000 01125298 0000013 0000012 2000000 05000000 011252910 0000014 0000013 2000000 05000000 0112529

Table 3 Comparison of three methods with 119886 = 05 119887 = minus30 120582 = 05 119896 = 001 and 119905 = 1

ℎPetrov-Galerkin Product approximation Collocation (Ismail [6])

119871infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU

01 0000051 0000027 0218 0000014 0000019 0203 0000000 0000003 0188005 0000013 0000007 0343 0000009 0000008 0328 0000000 0000003 03120025 0000003 0000004 0625 0000002 0000004 0609 0000000 0000003 0578

010

2030

4050

05

1015

200

01

02

03

04

05

x

t

u

Figure 1 Single soliton 119880 119886 = minus0125 119887 = minus30 and 120582 = 05

010

2030

4050

05

1015

200

00200400600801012014016

xt

v

Figure 2 Single soliton 119881 119886 = minus0125 119887 = minus30 and 120582 = 05

Table 4 Two solitons interaction (conserved quantities)

119879 1198681

1198682

1198683

000 6400000 minus3243013 minus2584118200 6399985 minus3243298 minus2586545400 6400034 minus3243016 minus2584155600 6399962 minus3243017 minus2584203800 6400234 minus3242978 minus2583492

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2

(43)

which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582

1= 10 120582

2= 06 119910

1= 10

and 1199102= 30 In Table 4 we present the conserved quantities

during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]

To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582

1= 10 120582

2= 06

Abstract and Applied Analysis 7

020

4060

80100

0

20

40

60

80

0123

minus1

x

t

u

Figure 3 Numerical solution 119880119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

020

4060

80100

010

2030

4050

6070

80

012

x

t

v

minus1

Figure 4 Numerical solution 119881119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

x

t

Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)

020

4060

80100

010

2030

4050

0

1

2

3

4

5

xt

u

minus1

Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =

0495

020

4060

80100

010

2030

4050

0

05

1

15

2

25

xt

v

minus05

Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =

0495

1199101= 10 119910

2= 30 119888 = 0 119889 = 100 ℎ = 005 and 119896 = 001 Our

results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12

53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform

119906 (119909 0) =

3

sum

119895=1

119906119895(119909 0) V (119909 0) =

3

sum

119895=1

V119895(119909 0) (44)

8 Abstract and Applied Analysis

020

4060

80100

010

2030

4050

6070

80

0123

x

t

u

minus1

Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

020

4060

80100

020

4060

80

0020406081

121416

xt

v

minus02

Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2 3

(45)

In this test we choose the set of parameters 119909119871= 0 119909

119877=

80 ℎ = 005 119896 = 001 119886 = 05 119887 = minus30 1205821= 10 120582

2= 06

1205823= 03 119910

1= 10 119910

2= 30 and 119910

3= 50

The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906

1interacts with the other two

solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario

020

4060

80

05

1015

2025

024

minus40

minus20

minus2

x

t

u

Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30

Table 5 Three solitons interaction the conserved quantities

119879 1198681

1198682

1198683

0 7600000 minus3351855 minus261028320 7600042 minus3352227 minus261314940 7600800 minus3352034 minus261134760 7599367 minus3351547 minus260897080 7600545 minus3351562 minus2602090

Table 6 Birth of solitons with 119886 = 05 and 119887 = minus30

119879 1198681

1198682

00 17724343 minus1253314250 17723816 minus12532956100 17723352 minus12530116150 17722782 minus12529239200 17722217 minus12529013250 17721734 minus12528983

54 Birth of Solitons In this test we choose the initial condi-tion

119906 (119909 0) = exp (minus0011199092)

V (119909 0) = exp (minus0011199092) (46)

with the following set of parameters 119886 = 05 119887 = minus30 0 le

119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved

6 Conclusion

In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm

Abstract and Applied Analysis 9

during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12

To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981

[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003

[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004

[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981

[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997

[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009

[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004

[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008

[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008

[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007

[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004

[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ

Equ pp 485ndash568 ElsevierNorth-Holland Amsterdam TheNetherlands 2008

[15] S Kutluay and Y Ucar ldquoA quadratic B-spline Galerkin approachfor solving a coupled KdV equationrdquo Mathematical Modellingand Analysis vol 18 no 1 pp 103ndash121 2013

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

communication and optical fibers this system has beensolved numerically by Ismail using finite difference and finiteelementmethods [11ndash13]TheCKdVhas been also considerednumerically by some researchers Halim et al [2 3] haveintroduced a numerical scheme for general CKdV systemsThe scheme is valid for solving an arbitrary number ofequations with arbitrary constant coefficients the methodis applied to the Hirota system and compared with itsknown explicit solution investigating the influence of initialconditions and grid sizes on accuracyWazwaz [14] produceda finite difference scheme for the periodic initial boundaryproblem of the CKdV system Ismail [6] solved this systemusing collocation method and quintic splines Kutluay andUcar [15] solved this system using a quadratic B-splineGalerkin approach

In this work we are going to solve the CKdV equationusing Petrov-Galerkin method [4] We choose the cubic119861-spline as test functions and the linear 119861-spline as trial(basis) functions Implicit midpoint rule is used in the timedirection Newtonrsquos method is used to solve the nonlinearblock pentadigonal system obtained from the schemes wehave derived The von Neumann stability analysis shows thatthe scheme is unconditionally stable Regarding the accuracythe scheme is of second order in space and time

The paper is organized as follows In Section 2 Petrov-Galerkin method is used to derive a numerical method fortheCKdVequation a coupled nonlinear pentadigonal systemis obtained Analysis of the method is given in Section 3 InSection 4 product approximation technique is used to derivea second method for solving the CKdV equation Numericalresults of various tests are contained in Section 5 We recapand sum up our conclusions in Section 6

2 Numerical Method

To derive numerical method for the CKdV system weconsider the initial boundary value problem

120597119906

120597119905minus 119886(

1205973119906

1205971199093+ 6119906

120597119906

120597119909) minus 2119887

120597V120597119909

= 0

(119909 119905) isin [119909119871 119909119877] times (0 119879]

120597V120597119905

+1205973V1205971199093

+ 3119906120597V120597119909

= 0 (119909 119905) isin [119909119871 119909119877] times (0 119879]

(4)

subject to the initial conditions119906 (119909 0) = 119892 (119909) V (119909 0) = 0 (5)

and the boundary conditions119906 (119909119871 119905) = 119906 (119909

119877 119905) = 0

119906119909(119909119871 119905) = 119906

119909(119909119877 119905) = 0

(6)

A standard weak formulation is obtained by multiplying (4)by a twice differentiable test function120595(119909) and integrating byparts to obtain

(119906119905 120595) minus 6119886 (119906119906

119909 120595) minus 119886 (119906

119909 120595119909119909) minus 2119887 (VV

119909 120595) = 0

(V119905 120595) + 3 (119906V

119909 120595) + (V

119909 120595119909119909) = 0

(7)

where ( ) denotes the usual 1198712inner product

(119891 119892) = int

119909119877

119909119871

119891 (119909) 119892 (119909) 119889119909 (8)

The space interval [119909119871 119909119877] is discretized by uniform (119873+

1) grid points Consider

119909119898= 119909119871+ 119898ℎ 119898 = 0 1 119873 (9)

where the grid spacing ℎ is given by ℎ = (119909119877minus 119909119871)119873 We

introduce finite elements in space in (7) Approximate theexact solutions of 119906(119909 119905) and V(119909 119905) by

119906 (119909 119905) =

119873

sum

119898=0

119880119898(119905) 120601119898(119909)

V (119909 119905) =119873

sum

119898=0

119881119898(119905) 120601119898(119909)

(10)

where the trial functions 120601119898(119909) = 120601((119909 minus 119909

119898)ℎ) 119898 =

0 1 119873 are the piecewise linear functions Consider

120601 (119909) =

1 + 119909 if minus 1 lt 119909 le 0

1 minus 119909 if 0 lt 119909 le 1

0 otherwise(11)

The unknown functions 119880119898(119905) 119881119898(119905) 119898 = 0 1 2 119873

are determined from the ordinary differential system Con-sider

(119880119905 120595119895) minus 6119886 (119880119880

119909 120595119895) minus 119886 (119880

119909 (120595119895)119909119909)

minus 2119887 (119881119881119909 120595119895) = 0

(119881119905 120595119895) + 3 (119880119881

119909 120595119895) + (119881

119909 (120595119895)119909119909) = 0

119895 = 0 1 2 119873

(12)

where the test functions120595119895(119909) = 120595((119909minus119909

119894)ℎ) 119895 = 1 2 119873

are the cubic 119861-spline with compact support Consider

120595 (119909) =1

4

(119909 + 2)3 if minus 2 le 119909 le minus1

[(2 + 119909)3minus 4(1 + 119909)

3] if minus 1 lt 119909 le 0

[(2 minus 119909)3minus 4(1 minus 119909)

3] if 0 lt 119909 le 1

(2 minus 119909)3 if 1 lt 119909 le 2

0 otherwise

(13)

Abstract and Applied Analysis 3

Direct calculation of the inner products given in (12)will lead us to the following system of first order ordinarydifferential system (ODES)

1

80(119898minus2

+ 26119898minus1

+ 66119898+ 26

119898+1+ 119898+2

)

minus6119886

80ℎ119865 (119880119898 119880119898) minus

2119887

80ℎ119865 (119881119898 119881119898)

minus3119886

4ℎ3(119880119898+2

minus 2119880119898+1

+ 2119880119898minus1

minus 119880119898minus2

) = 0

1

80(119898minus2

+ 26119898minus1

+ 66119898+ 26

119898+1+ 119898+2

)

+3

4ℎ3(119881119898+2

minus 2119881119898+1

+ 2119881119898minus1

minus 119881119898minus2

)

+3

80ℎ119865 (119880119898 119881119898) = 0

(14)

where119865 (119880119898 119881119898) = (minus119881

119898minus2+ 119881119898minus1

) (119880119898minus2

+ 4119880119898minus1

)

+ (minus119881119898minus1

+ 119881119898) (22119880

119898minus1+ 33119880

119898)

+ (minus119881119898+ 119881119898+1

) (33119880119898+ 22119880

119898+1)

+ (minus119881119898+1

+ 119881119898+2

) (4119880119898+1

+ 119880119898+2

)

(15)

Now to the ODES we assume 119880119899119898and 119881119899

119898to be the fully

discrete approximation to the exact solutions 119906(119909119898 119905119899) and

V(119909119898 119905119899) where 119905

119899= 119899119896 and 119896 is the time step size By using

the finite difference approximation

119898=119880119899+1

119898minus 119880119899

119898

119896 119880

119898=119880119899+1

119898+ 119880119899

119898

2= 119880lowast

119898=119881119899+1

119898minus 119881119899

119898

119896 119881

119898=119881119899+1

119898+ 119881119899

119898

2= 119881lowast

(16)

the ODES (14) will be reduced to the implicit nonlinear finitedifference scheme

(119880119899+1

119898minus2+ 26119880

119899+1

119898minus1+ 66119880

119899+1

119898+ 26119880

119899+1

119898+1+ 119880119899+1

119898+2)

minus (119880119899

119898minus2+ 26119880

119899

119898minus1+ 66119880

119899

119898+ 26119880

119899

119898+1+ 119880119899

119898+2)

minus 1199011(119880lowast

119898+2minus 2119880lowast

119898+1+ 2119880lowast

119898minus1minus 119880lowast

119898minus2)

+ 1199012119865 (119880lowast 119880lowast) + 1199013119865 (119881lowast 119881lowast) = 0

(17)

(119881119899+1

119898minus2+ 26119881

119899+1

119898minus1+ 66119881

119899+1

119898+ 26119881

119899+1

119898+1+ 119881119899+1

119898+2)

minus (119881119899

119898minus2+ 26119881

119899

119898minus1+ 66119881

119899

119898+ 26119881

119899

119898+1+ 119881119899

119898+2)

+ 1199014(119881lowast

119898+2minus 2119881lowast

119898+1+ 2119881lowast

119898minus1minus 119881lowast

119898minus2)

+ 1199015119865 (119880lowast 119881lowast) = 0 119898 = 1 2 119873

(18)

1199011=minus60119886119896

ℎ3 119901

2=minus6119886119896

ℎ 119901

3=minus2119887119896

ℎ

1199014=60119896

ℎ3 119901

5=3119896

ℎ

(19)

The system (17)-(18) is a nonlinear block penta diagonalsystem in the unknown vectors U119899+1 and V119899+1 and is solvedusing Newtonrsquos methodWe denote this method by Scheme 1

3 Analysis of the Method

In this section we will discuss the stability and the accuracyof Scheme 1

31 Stability of the Scheme To study the stability of Scheme 1von Neumann stability analysis will be used and this can beonly applied for linear finite difference schemes accordinglywe consider the linear version of the proposed method

(119880119899+1

119898minus2+ 26119880

119899+1

119898minus1+ 66119880

119899+1

119898+ 26119880

119899+1

119898+1+ 119880119899+1

119898+2)

minus (119880119899

119898minus2+ 26119880

119899

119898minus1+ 66119880

119899

119898+ 26119880

119899

119898+1+ 119880119899

119898+2)

+ 1199011(119880lowast

119898+2minus 2119880lowast

119898+1+ 2119880lowast

119898minus1minus 119880lowast

119898minus2)

+ 1199012119865 (119880119880

lowast) + 1199013119865 (119881119881

lowast) = 0

(119881119899+1

119898minus2+ 26119881

119899+1

119898minus1+ 66119881

119899+1

119898+ 26119881

119899+1

119898+1+ 119881119899+1

119898+2)

minus (119881119899

119898minus2+ 26119881

119899

119898minus1+ 66119881

119899

119898+ 26119881

119899

119898+1+ 119881119899

119898+2)

+ 1199014(119881lowast

119898+2minus 2119881lowast

119898+1+ 2119881lowast

119898minus1minus 119881lowast

119898minus2)

+ 1199015119865 (119880119881

lowast) = 0

(20)

where

119865 (119880119881) = 119880 [119881119898+2

+ 10119881119898+1

minus 10119881119898minus1

minus 119881119898minus2

] (21)

1199011015840

2= 51199012 11990110158403= 51199013 and 1199011015840

5= 51199015 119880 and 119881 are assumed

to be constant on the whole range We assume the solution ofthe linearized scheme (20) to be of the form

119880119899

119898= 119880119899119890119894120573119898ℎ

119881119899

119898= 119881119899119890119894120573119898ℎ

(22)

where120573 is a real constant Direct substitution of (22) into (20)will lead us to the following system

[1205741+ 11989411990111205742+ 1198941199011015840

21198801205743]119880119899+1

+ 1198941199011015840

31198811205743119881119899+1

= [1205741minus 11989411990111205742minus 1198941199011015840

21198801205743]119880119899minus 1198941199011015840

31198811205743119881119899

[1205741+ 11989411990141205742+ 1198941199011015840

51198801205743]119881119899+1

= [1205741minus 11989411990141205742minus 1198941199011015840

51198801205743]119881119899

(23)

where

1205741= 66 + 2 cos (2120573ℎ) + 52 cos (120573ℎ)

1205742= 2 sin (2120573ℎ) minus 4 sin (120573ℎ)

1205743= 2 sin (2120573ℎ) + 20 sin (120573ℎ)

(24)

The system (23) can be written in a matrix vector form as

Ψ119899+1

= 119861Ψ119899 (25)

4 Abstract and Applied Analysis

where Ψ = [119880 119881]119905 and 119861 is the (2 times 2)matrix Consider

119861 = [1205741+ 1198941198881

1198882

0 1205741+ 1198941198883

]

minus1

[1205741minus 1198941198881

minus1198882

0 1205741minus 1198941198883

] (26)

where

1198881= 11990111205742+ 1199011015840

21198801205743 119888

2= 1199011015840

31198811205743

1198883= 11990141205742+ 1199011015840

51198801205743

(27)

von Neumann stability condition for the system (25) statesthat for the scheme to be stable the maximum modulus ofthe eigenvalues of the amplification matrix 119861 should be lessthan or equal to one The eigenvalues of the matrix 119861 are

1205821=1205741minus 1198941198881

1205741+ 1198941198881

1205822=1205741minus 1198941198883

1205741+ 1198941198883

(28)

which hasmodulus equal to one and hence the scheme underconsideration is unconditionally stable in the linearizedsense

32 Accuracy of the Scheme In order to study the accuracy ofthe scheme we replace the numerical solution 119880119899

119898and 119881119899

119898in

(17) and (18) by the exact solutions 119906119899119898and V119899119898 By making use

of the following Taylor series expansions at the point (119909119898 119905119899)

1

119896[(119906119899+1

119898minus2+ 26119906

119899+1

119898minus1+ 66119906

119899+1

119898+ 26119906

119899+1

119898+1+ 119906119899+1

119898+2)

minus (119906119899

119898minus2+ 26119906

119899

119898minus1+ 66119906

119899

119898+ 26119906

119899

119898+1+ 119906119899

119898+2) ]

=3

2119906119905+3119896

4119906119905119905+3ℎ2

8119906119909119909119905

+1198962

4119906119905119905119905+ sdot sdot sdot

(29)

3

8ℎ3[119906119899+1

119898+2minus 2119906119899+1

119898+1+ 2119906119899+1

119898minus1minus 119906119899+1

119898minus2

+ 119906119899

119898+2minus 2119906119899

119898+1+ 2119906119899

119898minus1minus 119906119899

119898minus2]

=3

2119906119909119909119909

+3119896

4119906119909119909119909119905

+3ℎ2

8119906119909119909119909119909119909

+31198962

8119906119909119909119909119905119905

+ sdot sdot sdot

(30)

1

80ℎ119865 (119906lowast 119906lowast) =

3

2119906119906119909+3119896

4(119906119906119909)119905

+ ℎ2[3

8119906119906119909119909119909

+7

8119906119909119906119909119909]

+31198962

8[119906119909119906119905119905+ 119906119905119906119909119905+ 119906119906119909119905119905] + sdot sdot sdot

(31)

1

80ℎ119865 (119906lowast Vlowast) =

3

2119906V119909+3119896

4(119906V119909)119905

+ ℎ2[3

8119906V119909119909119909

+3

8V119909119906119909119909+1

2119906119909V119909119909]

+31198962

8[V119909119906119905119905+ 119906119905V119909119905+ 119906V119909119905119905] + sdot sdot sdot

(32)

and the CKdV system (1) we obtain the local truncation error(LTE) for the proposed scheme

LTE (17) = minus3ℎ2

2119906119909119906119909119909minus119887ℎ2

2V119906119909119909minus1198962

8119906119905119905119905

+91198962119886

4119906119905119906119909119905minus31198871198962

4V119905V119909119905+ sdot sdot sdot

LTE (18) = minus3ℎ2

2119906119909V119909119909minus1198962

8119906119905119905119905minus31198962

8119906119905V119909119905+ sdot sdot sdot

(33)

and hence the scheme is of second order in both directionsspace and time

4 Product Approximation Technique

A modified Petrov-Galerkin method for solving the CKdVsystem (1) can be achieved by using the product approxi-mation technique where we used special approximation tothe nonlinear terms in the differential system In order toapply this technique we rewrite the CKdV in the followingform

120597119906

120597119905= 119886(

1205973119906

1205971199093+ 3

120597 (1199062)

120597119909) + 119887

120597 (V2)120597119909

(34)

120597V120597119905

= minus1205973V1205971199093

minus 3119906120597V120597119909

(35)

The product approximation technique is used to approx-imate the nonlinear terms in (34) in the following man-ner

1199062(119909 119905) =

119873

sum

119898=1

1198802

119898(119905) 120601119898(119909)

V2 (119909 119905) =119873

sum

119898=1

1198812

119898(119905) 120601119898(119909)

(36)

By using the same procedure in deriving Scheme 1 and theapproximation (37) we can obtain after some manipulationsthe following scheme

1199011[(119880119899+1

119898minus2+ 26119880

119899+1

119898minus1+ 66119880

119899+1

119898+ 26119880

119899+1

119898+1+ 119880119899+1

119898+2)

minus (119880119899

119898minus2+ 26119880

119899

119898minus1+ 66119880

119899

119898+ 26119880

119899

119898+1+ 119880119899

119898+2) ]

minus 1199012[(119880lowast

119898+2)2+ 10(119880

lowast

119898+1)2minus 10(119880

lowast

119898minus1)2minus (119880lowast

119898minus2)2]

Abstract and Applied Analysis 5

Table 1 Petrov-Galerkin method

119879 119871infin(119880) 119871

infin(119881) 119868

11198682

1198683

0 0000000 0000000 2000000 05000000 01125292 0000015 0000004 2000000 05000000 01125294 0000021 0000005 2000000 05000000 01125296 0000023 0000006 2000000 05000000 01125298 0000024 0000007 2000000 05000000 011252910 0000025 0000008 2000000 05000000 0112529

minus 1199013(119880lowast

119898+2minus 2119880lowast

119898+1+ 2119880lowast

119898minus1minus 119880lowast

119898minus2)

minus 1199014[(119881lowast

119898+2)2+ 10(119881

lowast

119898+1)2minus 10(119881

lowast

119898minus1)2minus (119881lowast

119898minus2)2] = 0

(37)

1199011[(119881119899+1

119898minus2+ 26119881

119899+1

119898minus1+ 66119881

119899+1

119898+ 26119881

119899+1

119898+1+ 119881119899+1

119898+2)

minus (119881119899

119898minus2+ 26119881

119899

119898minus1+ 66119881

119899

119898+ 26119881

119899

119898+1+ 119881119899

119898+2) ]

+ 1199015(119881lowast

119898+2minus 2119881lowast

119898+1+ 2119881lowast

119898minus1minus 119881lowast

119898minus2)

+ 1199016[minus (119880lowast

119898minus2+ 4119880lowast

119898minus1) 119881lowast

119898minus2

+ (119880lowast

119898minus2minus 18119880

lowast

119898minus1minus 33119880

lowast

119898) 119881lowast

119898minus1

+ 22 (119880lowast

119898minus1minus 119880lowast

119898+1) 119881lowast

119898

+ (33119880lowast

119898+ 18119880

lowast

119898+1minus 119880lowast

119898+2) 119881lowast

119898+1

+ (4119880lowast

119898+1+ 119880lowast

119898+2) 119881lowast

119898+2] = 0

(38)

where

119880lowast

119898= 119880119899+1

119898+ 119880119899

119898 119881

lowast

119898= 119881119899+1

119898+ 119881119899

119898

1199011=

1

80 119901

2=3119886119896

64ℎ 119901

3=3119886119896

8ℎ3

1199014=

119887119896

64ℎ 119901

5=

3119896

8ℎ3 119901

6=

3119896

320ℎ

(39)

Again the method is unconditionally stable accordingto von Neumann stability analysis and it is of secondorder in both directions The scheme produced a nonlinearblock pentadiagonal system and its solution obtained usingNewtonrsquosmethodWehave noticed that the accuracy has beenimproved in the first equation (38) as we will see in the nextsection We will denote the scheme obtained by using theproduct approximation technique by Scheme 2

5 Numerical Results

To gain insight into the performance of the proposedschemes we perform different numerical tests like sin-gle soliton two and three solitons interaction and birthof solitons The conservation properties of the proposedschemes are examined by calculating 119868

1 1198682 and 119868

3using

the trapezoidal rule The 119871infin(119906) and 119871

infin(V) error norms are

defined as

119871infin(119906) =

1003817100381710038171003817U119899minus u1198991003817100381710038171003817infin = max

119898

1003816100381610038161003816119880119899

119898minus 119906119899

119898

1003816100381610038161003816

119871infin (V) =

1003817100381710038171003817V119899minus k1198991003817100381710038171003817infin = max

119898

1003816100381610038161003816119881119899

119898minus V119899119898

1003816100381610038161003816

(40)

are used to examine the accuracy of the proposed schemes

51 Single Soliton In the first test we choose the initialconditions

119906 (119909 0) = 21205822sech2 (120585) V (119909 0) =

1

2radic119908sech (120585)

120585 = 120582119909 +1

2 log (119908) 119908 =

minus119887

8 (4119886 + 1) 1205824

(41)

which represents a single soliton solution at 119905 = 0 To studythe behavior of numerical solution using Scheme 1 andScheme 2 we choose the set of parameters as ℎ = 005119896 = 001 tol = 10

minus7 119909119871= minus25 119909

119877= 25 119886 = minus0125

119887 = minus3 and 120582 = 05 The conserved quantities and theerror norms 119871

infin(119880) 119871

infin(119881) are displayed in Tables 1 and 2

for Scheme 1 and Scheme 2 respectively It is clear from thesetables that our schemes are highly accurate In addition theschemes preserve the conserved quantities exactly during theevolution of the numerical solution from 119905 = 0 to 119905 = 10 Theexecution time required to produce Table 1 is 2328 secondand 2171 second to produce Table 2 We have noticed thatScheme 2 has an upper hand over Scheme 1 with respect toaccuracy and CPU time In Figures 1 and 2 we display thenumerical solution of 119880119899

119898and 119881119899

119898for 119905 = 0 1 2 20

By choosing the set of values 119896 = 001 119886 = 05 119887 = minus30120582 = 05 and 119905 = 1 we perform a comparison of Scheme 1 andScheme 2 with Ismail [6] and we display this in Table 3 wecan easily see that the three methods produce highly accurateresults with some credits for collocation method

52 Two Solitons Interaction To study the interaction of twosolitons we choose the initial conditions as

119906 (119909 0) =

2

sum

119895=1

119906119895(119909 0) V (119909 0) =

2

sum

119895=1

V119895(119909 0) (42)

6 Abstract and Applied Analysis

Table 2 Product approximation technique

119879 119871infin(119880) 119871

infin(119881) 119868

11198682

1198683

0 0000000 0000000 2000000 05000000 01125292 0000004 0000005 2000000 05000000 01125294 0000008 0000007 2000000 05000000 01125296 0000010 0000009 2000000 05000000 01125298 0000013 0000012 2000000 05000000 011252910 0000014 0000013 2000000 05000000 0112529

Table 3 Comparison of three methods with 119886 = 05 119887 = minus30 120582 = 05 119896 = 001 and 119905 = 1

ℎPetrov-Galerkin Product approximation Collocation (Ismail [6])

119871infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU

01 0000051 0000027 0218 0000014 0000019 0203 0000000 0000003 0188005 0000013 0000007 0343 0000009 0000008 0328 0000000 0000003 03120025 0000003 0000004 0625 0000002 0000004 0609 0000000 0000003 0578

010

2030

4050

05

1015

200

01

02

03

04

05

x

t

u

Figure 1 Single soliton 119880 119886 = minus0125 119887 = minus30 and 120582 = 05

010

2030

4050

05

1015

200

00200400600801012014016

xt

v

Figure 2 Single soliton 119881 119886 = minus0125 119887 = minus30 and 120582 = 05

Table 4 Two solitons interaction (conserved quantities)

119879 1198681

1198682

1198683

000 6400000 minus3243013 minus2584118200 6399985 minus3243298 minus2586545400 6400034 minus3243016 minus2584155600 6399962 minus3243017 minus2584203800 6400234 minus3242978 minus2583492

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2

(43)

which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582

1= 10 120582

2= 06 119910

1= 10

and 1199102= 30 In Table 4 we present the conserved quantities

during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]

To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582

1= 10 120582

2= 06

Abstract and Applied Analysis 7

020

4060

80100

0

20

40

60

80

0123

minus1

x

t

u

Figure 3 Numerical solution 119880119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

020

4060

80100

010

2030

4050

6070

80

012

x

t

v

minus1

Figure 4 Numerical solution 119881119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

x

t

Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)

020

4060

80100

010

2030

4050

0

1

2

3

4

5

xt

u

minus1

Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =

0495

020

4060

80100

010

2030

4050

0

05

1

15

2

25

xt

v

minus05

Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =

0495

1199101= 10 119910

2= 30 119888 = 0 119889 = 100 ℎ = 005 and 119896 = 001 Our

results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12

53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform

119906 (119909 0) =

3

sum

119895=1

119906119895(119909 0) V (119909 0) =

3

sum

119895=1

V119895(119909 0) (44)

8 Abstract and Applied Analysis

020

4060

80100

010

2030

4050

6070

80

0123

x

t

u

minus1

Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

020

4060

80100

020

4060

80

0020406081

121416

xt

v

minus02

Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2 3

(45)

In this test we choose the set of parameters 119909119871= 0 119909

119877=

80 ℎ = 005 119896 = 001 119886 = 05 119887 = minus30 1205821= 10 120582

2= 06

1205823= 03 119910

1= 10 119910

2= 30 and 119910

3= 50

The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906

1interacts with the other two

solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario

020

4060

80

05

1015

2025

024

minus40

minus20

minus2

x

t

u

Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30

Table 5 Three solitons interaction the conserved quantities

119879 1198681

1198682

1198683

0 7600000 minus3351855 minus261028320 7600042 minus3352227 minus261314940 7600800 minus3352034 minus261134760 7599367 minus3351547 minus260897080 7600545 minus3351562 minus2602090

Table 6 Birth of solitons with 119886 = 05 and 119887 = minus30

119879 1198681

1198682

00 17724343 minus1253314250 17723816 minus12532956100 17723352 minus12530116150 17722782 minus12529239200 17722217 minus12529013250 17721734 minus12528983

54 Birth of Solitons In this test we choose the initial condi-tion

119906 (119909 0) = exp (minus0011199092)

V (119909 0) = exp (minus0011199092) (46)

with the following set of parameters 119886 = 05 119887 = minus30 0 le

119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved

6 Conclusion

In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm

Abstract and Applied Analysis 9

during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12

To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981

[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003

[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004

[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981

[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997

[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009

[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004

[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008

[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008

[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007

[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004

[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ

Equ pp 485ndash568 ElsevierNorth-Holland Amsterdam TheNetherlands 2008

[15] S Kutluay and Y Ucar ldquoA quadratic B-spline Galerkin approachfor solving a coupled KdV equationrdquo Mathematical Modellingand Analysis vol 18 no 1 pp 103ndash121 2013

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Direct calculation of the inner products given in (12)will lead us to the following system of first order ordinarydifferential system (ODES)

1

80(119898minus2

+ 26119898minus1

+ 66119898+ 26

119898+1+ 119898+2

)

minus6119886

80ℎ119865 (119880119898 119880119898) minus

2119887

80ℎ119865 (119881119898 119881119898)

minus3119886

4ℎ3(119880119898+2

minus 2119880119898+1

+ 2119880119898minus1

minus 119880119898minus2

) = 0

1

80(119898minus2

+ 26119898minus1

+ 66119898+ 26

119898+1+ 119898+2

)

+3

4ℎ3(119881119898+2

minus 2119881119898+1

+ 2119881119898minus1

minus 119881119898minus2

)

+3

80ℎ119865 (119880119898 119881119898) = 0

(14)

where119865 (119880119898 119881119898) = (minus119881

119898minus2+ 119881119898minus1

) (119880119898minus2

+ 4119880119898minus1

)

+ (minus119881119898minus1

+ 119881119898) (22119880

119898minus1+ 33119880

119898)

+ (minus119881119898+ 119881119898+1

) (33119880119898+ 22119880

119898+1)

+ (minus119881119898+1

+ 119881119898+2

) (4119880119898+1

+ 119880119898+2

)

(15)

Now to the ODES we assume 119880119899119898and 119881119899

119898to be the fully

discrete approximation to the exact solutions 119906(119909119898 119905119899) and

V(119909119898 119905119899) where 119905

119899= 119899119896 and 119896 is the time step size By using

the finite difference approximation

119898=119880119899+1

119898minus 119880119899

119898

119896 119880

119898=119880119899+1

119898+ 119880119899

119898

2= 119880lowast

119898=119881119899+1

119898minus 119881119899

119898

119896 119881

119898=119881119899+1

119898+ 119881119899

119898

2= 119881lowast

(16)

the ODES (14) will be reduced to the implicit nonlinear finitedifference scheme

(119880119899+1

119898minus2+ 26119880

119899+1

119898minus1+ 66119880

119899+1

119898+ 26119880

119899+1

119898+1+ 119880119899+1

119898+2)

minus (119880119899

119898minus2+ 26119880

119899

119898minus1+ 66119880

119899

119898+ 26119880

119899

119898+1+ 119880119899

119898+2)

minus 1199011(119880lowast

119898+2minus 2119880lowast

119898+1+ 2119880lowast

119898minus1minus 119880lowast

119898minus2)

+ 1199012119865 (119880lowast 119880lowast) + 1199013119865 (119881lowast 119881lowast) = 0

(17)

(119881119899+1

119898minus2+ 26119881

119899+1

119898minus1+ 66119881

119899+1

119898+ 26119881

119899+1

119898+1+ 119881119899+1

119898+2)

minus (119881119899

119898minus2+ 26119881

119899

119898minus1+ 66119881

119899

119898+ 26119881

119899

119898+1+ 119881119899

119898+2)

+ 1199014(119881lowast

119898+2minus 2119881lowast

119898+1+ 2119881lowast

119898minus1minus 119881lowast

119898minus2)

+ 1199015119865 (119880lowast 119881lowast) = 0 119898 = 1 2 119873

(18)

1199011=minus60119886119896

ℎ3 119901

2=minus6119886119896

ℎ 119901

3=minus2119887119896

ℎ

1199014=60119896

ℎ3 119901

5=3119896

ℎ

(19)

The system (17)-(18) is a nonlinear block penta diagonalsystem in the unknown vectors U119899+1 and V119899+1 and is solvedusing Newtonrsquos methodWe denote this method by Scheme 1

3 Analysis of the Method

In this section we will discuss the stability and the accuracyof Scheme 1

31 Stability of the Scheme To study the stability of Scheme 1von Neumann stability analysis will be used and this can beonly applied for linear finite difference schemes accordinglywe consider the linear version of the proposed method

(119880119899+1

119898minus2+ 26119880

119899+1

119898minus1+ 66119880

119899+1

119898+ 26119880

119899+1

119898+1+ 119880119899+1

119898+2)

minus (119880119899

119898minus2+ 26119880

119899

119898minus1+ 66119880

119899

119898+ 26119880

119899

119898+1+ 119880119899

119898+2)

+ 1199011(119880lowast

119898+2minus 2119880lowast

119898+1+ 2119880lowast

119898minus1minus 119880lowast

119898minus2)

+ 1199012119865 (119880119880

lowast) + 1199013119865 (119881119881

lowast) = 0

(119881119899+1

119898minus2+ 26119881

119899+1

119898minus1+ 66119881

119899+1

119898+ 26119881

119899+1

119898+1+ 119881119899+1

119898+2)

minus (119881119899

119898minus2+ 26119881

119899

119898minus1+ 66119881

119899

119898+ 26119881

119899

119898+1+ 119881119899

119898+2)

+ 1199014(119881lowast

119898+2minus 2119881lowast

119898+1+ 2119881lowast

119898minus1minus 119881lowast

119898minus2)

+ 1199015119865 (119880119881

lowast) = 0

(20)

where

119865 (119880119881) = 119880 [119881119898+2

+ 10119881119898+1

minus 10119881119898minus1

minus 119881119898minus2

] (21)

1199011015840

2= 51199012 11990110158403= 51199013 and 1199011015840

5= 51199015 119880 and 119881 are assumed

to be constant on the whole range We assume the solution ofthe linearized scheme (20) to be of the form

119880119899

119898= 119880119899119890119894120573119898ℎ

119881119899

119898= 119881119899119890119894120573119898ℎ

(22)

where120573 is a real constant Direct substitution of (22) into (20)will lead us to the following system

[1205741+ 11989411990111205742+ 1198941199011015840

21198801205743]119880119899+1

+ 1198941199011015840

31198811205743119881119899+1

= [1205741minus 11989411990111205742minus 1198941199011015840

21198801205743]119880119899minus 1198941199011015840

31198811205743119881119899

[1205741+ 11989411990141205742+ 1198941199011015840

51198801205743]119881119899+1

= [1205741minus 11989411990141205742minus 1198941199011015840

51198801205743]119881119899

(23)

where

1205741= 66 + 2 cos (2120573ℎ) + 52 cos (120573ℎ)

1205742= 2 sin (2120573ℎ) minus 4 sin (120573ℎ)

1205743= 2 sin (2120573ℎ) + 20 sin (120573ℎ)

(24)

The system (23) can be written in a matrix vector form as

Ψ119899+1

= 119861Ψ119899 (25)

4 Abstract and Applied Analysis

where Ψ = [119880 119881]119905 and 119861 is the (2 times 2)matrix Consider

119861 = [1205741+ 1198941198881

1198882

0 1205741+ 1198941198883

]

minus1

[1205741minus 1198941198881

minus1198882

0 1205741minus 1198941198883

] (26)

where

1198881= 11990111205742+ 1199011015840

21198801205743 119888

2= 1199011015840

31198811205743

1198883= 11990141205742+ 1199011015840

51198801205743

(27)

von Neumann stability condition for the system (25) statesthat for the scheme to be stable the maximum modulus ofthe eigenvalues of the amplification matrix 119861 should be lessthan or equal to one The eigenvalues of the matrix 119861 are

1205821=1205741minus 1198941198881

1205741+ 1198941198881

1205822=1205741minus 1198941198883

1205741+ 1198941198883

(28)

which hasmodulus equal to one and hence the scheme underconsideration is unconditionally stable in the linearizedsense

32 Accuracy of the Scheme In order to study the accuracy ofthe scheme we replace the numerical solution 119880119899

119898and 119881119899

119898in

(17) and (18) by the exact solutions 119906119899119898and V119899119898 By making use

of the following Taylor series expansions at the point (119909119898 119905119899)

1

119896[(119906119899+1

119898minus2+ 26119906

119899+1

119898minus1+ 66119906

119899+1

119898+ 26119906

119899+1

119898+1+ 119906119899+1

119898+2)

minus (119906119899

119898minus2+ 26119906

119899

119898minus1+ 66119906

119899

119898+ 26119906

119899

119898+1+ 119906119899

119898+2) ]

=3

2119906119905+3119896

4119906119905119905+3ℎ2

8119906119909119909119905

+1198962

4119906119905119905119905+ sdot sdot sdot

(29)

3

8ℎ3[119906119899+1

119898+2minus 2119906119899+1

119898+1+ 2119906119899+1

119898minus1minus 119906119899+1

119898minus2

+ 119906119899

119898+2minus 2119906119899

119898+1+ 2119906119899

119898minus1minus 119906119899

119898minus2]

=3

2119906119909119909119909

+3119896

4119906119909119909119909119905

+3ℎ2

8119906119909119909119909119909119909

+31198962

8119906119909119909119909119905119905

+ sdot sdot sdot

(30)

1

80ℎ119865 (119906lowast 119906lowast) =

3

2119906119906119909+3119896

4(119906119906119909)119905

+ ℎ2[3

8119906119906119909119909119909

+7

8119906119909119906119909119909]

+31198962

8[119906119909119906119905119905+ 119906119905119906119909119905+ 119906119906119909119905119905] + sdot sdot sdot

(31)

1

80ℎ119865 (119906lowast Vlowast) =

3

2119906V119909+3119896

4(119906V119909)119905

+ ℎ2[3

8119906V119909119909119909

+3

8V119909119906119909119909+1

2119906119909V119909119909]

+31198962

8[V119909119906119905119905+ 119906119905V119909119905+ 119906V119909119905119905] + sdot sdot sdot

(32)

and the CKdV system (1) we obtain the local truncation error(LTE) for the proposed scheme

LTE (17) = minus3ℎ2

2119906119909119906119909119909minus119887ℎ2

2V119906119909119909minus1198962

8119906119905119905119905

+91198962119886

4119906119905119906119909119905minus31198871198962

4V119905V119909119905+ sdot sdot sdot

LTE (18) = minus3ℎ2

2119906119909V119909119909minus1198962

8119906119905119905119905minus31198962

8119906119905V119909119905+ sdot sdot sdot

(33)

and hence the scheme is of second order in both directionsspace and time

4 Product Approximation Technique

A modified Petrov-Galerkin method for solving the CKdVsystem (1) can be achieved by using the product approxi-mation technique where we used special approximation tothe nonlinear terms in the differential system In order toapply this technique we rewrite the CKdV in the followingform

120597119906

120597119905= 119886(

1205973119906

1205971199093+ 3

120597 (1199062)

120597119909) + 119887

120597 (V2)120597119909

(34)

120597V120597119905

= minus1205973V1205971199093

minus 3119906120597V120597119909

(35)

The product approximation technique is used to approx-imate the nonlinear terms in (34) in the following man-ner

1199062(119909 119905) =

119873

sum

119898=1

1198802

119898(119905) 120601119898(119909)

V2 (119909 119905) =119873

sum

119898=1

1198812

119898(119905) 120601119898(119909)

(36)

By using the same procedure in deriving Scheme 1 and theapproximation (37) we can obtain after some manipulationsthe following scheme

1199011[(119880119899+1

119898minus2+ 26119880

119899+1

119898minus1+ 66119880

119899+1

119898+ 26119880

119899+1

119898+1+ 119880119899+1

119898+2)

minus (119880119899

119898minus2+ 26119880

119899

119898minus1+ 66119880

119899

119898+ 26119880

119899

119898+1+ 119880119899

119898+2) ]

minus 1199012[(119880lowast

119898+2)2+ 10(119880

lowast

119898+1)2minus 10(119880

lowast

119898minus1)2minus (119880lowast

119898minus2)2]

Abstract and Applied Analysis 5

Table 1 Petrov-Galerkin method

119879 119871infin(119880) 119871

infin(119881) 119868

11198682

1198683

0 0000000 0000000 2000000 05000000 01125292 0000015 0000004 2000000 05000000 01125294 0000021 0000005 2000000 05000000 01125296 0000023 0000006 2000000 05000000 01125298 0000024 0000007 2000000 05000000 011252910 0000025 0000008 2000000 05000000 0112529

minus 1199013(119880lowast

119898+2minus 2119880lowast

119898+1+ 2119880lowast

119898minus1minus 119880lowast

119898minus2)

minus 1199014[(119881lowast

119898+2)2+ 10(119881

lowast

119898+1)2minus 10(119881

lowast

119898minus1)2minus (119881lowast

119898minus2)2] = 0

(37)

1199011[(119881119899+1

119898minus2+ 26119881

119899+1

119898minus1+ 66119881

119899+1

119898+ 26119881

119899+1

119898+1+ 119881119899+1

119898+2)

minus (119881119899

119898minus2+ 26119881

119899

119898minus1+ 66119881

119899

119898+ 26119881

119899

119898+1+ 119881119899

119898+2) ]

+ 1199015(119881lowast

119898+2minus 2119881lowast

119898+1+ 2119881lowast

119898minus1minus 119881lowast

119898minus2)

+ 1199016[minus (119880lowast

119898minus2+ 4119880lowast

119898minus1) 119881lowast

119898minus2

+ (119880lowast

119898minus2minus 18119880

lowast

119898minus1minus 33119880

lowast

119898) 119881lowast

119898minus1

+ 22 (119880lowast

119898minus1minus 119880lowast

119898+1) 119881lowast

119898

+ (33119880lowast

119898+ 18119880

lowast

119898+1minus 119880lowast

119898+2) 119881lowast

119898+1

+ (4119880lowast

119898+1+ 119880lowast

119898+2) 119881lowast

119898+2] = 0

(38)

where

119880lowast

119898= 119880119899+1

119898+ 119880119899

119898 119881

lowast

119898= 119881119899+1

119898+ 119881119899

119898

1199011=

1

80 119901

2=3119886119896

64ℎ 119901

3=3119886119896

8ℎ3

1199014=

119887119896

64ℎ 119901

5=

3119896

8ℎ3 119901

6=

3119896

320ℎ

(39)

Again the method is unconditionally stable accordingto von Neumann stability analysis and it is of secondorder in both directions The scheme produced a nonlinearblock pentadiagonal system and its solution obtained usingNewtonrsquosmethodWehave noticed that the accuracy has beenimproved in the first equation (38) as we will see in the nextsection We will denote the scheme obtained by using theproduct approximation technique by Scheme 2

5 Numerical Results

To gain insight into the performance of the proposedschemes we perform different numerical tests like sin-gle soliton two and three solitons interaction and birthof solitons The conservation properties of the proposedschemes are examined by calculating 119868

1 1198682 and 119868

3using

the trapezoidal rule The 119871infin(119906) and 119871

infin(V) error norms are

defined as

119871infin(119906) =

1003817100381710038171003817U119899minus u1198991003817100381710038171003817infin = max

119898

1003816100381610038161003816119880119899

119898minus 119906119899

119898

1003816100381610038161003816

119871infin (V) =

1003817100381710038171003817V119899minus k1198991003817100381710038171003817infin = max

119898

1003816100381610038161003816119881119899

119898minus V119899119898

1003816100381610038161003816

(40)

are used to examine the accuracy of the proposed schemes

51 Single Soliton In the first test we choose the initialconditions

119906 (119909 0) = 21205822sech2 (120585) V (119909 0) =

1

2radic119908sech (120585)

120585 = 120582119909 +1

2 log (119908) 119908 =

minus119887

8 (4119886 + 1) 1205824

(41)

which represents a single soliton solution at 119905 = 0 To studythe behavior of numerical solution using Scheme 1 andScheme 2 we choose the set of parameters as ℎ = 005119896 = 001 tol = 10

minus7 119909119871= minus25 119909

119877= 25 119886 = minus0125

119887 = minus3 and 120582 = 05 The conserved quantities and theerror norms 119871

infin(119880) 119871

infin(119881) are displayed in Tables 1 and 2

for Scheme 1 and Scheme 2 respectively It is clear from thesetables that our schemes are highly accurate In addition theschemes preserve the conserved quantities exactly during theevolution of the numerical solution from 119905 = 0 to 119905 = 10 Theexecution time required to produce Table 1 is 2328 secondand 2171 second to produce Table 2 We have noticed thatScheme 2 has an upper hand over Scheme 1 with respect toaccuracy and CPU time In Figures 1 and 2 we display thenumerical solution of 119880119899

119898and 119881119899

119898for 119905 = 0 1 2 20

By choosing the set of values 119896 = 001 119886 = 05 119887 = minus30120582 = 05 and 119905 = 1 we perform a comparison of Scheme 1 andScheme 2 with Ismail [6] and we display this in Table 3 wecan easily see that the three methods produce highly accurateresults with some credits for collocation method

52 Two Solitons Interaction To study the interaction of twosolitons we choose the initial conditions as

119906 (119909 0) =

2

sum

119895=1

119906119895(119909 0) V (119909 0) =

2

sum

119895=1

V119895(119909 0) (42)

6 Abstract and Applied Analysis

Table 2 Product approximation technique

119879 119871infin(119880) 119871

infin(119881) 119868

11198682

1198683

0 0000000 0000000 2000000 05000000 01125292 0000004 0000005 2000000 05000000 01125294 0000008 0000007 2000000 05000000 01125296 0000010 0000009 2000000 05000000 01125298 0000013 0000012 2000000 05000000 011252910 0000014 0000013 2000000 05000000 0112529

Table 3 Comparison of three methods with 119886 = 05 119887 = minus30 120582 = 05 119896 = 001 and 119905 = 1

ℎPetrov-Galerkin Product approximation Collocation (Ismail [6])

119871infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU

01 0000051 0000027 0218 0000014 0000019 0203 0000000 0000003 0188005 0000013 0000007 0343 0000009 0000008 0328 0000000 0000003 03120025 0000003 0000004 0625 0000002 0000004 0609 0000000 0000003 0578

010

2030

4050

05

1015

200

01

02

03

04

05

x

t

u

Figure 1 Single soliton 119880 119886 = minus0125 119887 = minus30 and 120582 = 05

010

2030

4050

05

1015

200

00200400600801012014016

xt

v

Figure 2 Single soliton 119881 119886 = minus0125 119887 = minus30 and 120582 = 05

Table 4 Two solitons interaction (conserved quantities)

119879 1198681

1198682

1198683

000 6400000 minus3243013 minus2584118200 6399985 minus3243298 minus2586545400 6400034 minus3243016 minus2584155600 6399962 minus3243017 minus2584203800 6400234 minus3242978 minus2583492

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2

(43)

which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582

1= 10 120582

2= 06 119910

1= 10

and 1199102= 30 In Table 4 we present the conserved quantities

during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]

To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582

1= 10 120582

2= 06

Abstract and Applied Analysis 7

020

4060

80100

0

20

40

60

80

0123

minus1

x

t

u

Figure 3 Numerical solution 119880119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

020

4060

80100

010

2030

4050

6070

80

012

x

t

v

minus1

Figure 4 Numerical solution 119881119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

x

t

Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)

020

4060

80100

010

2030

4050

0

1

2

3

4

5

xt

u

minus1

Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =

0495

020

4060

80100

010

2030

4050

0

05

1

15

2

25

xt

v

minus05

Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =

0495

1199101= 10 119910

2= 30 119888 = 0 119889 = 100 ℎ = 005 and 119896 = 001 Our

results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12

53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform

119906 (119909 0) =

3

sum

119895=1

119906119895(119909 0) V (119909 0) =

3

sum

119895=1

V119895(119909 0) (44)

8 Abstract and Applied Analysis

020

4060

80100

010

2030

4050

6070

80

0123

x

t

u

minus1

Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

020

4060

80100

020

4060

80

0020406081

121416

xt

v

minus02

Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2 3

(45)

In this test we choose the set of parameters 119909119871= 0 119909

119877=

80 ℎ = 005 119896 = 001 119886 = 05 119887 = minus30 1205821= 10 120582

2= 06

1205823= 03 119910

1= 10 119910

2= 30 and 119910

3= 50

The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906

1interacts with the other two

solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario

020

4060

80

05

1015

2025

024

minus40

minus20

minus2

x

t

u

Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30

Table 5 Three solitons interaction the conserved quantities

119879 1198681

1198682

1198683

0 7600000 minus3351855 minus261028320 7600042 minus3352227 minus261314940 7600800 minus3352034 minus261134760 7599367 minus3351547 minus260897080 7600545 minus3351562 minus2602090

Table 6 Birth of solitons with 119886 = 05 and 119887 = minus30

119879 1198681

1198682

00 17724343 minus1253314250 17723816 minus12532956100 17723352 minus12530116150 17722782 minus12529239200 17722217 minus12529013250 17721734 minus12528983

54 Birth of Solitons In this test we choose the initial condi-tion

119906 (119909 0) = exp (minus0011199092)

V (119909 0) = exp (minus0011199092) (46)

with the following set of parameters 119886 = 05 119887 = minus30 0 le

119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved

6 Conclusion

In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm

Abstract and Applied Analysis 9

during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12

To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981

[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003

[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004

[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981

[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997

[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009

[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004

[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008

[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008

[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007

[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004

[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ

Equ pp 485ndash568 ElsevierNorth-Holland Amsterdam TheNetherlands 2008

[15] S Kutluay and Y Ucar ldquoA quadratic B-spline Galerkin approachfor solving a coupled KdV equationrdquo Mathematical Modellingand Analysis vol 18 no 1 pp 103ndash121 2013

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

where Ψ = [119880 119881]119905 and 119861 is the (2 times 2)matrix Consider

119861 = [1205741+ 1198941198881

1198882

0 1205741+ 1198941198883

]

minus1

[1205741minus 1198941198881

minus1198882

0 1205741minus 1198941198883

] (26)

where

1198881= 11990111205742+ 1199011015840

21198801205743 119888

2= 1199011015840

31198811205743

1198883= 11990141205742+ 1199011015840

51198801205743

(27)

von Neumann stability condition for the system (25) statesthat for the scheme to be stable the maximum modulus ofthe eigenvalues of the amplification matrix 119861 should be lessthan or equal to one The eigenvalues of the matrix 119861 are

1205821=1205741minus 1198941198881

1205741+ 1198941198881

1205822=1205741minus 1198941198883

1205741+ 1198941198883

(28)

which hasmodulus equal to one and hence the scheme underconsideration is unconditionally stable in the linearizedsense

32 Accuracy of the Scheme In order to study the accuracy ofthe scheme we replace the numerical solution 119880119899

119898and 119881119899

119898in

(17) and (18) by the exact solutions 119906119899119898and V119899119898 By making use

of the following Taylor series expansions at the point (119909119898 119905119899)

1

119896[(119906119899+1

119898minus2+ 26119906

119899+1

119898minus1+ 66119906

119899+1

119898+ 26119906

119899+1

119898+1+ 119906119899+1

119898+2)

minus (119906119899

119898minus2+ 26119906

119899

119898minus1+ 66119906

119899

119898+ 26119906

119899

119898+1+ 119906119899

119898+2) ]

=3

2119906119905+3119896

4119906119905119905+3ℎ2

8119906119909119909119905

+1198962

4119906119905119905119905+ sdot sdot sdot

(29)

3

8ℎ3[119906119899+1

119898+2minus 2119906119899+1

119898+1+ 2119906119899+1

119898minus1minus 119906119899+1

119898minus2

+ 119906119899

119898+2minus 2119906119899

119898+1+ 2119906119899

119898minus1minus 119906119899

119898minus2]

=3

2119906119909119909119909

+3119896

4119906119909119909119909119905

+3ℎ2

8119906119909119909119909119909119909

+31198962

8119906119909119909119909119905119905

+ sdot sdot sdot

(30)

1

80ℎ119865 (119906lowast 119906lowast) =

3

2119906119906119909+3119896

4(119906119906119909)119905

+ ℎ2[3

8119906119906119909119909119909

+7

8119906119909119906119909119909]

+31198962

8[119906119909119906119905119905+ 119906119905119906119909119905+ 119906119906119909119905119905] + sdot sdot sdot

(31)

1

80ℎ119865 (119906lowast Vlowast) =

3

2119906V119909+3119896

4(119906V119909)119905

+ ℎ2[3

8119906V119909119909119909

+3

8V119909119906119909119909+1

2119906119909V119909119909]

+31198962

8[V119909119906119905119905+ 119906119905V119909119905+ 119906V119909119905119905] + sdot sdot sdot

(32)

and the CKdV system (1) we obtain the local truncation error(LTE) for the proposed scheme

LTE (17) = minus3ℎ2

2119906119909119906119909119909minus119887ℎ2

2V119906119909119909minus1198962

8119906119905119905119905

+91198962119886

4119906119905119906119909119905minus31198871198962

4V119905V119909119905+ sdot sdot sdot

LTE (18) = minus3ℎ2

2119906119909V119909119909minus1198962

8119906119905119905119905minus31198962

8119906119905V119909119905+ sdot sdot sdot

(33)

and hence the scheme is of second order in both directionsspace and time

4 Product Approximation Technique

A modified Petrov-Galerkin method for solving the CKdVsystem (1) can be achieved by using the product approxi-mation technique where we used special approximation tothe nonlinear terms in the differential system In order toapply this technique we rewrite the CKdV in the followingform

120597119906

120597119905= 119886(

1205973119906

1205971199093+ 3

120597 (1199062)

120597119909) + 119887

120597 (V2)120597119909

(34)

120597V120597119905

= minus1205973V1205971199093

minus 3119906120597V120597119909

(35)

The product approximation technique is used to approx-imate the nonlinear terms in (34) in the following man-ner

1199062(119909 119905) =

119873

sum

119898=1

1198802

119898(119905) 120601119898(119909)

V2 (119909 119905) =119873

sum

119898=1

1198812

119898(119905) 120601119898(119909)

(36)

By using the same procedure in deriving Scheme 1 and theapproximation (37) we can obtain after some manipulationsthe following scheme

1199011[(119880119899+1

119898minus2+ 26119880

119899+1

119898minus1+ 66119880

119899+1

119898+ 26119880

119899+1

119898+1+ 119880119899+1

119898+2)

minus (119880119899

119898minus2+ 26119880

119899

119898minus1+ 66119880

119899

119898+ 26119880

119899

119898+1+ 119880119899

119898+2) ]

minus 1199012[(119880lowast

119898+2)2+ 10(119880

lowast

119898+1)2minus 10(119880

lowast

119898minus1)2minus (119880lowast

119898minus2)2]

Abstract and Applied Analysis 5

Table 1 Petrov-Galerkin method

119879 119871infin(119880) 119871

infin(119881) 119868

11198682

1198683

0 0000000 0000000 2000000 05000000 01125292 0000015 0000004 2000000 05000000 01125294 0000021 0000005 2000000 05000000 01125296 0000023 0000006 2000000 05000000 01125298 0000024 0000007 2000000 05000000 011252910 0000025 0000008 2000000 05000000 0112529

minus 1199013(119880lowast

119898+2minus 2119880lowast

119898+1+ 2119880lowast

119898minus1minus 119880lowast

119898minus2)

minus 1199014[(119881lowast

119898+2)2+ 10(119881

lowast

119898+1)2minus 10(119881

lowast

119898minus1)2minus (119881lowast

119898minus2)2] = 0

(37)

1199011[(119881119899+1

119898minus2+ 26119881

119899+1

119898minus1+ 66119881

119899+1

119898+ 26119881

119899+1

119898+1+ 119881119899+1

119898+2)

minus (119881119899

119898minus2+ 26119881

119899

119898minus1+ 66119881

119899

119898+ 26119881

119899

119898+1+ 119881119899

119898+2) ]

+ 1199015(119881lowast

119898+2minus 2119881lowast

119898+1+ 2119881lowast

119898minus1minus 119881lowast

119898minus2)

+ 1199016[minus (119880lowast

119898minus2+ 4119880lowast

119898minus1) 119881lowast

119898minus2

+ (119880lowast

119898minus2minus 18119880

lowast

119898minus1minus 33119880

lowast

119898) 119881lowast

119898minus1

+ 22 (119880lowast

119898minus1minus 119880lowast

119898+1) 119881lowast

119898

+ (33119880lowast

119898+ 18119880

lowast

119898+1minus 119880lowast

119898+2) 119881lowast

119898+1

+ (4119880lowast

119898+1+ 119880lowast

119898+2) 119881lowast

119898+2] = 0

(38)

where

119880lowast

119898= 119880119899+1

119898+ 119880119899

119898 119881

lowast

119898= 119881119899+1

119898+ 119881119899

119898

1199011=

1

80 119901

2=3119886119896

64ℎ 119901

3=3119886119896

8ℎ3

1199014=

119887119896

64ℎ 119901

5=

3119896

8ℎ3 119901

6=

3119896

320ℎ

(39)

Again the method is unconditionally stable accordingto von Neumann stability analysis and it is of secondorder in both directions The scheme produced a nonlinearblock pentadiagonal system and its solution obtained usingNewtonrsquosmethodWehave noticed that the accuracy has beenimproved in the first equation (38) as we will see in the nextsection We will denote the scheme obtained by using theproduct approximation technique by Scheme 2

5 Numerical Results

To gain insight into the performance of the proposedschemes we perform different numerical tests like sin-gle soliton two and three solitons interaction and birthof solitons The conservation properties of the proposedschemes are examined by calculating 119868

1 1198682 and 119868

3using

the trapezoidal rule The 119871infin(119906) and 119871

infin(V) error norms are

defined as

119871infin(119906) =

1003817100381710038171003817U119899minus u1198991003817100381710038171003817infin = max

119898

1003816100381610038161003816119880119899

119898minus 119906119899

119898

1003816100381610038161003816

119871infin (V) =

1003817100381710038171003817V119899minus k1198991003817100381710038171003817infin = max

119898

1003816100381610038161003816119881119899

119898minus V119899119898

1003816100381610038161003816

(40)

are used to examine the accuracy of the proposed schemes

51 Single Soliton In the first test we choose the initialconditions

119906 (119909 0) = 21205822sech2 (120585) V (119909 0) =

1

2radic119908sech (120585)

120585 = 120582119909 +1

2 log (119908) 119908 =

minus119887

8 (4119886 + 1) 1205824

(41)

which represents a single soliton solution at 119905 = 0 To studythe behavior of numerical solution using Scheme 1 andScheme 2 we choose the set of parameters as ℎ = 005119896 = 001 tol = 10

minus7 119909119871= minus25 119909

119877= 25 119886 = minus0125

119887 = minus3 and 120582 = 05 The conserved quantities and theerror norms 119871

infin(119880) 119871

infin(119881) are displayed in Tables 1 and 2

for Scheme 1 and Scheme 2 respectively It is clear from thesetables that our schemes are highly accurate In addition theschemes preserve the conserved quantities exactly during theevolution of the numerical solution from 119905 = 0 to 119905 = 10 Theexecution time required to produce Table 1 is 2328 secondand 2171 second to produce Table 2 We have noticed thatScheme 2 has an upper hand over Scheme 1 with respect toaccuracy and CPU time In Figures 1 and 2 we display thenumerical solution of 119880119899

119898and 119881119899

119898for 119905 = 0 1 2 20

By choosing the set of values 119896 = 001 119886 = 05 119887 = minus30120582 = 05 and 119905 = 1 we perform a comparison of Scheme 1 andScheme 2 with Ismail [6] and we display this in Table 3 wecan easily see that the three methods produce highly accurateresults with some credits for collocation method

52 Two Solitons Interaction To study the interaction of twosolitons we choose the initial conditions as

119906 (119909 0) =

2

sum

119895=1

119906119895(119909 0) V (119909 0) =

2

sum

119895=1

V119895(119909 0) (42)

6 Abstract and Applied Analysis

Table 2 Product approximation technique

119879 119871infin(119880) 119871

infin(119881) 119868

11198682

1198683

0 0000000 0000000 2000000 05000000 01125292 0000004 0000005 2000000 05000000 01125294 0000008 0000007 2000000 05000000 01125296 0000010 0000009 2000000 05000000 01125298 0000013 0000012 2000000 05000000 011252910 0000014 0000013 2000000 05000000 0112529

Table 3 Comparison of three methods with 119886 = 05 119887 = minus30 120582 = 05 119896 = 001 and 119905 = 1

ℎPetrov-Galerkin Product approximation Collocation (Ismail [6])

119871infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU

01 0000051 0000027 0218 0000014 0000019 0203 0000000 0000003 0188005 0000013 0000007 0343 0000009 0000008 0328 0000000 0000003 03120025 0000003 0000004 0625 0000002 0000004 0609 0000000 0000003 0578

010

2030

4050

05

1015

200

01

02

03

04

05

x

t

u

Figure 1 Single soliton 119880 119886 = minus0125 119887 = minus30 and 120582 = 05

010

2030

4050

05

1015

200

00200400600801012014016

xt

v

Figure 2 Single soliton 119881 119886 = minus0125 119887 = minus30 and 120582 = 05

Table 4 Two solitons interaction (conserved quantities)

119879 1198681

1198682

1198683

000 6400000 minus3243013 minus2584118200 6399985 minus3243298 minus2586545400 6400034 minus3243016 minus2584155600 6399962 minus3243017 minus2584203800 6400234 minus3242978 minus2583492

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2

(43)

which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582

1= 10 120582

2= 06 119910

1= 10

and 1199102= 30 In Table 4 we present the conserved quantities

during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]

To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582

1= 10 120582

2= 06

Abstract and Applied Analysis 7

020

4060

80100

0

20

40

60

80

0123

minus1

x

t

u

Figure 3 Numerical solution 119880119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

020

4060

80100

010

2030

4050

6070

80

012

x

t

v

minus1

Figure 4 Numerical solution 119881119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

x

t

Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)

020

4060

80100

010

2030

4050

0

1

2

3

4

5

xt

u

minus1

Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =

0495

020

4060

80100

010

2030

4050

0

05

1

15

2

25

xt

v

minus05

Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =

0495

1199101= 10 119910

2= 30 119888 = 0 119889 = 100 ℎ = 005 and 119896 = 001 Our

results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12

53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform

119906 (119909 0) =

3

sum

119895=1

119906119895(119909 0) V (119909 0) =

3

sum

119895=1

V119895(119909 0) (44)

8 Abstract and Applied Analysis

020

4060

80100

010

2030

4050

6070

80

0123

x

t

u

minus1

Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

020

4060

80100

020

4060

80

0020406081

121416

xt

v

minus02

Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2 3

(45)

In this test we choose the set of parameters 119909119871= 0 119909

119877=

80 ℎ = 005 119896 = 001 119886 = 05 119887 = minus30 1205821= 10 120582

2= 06

1205823= 03 119910

1= 10 119910

2= 30 and 119910

3= 50

The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906

1interacts with the other two

solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario

020

4060

80

05

1015

2025

024

minus40

minus20

minus2

x

t

u

Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30

Table 5 Three solitons interaction the conserved quantities

119879 1198681

1198682

1198683

0 7600000 minus3351855 minus261028320 7600042 minus3352227 minus261314940 7600800 minus3352034 minus261134760 7599367 minus3351547 minus260897080 7600545 minus3351562 minus2602090

Table 6 Birth of solitons with 119886 = 05 and 119887 = minus30

119879 1198681

1198682

00 17724343 minus1253314250 17723816 minus12532956100 17723352 minus12530116150 17722782 minus12529239200 17722217 minus12529013250 17721734 minus12528983

54 Birth of Solitons In this test we choose the initial condi-tion

119906 (119909 0) = exp (minus0011199092)

V (119909 0) = exp (minus0011199092) (46)

with the following set of parameters 119886 = 05 119887 = minus30 0 le

119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved

6 Conclusion

In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm

Abstract and Applied Analysis 9

during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12

To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981

[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003

[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004

[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981

[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997

[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009

[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004

[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008

[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008

[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007

[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004

[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ

Equ pp 485ndash568 ElsevierNorth-Holland Amsterdam TheNetherlands 2008

[15] S Kutluay and Y Ucar ldquoA quadratic B-spline Galerkin approachfor solving a coupled KdV equationrdquo Mathematical Modellingand Analysis vol 18 no 1 pp 103ndash121 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Table 1 Petrov-Galerkin method

119879 119871infin(119880) 119871

infin(119881) 119868

11198682

1198683

0 0000000 0000000 2000000 05000000 01125292 0000015 0000004 2000000 05000000 01125294 0000021 0000005 2000000 05000000 01125296 0000023 0000006 2000000 05000000 01125298 0000024 0000007 2000000 05000000 011252910 0000025 0000008 2000000 05000000 0112529

minus 1199013(119880lowast

119898+2minus 2119880lowast

119898+1+ 2119880lowast

119898minus1minus 119880lowast

119898minus2)

minus 1199014[(119881lowast

119898+2)2+ 10(119881

lowast

119898+1)2minus 10(119881

lowast

119898minus1)2minus (119881lowast

119898minus2)2] = 0

(37)

1199011[(119881119899+1

119898minus2+ 26119881

119899+1

119898minus1+ 66119881

119899+1

119898+ 26119881

119899+1

119898+1+ 119881119899+1

119898+2)

minus (119881119899

119898minus2+ 26119881

119899

119898minus1+ 66119881

119899

119898+ 26119881

119899

119898+1+ 119881119899

119898+2) ]

+ 1199015(119881lowast

119898+2minus 2119881lowast

119898+1+ 2119881lowast

119898minus1minus 119881lowast

119898minus2)

+ 1199016[minus (119880lowast

119898minus2+ 4119880lowast

119898minus1) 119881lowast

119898minus2

+ (119880lowast

119898minus2minus 18119880

lowast

119898minus1minus 33119880

lowast

119898) 119881lowast

119898minus1

+ 22 (119880lowast

119898minus1minus 119880lowast

119898+1) 119881lowast

119898

+ (33119880lowast

119898+ 18119880

lowast

119898+1minus 119880lowast

119898+2) 119881lowast

119898+1

+ (4119880lowast

119898+1+ 119880lowast

119898+2) 119881lowast

119898+2] = 0

(38)

where

119880lowast

119898= 119880119899+1

119898+ 119880119899

119898 119881

lowast

119898= 119881119899+1

119898+ 119881119899

119898

1199011=

1

80 119901

2=3119886119896

64ℎ 119901

3=3119886119896

8ℎ3

1199014=

119887119896

64ℎ 119901

5=

3119896

8ℎ3 119901

6=

3119896

320ℎ

(39)

Again the method is unconditionally stable accordingto von Neumann stability analysis and it is of secondorder in both directions The scheme produced a nonlinearblock pentadiagonal system and its solution obtained usingNewtonrsquosmethodWehave noticed that the accuracy has beenimproved in the first equation (38) as we will see in the nextsection We will denote the scheme obtained by using theproduct approximation technique by Scheme 2

5 Numerical Results

To gain insight into the performance of the proposedschemes we perform different numerical tests like sin-gle soliton two and three solitons interaction and birthof solitons The conservation properties of the proposedschemes are examined by calculating 119868

1 1198682 and 119868

3using

the trapezoidal rule The 119871infin(119906) and 119871

infin(V) error norms are

defined as

119871infin(119906) =

1003817100381710038171003817U119899minus u1198991003817100381710038171003817infin = max

119898

1003816100381610038161003816119880119899

119898minus 119906119899

119898

1003816100381610038161003816

119871infin (V) =

1003817100381710038171003817V119899minus k1198991003817100381710038171003817infin = max

119898

1003816100381610038161003816119881119899

119898minus V119899119898

1003816100381610038161003816

(40)

are used to examine the accuracy of the proposed schemes

51 Single Soliton In the first test we choose the initialconditions

119906 (119909 0) = 21205822sech2 (120585) V (119909 0) =

1

2radic119908sech (120585)

120585 = 120582119909 +1

2 log (119908) 119908 =

minus119887

8 (4119886 + 1) 1205824

(41)

which represents a single soliton solution at 119905 = 0 To studythe behavior of numerical solution using Scheme 1 andScheme 2 we choose the set of parameters as ℎ = 005119896 = 001 tol = 10

minus7 119909119871= minus25 119909

119877= 25 119886 = minus0125

119887 = minus3 and 120582 = 05 The conserved quantities and theerror norms 119871

infin(119880) 119871

infin(119881) are displayed in Tables 1 and 2

for Scheme 1 and Scheme 2 respectively It is clear from thesetables that our schemes are highly accurate In addition theschemes preserve the conserved quantities exactly during theevolution of the numerical solution from 119905 = 0 to 119905 = 10 Theexecution time required to produce Table 1 is 2328 secondand 2171 second to produce Table 2 We have noticed thatScheme 2 has an upper hand over Scheme 1 with respect toaccuracy and CPU time In Figures 1 and 2 we display thenumerical solution of 119880119899

119898and 119881119899

119898for 119905 = 0 1 2 20

By choosing the set of values 119896 = 001 119886 = 05 119887 = minus30120582 = 05 and 119905 = 1 we perform a comparison of Scheme 1 andScheme 2 with Ismail [6] and we display this in Table 3 wecan easily see that the three methods produce highly accurateresults with some credits for collocation method

52 Two Solitons Interaction To study the interaction of twosolitons we choose the initial conditions as

119906 (119909 0) =

2

sum

119895=1

119906119895(119909 0) V (119909 0) =

2

sum

119895=1

V119895(119909 0) (42)

6 Abstract and Applied Analysis

Table 2 Product approximation technique

119879 119871infin(119880) 119871

infin(119881) 119868

11198682

1198683

0 0000000 0000000 2000000 05000000 01125292 0000004 0000005 2000000 05000000 01125294 0000008 0000007 2000000 05000000 01125296 0000010 0000009 2000000 05000000 01125298 0000013 0000012 2000000 05000000 011252910 0000014 0000013 2000000 05000000 0112529

Table 3 Comparison of three methods with 119886 = 05 119887 = minus30 120582 = 05 119896 = 001 and 119905 = 1

ℎPetrov-Galerkin Product approximation Collocation (Ismail [6])

119871infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU

01 0000051 0000027 0218 0000014 0000019 0203 0000000 0000003 0188005 0000013 0000007 0343 0000009 0000008 0328 0000000 0000003 03120025 0000003 0000004 0625 0000002 0000004 0609 0000000 0000003 0578

010

2030

4050

05

1015

200

01

02

03

04

05

x

t

u

Figure 1 Single soliton 119880 119886 = minus0125 119887 = minus30 and 120582 = 05

010

2030

4050

05

1015

200

00200400600801012014016

xt

v

Figure 2 Single soliton 119881 119886 = minus0125 119887 = minus30 and 120582 = 05

Table 4 Two solitons interaction (conserved quantities)

119879 1198681

1198682

1198683

000 6400000 minus3243013 minus2584118200 6399985 minus3243298 minus2586545400 6400034 minus3243016 minus2584155600 6399962 minus3243017 minus2584203800 6400234 minus3242978 minus2583492

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2

(43)

which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582

1= 10 120582

2= 06 119910

1= 10

and 1199102= 30 In Table 4 we present the conserved quantities

during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]

To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582

1= 10 120582

2= 06

Abstract and Applied Analysis 7

020

4060

80100

0

20

40

60

80

0123

minus1

x

t

u

Figure 3 Numerical solution 119880119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

020

4060

80100

010

2030

4050

6070

80

012

x

t

v

minus1

Figure 4 Numerical solution 119881119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

x

t

Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)

020

4060

80100

010

2030

4050

0

1

2

3

4

5

xt

u

minus1

Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =

0495

020

4060

80100

010

2030

4050

0

05

1

15

2

25

xt

v

minus05

Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =

0495

1199101= 10 119910

2= 30 119888 = 0 119889 = 100 ℎ = 005 and 119896 = 001 Our

results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12

53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform

119906 (119909 0) =

3

sum

119895=1

119906119895(119909 0) V (119909 0) =

3

sum

119895=1

V119895(119909 0) (44)

8 Abstract and Applied Analysis

020

4060

80100

010

2030

4050

6070

80

0123

x

t

u

minus1

Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

020

4060

80100

020

4060

80

0020406081

121416

xt

v

minus02

Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2 3

(45)

In this test we choose the set of parameters 119909119871= 0 119909

119877=

80 ℎ = 005 119896 = 001 119886 = 05 119887 = minus30 1205821= 10 120582

2= 06

1205823= 03 119910

1= 10 119910

2= 30 and 119910

3= 50

The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906

1interacts with the other two

solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario

020

4060

80

05

1015

2025

024

minus40

minus20

minus2

x

t

u

Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30

Table 5 Three solitons interaction the conserved quantities

119879 1198681

1198682

1198683

0 7600000 minus3351855 minus261028320 7600042 minus3352227 minus261314940 7600800 minus3352034 minus261134760 7599367 minus3351547 minus260897080 7600545 minus3351562 minus2602090

Table 6 Birth of solitons with 119886 = 05 and 119887 = minus30

119879 1198681

1198682

00 17724343 minus1253314250 17723816 minus12532956100 17723352 minus12530116150 17722782 minus12529239200 17722217 minus12529013250 17721734 minus12528983

54 Birth of Solitons In this test we choose the initial condi-tion

119906 (119909 0) = exp (minus0011199092)

V (119909 0) = exp (minus0011199092) (46)

with the following set of parameters 119886 = 05 119887 = minus30 0 le

119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved

6 Conclusion

In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm

Abstract and Applied Analysis 9

during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12

To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981

[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003

[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004

[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981

[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997

[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009

[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004

[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008

[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008

[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007

[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004

[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ

Equ pp 485ndash568 ElsevierNorth-Holland Amsterdam TheNetherlands 2008

[15] S Kutluay and Y Ucar ldquoA quadratic B-spline Galerkin approachfor solving a coupled KdV equationrdquo Mathematical Modellingand Analysis vol 18 no 1 pp 103ndash121 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Table 2 Product approximation technique

119879 119871infin(119880) 119871

infin(119881) 119868

11198682

1198683

0 0000000 0000000 2000000 05000000 01125292 0000004 0000005 2000000 05000000 01125294 0000008 0000007 2000000 05000000 01125296 0000010 0000009 2000000 05000000 01125298 0000013 0000012 2000000 05000000 011252910 0000014 0000013 2000000 05000000 0112529

Table 3 Comparison of three methods with 119886 = 05 119887 = minus30 120582 = 05 119896 = 001 and 119905 = 1

ℎPetrov-Galerkin Product approximation Collocation (Ismail [6])

119871infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU 119871

infin(119880119899

119898) 119871

infin(119881119899

119898) CPU

01 0000051 0000027 0218 0000014 0000019 0203 0000000 0000003 0188005 0000013 0000007 0343 0000009 0000008 0328 0000000 0000003 03120025 0000003 0000004 0625 0000002 0000004 0609 0000000 0000003 0578

010

2030

4050

05

1015

200

01

02

03

04

05

x

t

u

Figure 1 Single soliton 119880 119886 = minus0125 119887 = minus30 and 120582 = 05

010

2030

4050

05

1015

200

00200400600801012014016

xt

v

Figure 2 Single soliton 119881 119886 = minus0125 119887 = minus30 and 120582 = 05

Table 4 Two solitons interaction (conserved quantities)

119879 1198681

1198682

1198683

000 6400000 minus3243013 minus2584118200 6399985 minus3243298 minus2586545400 6400034 minus3243016 minus2584155600 6399962 minus3243017 minus2584203800 6400234 minus3242978 minus2583492

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2

(43)

which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582

1= 10 120582

2= 06 119910

1= 10

and 1199102= 30 In Table 4 we present the conserved quantities

during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]

To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582

1= 10 120582

2= 06

Abstract and Applied Analysis 7

020

4060

80100

0

20

40

60

80

0123

minus1

x

t

u

Figure 3 Numerical solution 119880119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

020

4060

80100

010

2030

4050

6070

80

012

x

t

v

minus1

Figure 4 Numerical solution 119881119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

x

t

Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)

020

4060

80100

010

2030

4050

0

1

2

3

4

5

xt

u

minus1

Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =

0495

020

4060

80100

010

2030

4050

0

05

1

15

2

25

xt

v

minus05

Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =

0495

1199101= 10 119910

2= 30 119888 = 0 119889 = 100 ℎ = 005 and 119896 = 001 Our

results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12

53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform

119906 (119909 0) =

3

sum

119895=1

119906119895(119909 0) V (119909 0) =

3

sum

119895=1

V119895(119909 0) (44)

8 Abstract and Applied Analysis

020

4060

80100

010

2030

4050

6070

80

0123

x

t

u

minus1

Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

020

4060

80100

020

4060

80

0020406081

121416

xt

v

minus02

Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2 3

(45)

In this test we choose the set of parameters 119909119871= 0 119909

119877=

80 ℎ = 005 119896 = 001 119886 = 05 119887 = minus30 1205821= 10 120582

2= 06

1205823= 03 119910

1= 10 119910

2= 30 and 119910

3= 50

The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906

1interacts with the other two

solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario

020

4060

80

05

1015

2025

024

minus40

minus20

minus2

x

t

u

Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30

Table 5 Three solitons interaction the conserved quantities

119879 1198681

1198682

1198683

0 7600000 minus3351855 minus261028320 7600042 minus3352227 minus261314940 7600800 minus3352034 minus261134760 7599367 minus3351547 minus260897080 7600545 minus3351562 minus2602090

Table 6 Birth of solitons with 119886 = 05 and 119887 = minus30

119879 1198681

1198682

00 17724343 minus1253314250 17723816 minus12532956100 17723352 minus12530116150 17722782 minus12529239200 17722217 minus12529013250 17721734 minus12528983

54 Birth of Solitons In this test we choose the initial condi-tion

119906 (119909 0) = exp (minus0011199092)

V (119909 0) = exp (minus0011199092) (46)

with the following set of parameters 119886 = 05 119887 = minus30 0 le

119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved

6 Conclusion

In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm

Abstract and Applied Analysis 9

during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12

To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981

[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003

[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004

[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981

[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997

[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009

[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004

[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008

[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008

[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007

[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004

[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ

Equ pp 485ndash568 ElsevierNorth-Holland Amsterdam TheNetherlands 2008

[15] S Kutluay and Y Ucar ldquoA quadratic B-spline Galerkin approachfor solving a coupled KdV equationrdquo Mathematical Modellingand Analysis vol 18 no 1 pp 103ndash121 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

020

4060

80100

0

20

40

60

80

0123

minus1

x

t

u

Figure 3 Numerical solution 119880119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

020

4060

80100

010

2030

4050

6070

80

012

x

t

v

minus1

Figure 4 Numerical solution 119881119899 with parameters 120582

1= 10 120582

2=

06 1199101= 10 and 119910

2= 30

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

x

t

Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)

020

4060

80100

010

2030

4050

0

1

2

3

4

5

xt

u

minus1

Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =

0495

020

4060

80100

010

2030

4050

0

05

1

15

2

25

xt

v

minus05

Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =

0495

1199101= 10 119910

2= 30 119888 = 0 119889 = 100 ℎ = 005 and 119896 = 001 Our

results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12

53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform

119906 (119909 0) =

3

sum

119895=1

119906119895(119909 0) V (119909 0) =

3

sum

119895=1

V119895(119909 0) (44)

8 Abstract and Applied Analysis

020

4060

80100

010

2030

4050

6070

80

0123

x

t

u

minus1

Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

020

4060

80100

020

4060

80

0020406081

121416

xt

v

minus02

Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2 3

(45)

In this test we choose the set of parameters 119909119871= 0 119909

119877=

80 ℎ = 005 119896 = 001 119886 = 05 119887 = minus30 1205821= 10 120582

2= 06

1205823= 03 119910

1= 10 119910

2= 30 and 119910

3= 50

The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906

1interacts with the other two

solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario

020

4060

80

05

1015

2025

024

minus40

minus20

minus2

x

t

u

Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30

Table 5 Three solitons interaction the conserved quantities

119879 1198681

1198682

1198683

0 7600000 minus3351855 minus261028320 7600042 minus3352227 minus261314940 7600800 minus3352034 minus261134760 7599367 minus3351547 minus260897080 7600545 minus3351562 minus2602090

Table 6 Birth of solitons with 119886 = 05 and 119887 = minus30

119879 1198681

1198682

00 17724343 minus1253314250 17723816 minus12532956100 17723352 minus12530116150 17722782 minus12529239200 17722217 minus12529013250 17721734 minus12528983

54 Birth of Solitons In this test we choose the initial condi-tion

119906 (119909 0) = exp (minus0011199092)

V (119909 0) = exp (minus0011199092) (46)

with the following set of parameters 119886 = 05 119887 = minus30 0 le

119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved

6 Conclusion

In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm

Abstract and Applied Analysis 9

during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12

To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981

[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003

[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004

[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981

[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997

[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009

[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004

[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008

[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008

[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007

[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004

[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ

Equ pp 485ndash568 ElsevierNorth-Holland Amsterdam TheNetherlands 2008

[15] S Kutluay and Y Ucar ldquoA quadratic B-spline Galerkin approachfor solving a coupled KdV equationrdquo Mathematical Modellingand Analysis vol 18 no 1 pp 103ndash121 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

020

4060

80100

010

2030

4050

6070

80

0123

x

t

u

minus1

Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

020

4060

80100

020

4060

80

0020406081

121416

xt

v

minus02

Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582

1= 1 120582

2= 06 and 120582

3= 03

where

119906119895 (119909 0) = 2120582

2

119895sech2 (120585

119895) V

119895 (119909 0) =1

2radic119908119895

sech (120585119895)

120585119895= 120582119895(119909 minus 119910

119895) +

1

2 log (119908119895)

119908119895=

minus119887

8 (4119886 + 1) 1205824

119895

119895 = 1 2 3

(45)

In this test we choose the set of parameters 119909119871= 0 119909

119877=

80 ℎ = 005 119896 = 001 119886 = 05 119887 = minus30 1205821= 10 120582

2= 06

1205823= 03 119910

1= 10 119910

2= 30 and 119910

3= 50

The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906

1interacts with the other two

solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario

020

4060

80

05

1015

2025

024

minus40

minus20

minus2

x

t

u

Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30

Table 5 Three solitons interaction the conserved quantities

119879 1198681

1198682

1198683

0 7600000 minus3351855 minus261028320 7600042 minus3352227 minus261314940 7600800 minus3352034 minus261134760 7599367 minus3351547 minus260897080 7600545 minus3351562 minus2602090

Table 6 Birth of solitons with 119886 = 05 and 119887 = minus30

119879 1198681

1198682

00 17724343 minus1253314250 17723816 minus12532956100 17723352 minus12530116150 17722782 minus12529239200 17722217 minus12529013250 17721734 minus12528983

54 Birth of Solitons In this test we choose the initial condi-tion

119906 (119909 0) = exp (minus0011199092)

V (119909 0) = exp (minus0011199092) (46)

with the following set of parameters 119886 = 05 119887 = minus30 0 le

119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved

6 Conclusion

In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm

Abstract and Applied Analysis 9

during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12

To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981

[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003

[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004

[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981

[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997

[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009

[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004

[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008

[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008

[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007

[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004

[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ

Equ pp 485ndash568 ElsevierNorth-Holland Amsterdam TheNetherlands 2008

[15] S Kutluay and Y Ucar ldquoA quadratic B-spline Galerkin approachfor solving a coupled KdV equationrdquo Mathematical Modellingand Analysis vol 18 no 1 pp 103ndash121 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12

To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981

[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003

[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004

[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981

[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997

[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009

[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004

[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008

[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008

[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007

[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004

[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ

Equ pp 485ndash568 ElsevierNorth-Holland Amsterdam TheNetherlands 2008

[15] S Kutluay and Y Ucar ldquoA quadratic B-spline Galerkin approachfor solving a coupled KdV equationrdquo Mathematical Modellingand Analysis vol 18 no 1 pp 103ndash121 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of