Research ArticleA Galerkin Finite Element Method for Numerical Solutions ofthe Modified Regularized Long Wave Equation

Liquan Mei12 Yali Gao1 and Zhangxin Chen2

1 Center for Computational Geosciences School of Mathematics and Statistics Xirsquoan Jiaotong University Xirsquoan 710049 China2Department of Chemical amp Petroleum Engineering Schulich School of Engineering University of Calgary CalgaryAB Canada T2N 1N4

Correspondence should be addressed to Liquan Mei lqmeimailxjtueducn

Received 26 March 2014 Accepted 31 May 2014 Published 19 June 2014

Academic Editor Ljubisa Kocinac

Copyright copy 2014 Liquan Mei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A Galerkin method for a modified regularized long wave equation is studied using finite elements in space the Crank-Nicolsonscheme and the Runge-Kutta scheme in time In addition an extrapolation technique is used to transform a nonlinear system intoa linear system in order to improve the time accuracy of this method A Fourier stability analysis for the method is shown to bemarginally stable Three invariants of motion are investigated Numerical experiments are presented to check the theoretical studyof this method

1 Introduction

Much physical phenomena are described by nonlinear partialdifferential equations Most of these equations do not have ananalytical solution or it is extremely difficult and expensive tocompute their analytical solutions Hence a numerical studyof these nonlinear partial differential equations is importantin practice The regularized long wave (RLW) equation

119906119905+ 119906119909+ 119906119906119909minus 120583119906119909119909119905

= 0 (1)

where 120583 is a positive constant is a nonlinear evolutionequation which was originally introduced by Peregrine [1]in describing the behavior of an undular bore and studiedlater by Benjamin et al [2] This equation plays an importantrole in describing physical phenomena in various disciplinessuch as the nonlinear transverse waves in shallow water ion-acoustic waves in plasma magnetohydrodynamics waves inplasma longitudinal dispersive waves in elastic rods andpressure waves in liquidrsquos gas bubblesMany numerical meth-ods for the RLW equation have been proposed such as finitedifferencemethods [3 4] the Galerkin finite elementmethod[5ndash8] the least squares method [9ndash11] various collocationmethods with quadratic B-splines [12] cubic B-splines [13]and septic splines [14] meshfree method [15 16] and anexplicit multistep method [17]

The RLW equation is a special case of the generalizedregularized long wave (GRLW) equation

119906119905+ 119906119909+ 120575119906119901119906119909minus 120583119906119909119909119905

= 0 (2)

where 120575 and 120583 are positive constants and 119901 is a positive inte-ger Somenumericalmethods [18ndash24] for theGRLWequationhave been also presented such as a finite difference method[18] a decomposition method [20] and a sinc-collocationmethod [23] Another special case of the GRLW equation iscalled the modified regularized long wave (MRLW) equationin which 119901 = 2 [25] Some authors have studied the MRLWequation using various numerical methods such as a finitedifference method [26] a collocation method [27] a splinemethod [28 29] and the Adomian decomposition method[30]

In this paper we study a Galerkin method for theMRLW equation by using linear finite elements in space andextrapolation to remove the nonlinear term We discuss theproperties of this method and compare its accuracy withprevious studies The interaction of two and three solitonsis also studied Moreover the propagation of the Maxwellianinitial condition is simulated

The outline of this paper is as follows In the nextsection the governing equation its analytical solution and

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 438289 11 pageshttpdxdoiorg1011552014438289

2 Abstract and Applied Analysis

three invariants are given In Section 3 we propose thenumerical method including semidiscretization in space andfull discretization in space and time In Section 4 we presenta stability analysis In Section 5 some numerical experimentsare presented Finally we give a brief conclusion in Section 6

2 Governing Equation

We assumeΩ = (119886 119887) to be an open bounded subset of119877 and119879 gt 0 to be a final time We set Ω

119879= Ω times (0 119879) 119869 = (0 119879]

and consider the following MRLW equation

119906119905+ 119906119909+ 61199062119906119909minus 120583119906119909119909119905

= 0 in Ω119879

119906 = 0 on 120597Ω times [0 119879]

119906 = 1199060 on Ω times 119905 = 0

(3)

where 1199060 Ω rarr 119877 is the initial datum 120583 is a positive

constant and the subscripts 119909 and 119905 denote differentiation inspace and time respectively The physical boundary condi-tion requires 119906 rarr 0 as 119909 rarr plusmninfin

The MRLW equation (3) has the exact solution [25]119906 (119909 119905) = radic119888 sech (119901 (119909 minus (119888 + 1) 119905 minus 119909

0)) (4)

where 119901 = radic119888120583(119888 + 1) and 1199090and 119888 are arbitrary constants

Furthermore (3) possesses three invariants of motion corre-sponding to conservation of mass momentum and energy[25]

1198681= int

119887

119886

119906 119889119909

1198682= int

119887

119886

(1199062+ 120583119906119909

2) 119889119909

1198683= int

119887

119886

(1199064minus 120583119906119909

2) 119889119909

(5)

These invariants are used to check the conservative propertiesof a numerical method especially for problems without ananalytical solution and during collision of solitons

3 Numerical Methods

Set 119881 = 1198671

0(Ω) = V isin 1198671(Ω) V|

120597Ω= 0 Applying Greenrsquos

formula to problem (3) and using the boundary condition inthe definition of119881 we derive the variational form of problem(3) Find 119906 119869 rarr 119881 such that

(

120597119906

120597119905

V) + 119861 (119906 V) + 119862 (119906 V) + 119863 (119906 V) = 0 forallV isin 119881 119905 isin 119869

(119906 (119909 0) V) = (1199060(119909) V) forallV isin 119881 119909 isin Ω

(6)where

119861 (119906 V) = int119887

119886

120583119906119909119905sdot V119909119889119909

119862 (119906 V) = int119887

119886

119906119909sdot V 119889119909

119863 (119906 V) = int119887

119886

61199062119906119909sdot V 119889119909

(7)

31 Semidiscretization in Space Let us consider a uniform 1119863

meshwith themesh sizeℎ = 119909119894+1minus119909119894 119894 = 0 1 119873minus1 which

consists of119873 + 1 points 119886 = 1199090lt 1199091lt sdot sdot sdot lt 119909

119873minus1lt 119909119873= 119887

Then we obtain119873minus1

sum

119894=0

(int

119909119894+1

119909119894

120597119906

120597119905

sdot V 119889119909 + int119909119894+1

119909119894

120583119906119909119905sdot V119909119889119909

+int

119909119894+1

119909119894

119906119909sdot V 119889119909 + int

119909119894+1

119909119894

61199062119906119909sdot V 119889119909) = 0

(8)

We make the transformation 119909 = 119909119894+ 120578ℎ 0 le 120578 le 1 119894 =

0 1 119873minus1 in order to transfer an element into a standardinterval From (8) we derive the following equation

int

1

0

120597119906

120597119905

sdot V 119889120578 +120583

ℎ2int

1

0

119906120578119905sdot V120578119889120578 +

1

ℎ

int

1

0

119906120578sdot V 119889120578

+

6

ℎ

int

1

0

1199062119906120578sdot V 119889120578 = 0

(9)

Now we define the finite dimensional subspace 119881ℎsub 119881

119881ℎ= span119871

1 1198712 where

1198711= 120578 119871

2= 1 minus 120578 (10)

are the linear basis functions on each element Then thesemidiscrete scheme for problem (3) is formulated as followsFind 119906

ℎ 119869 rarr 119881

ℎsuch that

(

120597119906ℎ

120597119905

V) + 119861 (119906ℎ V) + 119862 (119906

ℎ V) + 119863 (119906

ℎ V) = 0

forallV isin 119881ℎ 119905 isin 119869

(119906ℎ(119909 0) V) = (119906ℎ

0(119909) V) forallV isin 119881

ℎ 119909 isin Ω

(11)

The variation of 119906 over the element [119909119894 119909119894+1] 119894 =

0 1 119873 minus 1 is expressed as

119906119890=

2

sum

119895=1

119871119895(119909) 119906119895(119905) (12)

For 119895 = 1 2 we choose V = 119871119895in problem (11) and

substitute (12) into (11) Then an elementrsquos contribution isobtained in the form of

2

sum

119894=1

(int

1

0

119871119894sdot 119871119895119889119909 +

120583

ℎ2int

1

0

1198711015840

119894sdot 1198711015840

119895119889119909)

120597119906119894

120597119905

+

1

ℎ

(int

1

0

1198711015840

119894sdot 119871119895119889119909)119906

119894

+(

6

ℎ

int

1

0

(

2

sum

119894=1

119906119894119871119894)

2

1198711015840

119894sdot 119871119895119889119909)119906

119894

= 0

(13)

where the symbol ( 1015840) denotes differentiation with respect to120578 which in matrix form is given by

(119860119890+ 119861119890)

120597119906119890

120597119905

+ 119862119890119906119890+ 119863119890(119906119890) 119906119890= 0 (14)

Abstract and Applied Analysis 3

where 119906119890 = (1199061 1199062)119879 are relevant nodal parameters The

element matrices are

119860119890

119895119896= int

1

0

119871119895119871119896119889120578

119861119890

119895119896=

120583

ℎ2int

1

0

1198711015840

1198951198711015840

119896119889120578

119862119890

119895119896=

1

ℎ

int

1

0

1198711015840

119895119871119896119889120578

119863119890

119895119896=

6

ℎ

int

1

0

(

2

sum

119898=1

119906119898119871119898)

2

1198711015840

119895119871119896119889120578

(15)

For the element [119909119894 119909119894+1] (119894 = 0 1 119873minus 1) the indices

119895 and 119896 take only the values 1 and 2 so that the matrices 119860119890119861119890 119862119890 and119863119890 are 2 times 2

119860119890= (

1

3

1

6

1

6

1

3

)

119861119890

119895119896=

120583

ℎ2(

1 minus1

minus1 1)

119862119890

119895119896=

1

ℎ

(

minus

1

2

1

2

minus

1

2

1

2

)

119863119890

119895119896

=

6

ℎ

times(

minus(

1

4

1199062

1+

1

6

11990611199062+

1

12

1199062

2)

1

4

1199062

1+

1

6

11990611199062+

1

12

1199062

2

minus(

1

12

1199062

1+

1

6

11990611199062+

1

4

1199062

2)

1

12

1199062

1+

1

6

11990611199062+

1

4

1199062

2

)

(16)

Assembling contributions from all elements leads to thefollowing matrix form of the coupled nonlinear ordinarydifferential equations

(119860 + 119861)

120597119906

120597119905

+ 119862119906 + 119863 (119906) 119906 = 0 (17)

where 119906 = (1199060 1199061 119906

119873)119879 contains all the nodal parame-

tersThe four assembledmatrices are tridiagonalThe generalrow for each matrix has the following form

119860 (

1

6

2

3

1

6

)

119861

120583

ℎ2(minus1 2 minus1)

119862

1

2ℎ

(minus1 0 1)

119863

6

ℎ

(minus

1

12

(1199062

119898minus1+ 2119906119898minus1

119906119898+ 31199062

119898)

1

12

(1199062

119898minus1minus 1199062

119898+1) +

1

6

119906119898(119906119898minus1

minus 119906119898+1

)

1

12

(31199062

119898+ 2119906119898119906119898+1

+ 1199062

119898+1))

(18)

32 Full Discretization We now consider a fully discretescheme Let 0 = 1199050 lt 1199051 lt sdot sdot sdot lt 119905119872 = 119879 be a uniformpartitionof 119869 into subintervals 119869119899 = (119905119899 119905119899+1) 119899 = 0 1 119872 minus 1 withlength Δ119905 = 119905

119899+1minus 119905119899= 119879119872 For a generic function V of

time set V119899 = V(119905119899)Weuse theCrank-Nicolson discretizationmethod in (17) with

120597119906

120597119905

=

119906119899minus 119906119899minus1

Δ119905

119899=

119906119899+ 119906119899minus1

2

(19)

Then (17) can be written as

(119860 + 119861)

120597119906

120597119905

+ 119862119899+ 119863 (

119899) 119899= 0 (20)

with 1199060 = 119906ℎ

0 This equation is symmetric about the point

119905 = 119905119899minus12

and one should therefore expect second orderaccuracy in time However the linearization decreases theorder of the time discretization error to119874(Δ119905)This drawbackcan be overcome by using an extrapolation technique in thelinearization of the nonlinear coefficient 119863 We choose 119906119899 =(32)119906

119899minus1minus (12)119906

119899minus2 for 119899 ge 2 and define

(119860 + 119861)

120597119906

120597119905

+ 119862119899+ 119863 (119906

119899) 119899= 0 (21)

Now the Crank-Nicolson method can be shown to producean error of order119874((Δ119905)2) Combining with the error of finiteelementmethod the Crank-Nicolson Galerkin finite elementmethod of (6) produces an error of order119874((Δ119905)2+ℎ2) overall[17]

We can obtain the following recurrence relation at point119909119898

[

1

6

minus 119887 minus

Δ119905

4ℎ

minus

6Δ119905

ℎ

(

1

24

V2119898minus1

+

1

12

V119898minus1

V119898+

1

8

V2119898)] 119906119899

119898minus1

+ [

2

3

+ 2119887 +

6Δ119905

ℎ

(

1

24

(V2119898minus1

minus V2119898+1

)

+

1

12

(V119898minus1

minus V119898+1

) V119898)] 119906119899

119898

+ [

1

6

minus 119887 +

Δ119905

4ℎ

+

6Δ119905

ℎ

(

1

8

V2119898+

1

12

V119898V119898+1

+

1

24

V2119898+1

)] 119906119899

119898+1

= [

1

6

minus 119887 +

Δ119905

4ℎ

+

6Δ119905

ℎ

(

1

24

V2119898minus1

+

1

12

V119898minus1

V119898+

1

8

V2119898)] 119906119899minus1

119898minus1

+ [

2

3

+ 2119887 minus

6Δ119905

ℎ

(

1

24

(V2119898minus1

minus V2119898+1

)

+

1

12

(V119898minus1

minus V119898+1

) V119898)] 119906119899minus1

119898

+ [

1

6

minus 119887 minus

Δ119905

4ℎ

minus

6Δ119905

ℎ

(

1

8

V2119898+

1

12

V119898V119898+1

+

1

24

V2119898+1

)] 119906119899minus1

119898+1

(22)

4 Abstract and Applied Analysis

where V119898and 119887 are given by

V119898=

3

2

119906119899minus1

119898minus

1

2

119906119899minus2

119898

119887 =

120583

ℎ2

(23)

As (22) is used only for 119899 ge 2 we must compute 1199061 byanother method For this we choose a predictor-correctormethod Let 119899 = 1 in (21) we can get the first approximation11990610 with 119906

1 replaced by 1199060 Then we use (12)(11990610 + 1199060)

to substitute 1199061 in (21) and regard the result as the finalapproximation 1199061 Or more precisely the concrete procedureis as follows First set

1199060= 1199060

ℎ (24)

then we calculate

(119860 + 119861)

11990610minus 1199060

Δ119905

+ 119862

11990610+ 1199060

2

+ 119863 (1199060)

11990610+ 1199060

2

= 0

(25)

finally we have

(119860 + 119861)

1199061minus 1199060

Δ119905

+ 119862

1199061+ 1199060

2

+ 119863(

11990610+ 1199060

2

)

1199061+ 1199060

2

= 0

(26)

Thus we can use theThomas algorithm to solve the linearalgebraic equations

In order to improve the accuracy in time we also useRunge-Kutta discretizationmethodThe fourth order Runge-Kutta discretization method in (17) can be described asfollows

119906119899+1

= 119906119899+

1198701+ 21198702+ 21198703+ 1198704

6

minus (119860 + 119861)1198701= [119862119906

119899+ 119863 (119906

119899) 119906119899] Δ119905

minus (119860 + 119861)1198702

= [119862(119906119899+

1

2

1198701) + 119863(119906

119899+

1

2

1198701)(119906119899+

1

2

1198701)]Δ119905

minus (119860 + 119861)1198703

= [119862(119906119899+

1

2

1198702) + 119863(119906

119899+

1

2

1198702)(119906119899+

1

2

1198702)]Δ119905

minus (119860 + 119861)1198704

= [119862 (119906119899+ 1198703) + 119863 (119906

119899+ 1198703) (119906119899+ 1198703)] Δ119905

(27)

4 Stability Analysis

A linear stability analysis is made of the growth factor 119892 of theerror 120576119899

119895in a typical Fourier mode of amplitude 120576119899

120576119899

119895= 119890119894119895119896ℎ

120576119899 (28)

where 119896 is a mode number and ℎ is the element size The vonNeumann stability method can only be applied to a linearscheme so we linearize the nonlinear term by assuming that119906 in this term is locally constant Substituting (28) into (22)gives

120576119899+1

= 119892 120576119899 (29)

119892 =

120572 + 120573119894

120572 minus 120573119894

(30)

where

120572 = (

1

3

minus 2119887) cos (119896ℎ) + 2

3

+ 2119887

120573 =

Δ119905

2ℎ

(1 + 120576119862) sin (119896ℎ) (31)

Taking the modulus of (30) gives

10038161003816100381610038161198921003816100381610038161003816= radic119892119892 = 1 (32)

Therefore our method is marginally stable

5 Numerical Experiments

In this section we present some numerical tests to checkthe efficiency and accuracy of our method which are thepropagation of single soliton and collision of two and threesolitons at different time levels Finally we investigate thedevelopment of theMaxwellian initial condition into solitarywaves

To illustrate the accuracy of the present method we use1198712- and 119871

infin-error norms

1198712=

10038171003817100381710038171003817119906exact119899

minus 119906119899100381710038171003817100381710038172= (ℎ

119873

sum

119894=1

10038161003816100381610038161003816119906exact119899119895

minus 119906119899

119895

10038161003816100381610038161003816

2

)

12

119871infin=

10038171003817100381710038171003817119906exact119899

minus 11990611989910038171003817100381710038171003817infin

= max119895

10038161003816100381610038161003816119906exact119899119895

minus 119906119899

119895

10038161003816100381610038161003816

(33)

to compare the numerical solution with the exact solutionwhich show the mean and maximum differences betweenthe numerical and analytical solutions The quantities 119868

1 1198682

and 1198683measure the conservation laws of our method during

propagation

51 Single Soliton The analytical values of the three variantsare

1198681=

120587radic119888

119901

1198682=

2119888

119901

+

2120583119901119888

3

1198683=

41198882

3119901

minus

2120583119901119888

3

(34)

For the purpose of comparison with the previous work[28] we choose the parameters 119909

0= 40 119888 = 005 ℎ =

119896 = 02 119879 = 10 and 0 le 119909 le 100 Then the amplitudeis 07071068 and the analytical values of three invariants are1198681= 3219174470 119868

2= 0465531499 and 119868

3= 0008001323

The relevant numerical results which applied the Crank-Nicolson scheme in (22) and the Runge-Kutta scheme in

Abstract and Applied Analysis 5

0 20 40 60 80 100

0

005

01

015

02

minus005

T = 0

T = 6

T = 10

Figure 1 Solitary wave with 119888 = 005 ℎ = 02 119896 = 02 and 1199090= 40

0 le 119909 le 100

0

50

100

24

68

10

0

05

1

15

2

Error-axis

minus1

minus05

times10minus4

t-axisx-ax

is

Figure 2 Error graph for 119888 = 005 ℎ = 02 119896 = 02 and1199090= 40 0 le

119909 le 100

(27) are presented in Tables 1 and 2 respectively The profilesof solitary waves at time = 5 and time = 10 are given inFigure 1 and the error distribution in the wave profile in thethree-dimensional view at different time level is illustratedin Figure 2 Table 3 presents the variants 119871

infinnorm and 119871

2

norm at time = 10 against the quadratic B-splines and thecubic B-splines in [28]

We also choose the parameters 119888 = 03 ℎ = 01 119896 = 001and 119909

0= 40 with the range [0 100] Then the amplitude

is 054772 and the analytical values of three invariants are1198681= 3581966678 119868

2= 1345076492 and 119868

3= 0153723028

Tables 4 and 5 show the results about the three invariants the1198712norm and the 119871

infinnorm at different times and compare

these results with those by the collocation method with

0 20 40 60 80 100

0

01

02

03

04

05

06

T = 2

T = 6

T = 10

minus01

Figure 3 Solitary wave with 119888 = 03 ℎ = 01 119896 = 001 and 1199090=

40 0 le 119909 le 100

0

2

4

0

50

100

x-axis

24

68

10

t-axis

minus4

minus2

Error-axis

times10minus4

Figure 4 Error graph for 119888 = 03 ℎ = 01 119896 = 001 and1199090= 40 0 le

119909 le 100

cubic B-splines in [27] using the Crank-Nicolson scheme in(22) and the Runge-Kutta scheme in (27) respectively Thechanges of the variants are satisfactorily small though ourresult for this case is notmore accurate than the result in [27]where the simulation is done up to 119905 = 10 The profile of theapproximate solution is given in Figures 3 and 4 shows theerror profile of the single solitary waves

The Runge-Kutta Galerkin method algorithm has beenrun for different space steps with a fixed time step 119896 = 001The error and convergence order of space for the 119871

2- and

119871infin-norm are recorded in Tables 6 and 7 respectively which

verify that the 1198712- and 119871

infin-error can achieve the order119874(ℎ2)

in space And the algorithm has also been run for differenttime steps with a fixed space step ℎ = 01 The error andconvergence order of time for the 119871

2- and 119871

infin-norm are

recorded in Tables 8 and 9 respectively

6 Abstract and Applied Analysis

Table 1 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007784E minus 03 0 02 3218839 04655233 8007697E minus 03 083581119864 minus 04 045779119864 minus 04

4 3218854 04655215 8007602E minus 03 136403119864 minus 04 045846119864 minus 04

6 3218842 04655196 8007507E minus 03 192290119864 minus 04 068890119864 minus 04

8 3218819 04655177 8007410E minus 03 247879119864 minus 04 091724119864 minus 04

10 3218794 04655159 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Table 2 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007778E minus 03 0 02 3218977 04655250 8007779E minus 03 2613735119864 minus 05 4577868119864 minus 05

4 3219083 04655250 8007780E minus 03 3432730119864 minus 05 2894973119864 minus 05

6 3219165 04655250 8007780E minus 03 4744950119864 minus 05 2214867119864 minus 05

8 3219229 04655250 8007780E minus 03 6052811119864 minus 05 2298029119864 minus 05

10 3219274 04655250 8007780E minus 03 7200630119864 minus 05 2346961119864 minus 05

Table 3 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 0 le 119909 le 100 and time = 10

Method 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

Our scheme in (22) 3218794 04655160 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Our scheme in (27) 3219274 04655250 8007780E minus 03 0720063119864 minus 04 0234696119864 minus 04

Quadratic [28] 3215653 04655665 8004883E minus 03 199288119864 minus 04 481156119864 minus 04

Cubic [28] 3215189 04655136 7999173E minus 03 453811119864 minus 04 625887119864 minus 04

Table 4 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson

time discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 142310119864 minus 04 045779119864 minus 04

4 3581966 1344973 01538262 265355119864 minus 04 153988119864 minus 04

6 3581966 1344973 01538261 374673119864 minus 04 200152119864 minus 04

8 3581966 1344972 01538260 477620119864 minus 04 242916119864 minus 04

10 3581966 1344972 01538260 577512119864 minus 04 284373119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Table 5 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 1396685119864 minus 04 0920780119864 minus 04

4 3581967 1344973 01538262 2592545119864 minus 04 1512530119864 minus 04

6 3581967 1344973 01538262 3647701119864 minus 04 1956752119864 minus 04

8 3581967 1344973 01538262 4638980119864 minus 04 2368004119864 minus 04

10 3581967 1344973 01538262 5600205119864 minus 04 2766940119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Abstract and Applied Analysis 7

Table 6 The order of convergence in space for 1198712error norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δℎ = 04

Order Order(0102) (0204)

2 1396685 times 10minus4 5598770 times 10minus4 2258793 times 10minus3 20031 201244 2592545 times 10minus4 1038054 times 10minus3 4168527 times 10minus3 20014 200576 3647701 times 10minus4 1459612 times 10minus3 5846531 times 10minus3 20005 200208 4638980 times 10minus4 1855582 times 10minus3 7421648 times 10minus3 20000 1999910 5600205 times 10minus4 2239531 times 10minus3 8948683 times 10minus3 19996 19985

Table 7 The order of convergence in space for 119871infinerror norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δ119905 = 04

Order Order(0102) (0204)

2 0920780 times 10minus4 3689931 times 10minus4 1480253 times 10minus3 20027 200424 1512530 times 10minus4 6055462 times 10minus4 2430139 times 10minus3 20012 200476 1956752 times 10minus4 7811765 times 10minus4 3126531 times 10minus3 19972 200088 2368004 times 10minus4 9454389 times 10minus4 3768796 times 10minus3 19973 1995010 2766940 times 10minus4 1104758 times 10minus3 4416067 times 10minus3 19974 19990

Table 8 The order of convergence in time for 1198712error norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00511878 000418182 4219760 times 10minus4 3613597 33088994 00835505 000806197 8133115 times 10minus4 3309253 33092536 01027137 001176588 1180928 times 10minus3 3125948 33166178 01141693 001561415 1547231 times 10minus3 2870249 333509410 01220153 001982141 1925846 times 10minus3 2621930 3363496

Table 9 The order of convergence in time for 119871infinerror norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00318115 2528470 times 10minus3 2505105 times 10minus4 3653213 33353224 00498808 4159059 times 10minus3 4456332 times 10minus4 3584155 32223286 00594582 5587773 times 10minus3 5905399 times 10minus4 3411531 32421678 00641225 7106547 times 10minus3 7270440 times 10minus4 3173611 328903410 00661177 8861734 times 10minus3 8671150 times 10minus4 2899376 3353294

52 Collision of Two Solitons In this section we study theinteraction of two solitary waves with the initial conditionsgiven by a linear sum of two separate solitary waves of variousamplitudes

119906 (119909 0) =

2

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (35)

where 119860119894= radic119888119894

119901119894= radic119888

119894120583(119888119894+ 1) 119894 = 1 2 and 119909

119894and

119888119894(119894 = 1 2) are arbitrary constants The analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

1198682=

21198881

1199011

+

21198882

1199012

+

212058311988811199011

3

+

212058311990121198882

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

minus

212058311990111198881

3

minus

212058311990121198882

3

(36)

In our numerical scheme we choose ℎ = 01 119896 = 001 1198881= 1

1198882= 05 119909

1= 10 and 119909

2= 40 with interval [0 100] Then

the amplitudes are in ratio 2 1 where 21198602= 1198601and the

analytical values for the conservation are 1198681= 82905324294

1198682= 5224332548 and 119868

3= 1799113742 The changes of

the invariants are satisfactorily small since the changes ofthe invariants 119868

1 1198682 and 119868

3are 13295 times 10minus4 22259 times 10minus5

and 68989 times 10minus5 respectively The results are recorded inTable 10 A graph of the two solitary waves collision plottedin a three-dimensional view at some discrete times is shownin Figure 5

53 Collision of Three Solitons In this section we study theinteraction of three solitary waves with the initial conditionsgiven by a linear sum of three separate solitary waves ofvarious amplitudes

119906 (119909 0) =

3

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (37)

8 Abstract and Applied Analysis

Table 10 Invariants and error norms for two solitary waves 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 ℎ = 01 119896 = 001 and 0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 8288129 8288129 5222934 5222933 1800512 18005132 8289522 8293010 5222910 5222951 1800479 18005074 8289462 8295348 5222887 5222954 1800458 18225076 8289304 8297822 5222864 5222956 1800433 18005078 8289233 8300322 5222840 5222958 1800411 180050810 8289231 8302630 5222817 5222960 1800388 1800509

Table 11 Invariants and error norms for three solitary waves 1198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 119909

3= 40 ℎ = 01 119896 = 001 and

0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 11800513 11800513 6490318 6490318 1997082 19970824 11799364 11807800 6485747 6490319 1992012 19966238 11797850 11812825 6483120 6490298 1988853 199607612 11797570 11817192 6482677 6490279 1988441 199636016 11791837 11820521 6474939 6490305 1982852 199687520 11788577 11823320 6469884 6490341 1977926 1996758

0

02

04

06

08

1

u-axis

0

50

100

x-axis

24

68

10

t-axis

minus02

Figure 5 Interaction of two solitons at times from 119905 = 2 4 6 8 10

for 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 and 0 le 119909 le 100

where 119860119894= radic119888119894

119901119894= radic119888119894120583(119888119894+ 1) 119894 = 1 2 3 and 119909

119894and

119888119894(119894 = 1 2 3) are arbitrary constantsThe analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

+

120587radic1198883

1199013

1198682=

21198881

1199011

+

21198882

1199012

+

21198883

1199013

+

212058311988811199011

3

+

212058311990121198882

3

+

212058311990131198883

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

+

41198883

2

31199013

minus

212058311990111198881

3

minus

212058311990121198882

3

minus

212058311990131198883

3

(38)

In this case we choose paraments as ℎ = 01 119896 = 0051198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 and 119909

3= 40

with interval [0 100]Then the amplitudes are in ratio 4 2 1

where 1198601= 2119860

2= 4119860

3and the analytical values for the

conservation are 1198681= 11802939794 119868

2= 6416902131

and 1198683= 1910917141 The changes of the invariants from

the initial variants approach zero throughout and agree withthe analytical values for the three invariants as presentedin Table 11 which indicates that our scheme is satisfactorilyconservativeThe geometry of the initial state and the profilesat time 119905 = 10 119905 = 20 and 119905 = 35 is shown graphically inFigure 6 respectively

54 The Maxwellian Initial Condition Finally we considerthe development of the Maxwellian initial condition

119906 (119909 0) = 119890minus(119909minus40)

2 (39)

into a train of solitary waves We choose different values 120583 =10 120583 = 05 and 120583 = 01 in our numerical scheme Thecomparison of the three variants 119868

1 1198682 and 119868

3with the earlier

result in [29] is presented in Table 12 here Crank-Nicolsonmethod is used for time discretization For 120583 = 10 thechanges of the variants 119868

1 1198682 and 119868

3with respect to the initial

values are 39492 times 10minus6 69575 times 10minus4 and 31509 times 10minus3respectively whereas they are 28210 times 10minus6 260000 times 10minus4and 70771times 10minus4 in [29] For 120583 = 05 the variants are changedby 78982 times 10minus6 68214 times 10minus5 and 61410 times 10minus4 in this caseThedevelopment of theMaxwellian initial condition is shownin Figure 7 with different parameter values respectively Thesmaller 120583 there is the more the number of solitary waves willform The simulations are done up to time = 10 in this caseFrom the previous work [25] the total number of solitarywaves and the values 120583 have the approximate relation

119873 cong [

1

5radic120583

] (40)

Abstract and Applied Analysis 9

Table 12 Values of 1198681 1198682 and 119868

3for Maxwellian initial condition when ℎ = 01 and 119896 = 001 and the space interval [0 100]

120583 Time 1198681

1198682

1198683

1198681[29] 119868

2[29] 119868

3[29]

1

2 1772528 2499467 minus03633570 1772449 2506352 minus036681494 1772544 2499870 minus03643380 1772446 2506235 minus036669746 1772532 2500822 minus03657774 1772447 2506171 minus036663268 1772524 2500977 minus03661331 1772446 2506123 minus0366586010 1772521 2501206 minus03665079 1772444 22060926 minus03665553

05

2 1772534 1876446 02597450 1772451 1879888 025964944 1772534 1876534 02587968 1772446 1879855 025968286 1772528 1876600 02583802 1772449 1879841 025969738 1772523 1876581 02582386 1772450 1879834 0259705010 1772520 1876574 02581499 1772449 1879828 02597092

01

2 1770991 1373320 07687222 1772452 1378607 076087774 1769863 1371236 07658462 1772451 1378577 076083646 1768742 1369139 07630269 1772451 1378546 076079378 1767630 1367058 07602389 1772451 1378515 0760752910 1766526 1364993 07574795 1772453 1378483 07607117

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(a) Time = 0

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(b) Time = 10

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(c) Time = 20

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(d) Time = 35

Figure 6 Interaction of three solitary waves

10 Abstract and Applied Analysis

0 20 40 60 80 100

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

(a) 120583 = 10

0 20 40 60 80 100minus04

minus02

0

02

04

06

08

1

(b) 120583 = 05

0 20 40 60 80 100minus02

0

02

04

06

08

1

(c) 120583 = 01

Figure 7 Maxwellian initial condition with different values 120583

6 Conclusion

In this paper we have developed a Galerkin linear finiteelementmethod to investigate the propagation of solitons andtheir interactions governed by the nonlinear MRLW equa-tion An extrapolation technique has been used to improvethe accuracy in time for this numerical method A linearstability analysis shows that this method is marginally stableThe high efficiency and accuracy of our method is testedby the numerical experiments propagation of single solitoncollision of two and three solitons and development of theMaxwellian initial condition into solitary waves Moreoverthis method numerically satisfies the conservation laws ofmass momentum and energy

Conflict of Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFund of China (11371289) and the China Scholarship Council(201206285005)

References

[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966

[2] T B Benjamin J L Bona and J J Mahony ldquoModel equationsfor long waves in nonlinear dispersive systemsrdquo PhilosophicalTransactions of the Royal Society of London A vol 272 no 1220pp 47ndash78 1972

[3] S Kutluay and A Esen ldquoA finite difference solution of theregularized long-wave equationrdquo Mathematical Problems inEngineering vol 2006 Article ID 85743 14 pages 2006

Abstract and Applied Analysis 11

[4] J Lin Z Xie and J Zhou ldquoHigh-order compact differencescheme for the regularized long wave equationrdquo Communica-tions in Numerical Methods in Engineering vol 23 no 2 pp135ndash156 2007

[5] P Avilez-Valente and F J Seabra-Santos ldquoA Petrov-Galerkinfinite element scheme for the regularized long wave equationrdquoComputational Mechanics vol 34 no 4 pp 256ndash270 2004

[6] A Esen and S Kutluay ldquoApplication of a lumped Galerkinmethod to the regularized long wave equationrdquo Applied Mathe-matics and Computation vol 174 no 2 pp 833ndash845 2006

[7] Y Liu H Li Y Du and J Wang ldquoExplicit multistep mixedfinite element method for RLW equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 768976 12 pages 2013

[8] B Saka and I Dag ldquoA numerical solution of the RLW equationby Galerkin method using quartic B-splinesrdquo Communicationsin Numerical Methods in Engineering with Biomedical Applica-tions vol 24 no 11 pp 1339ndash1361 2008

[9] I Dag ldquoLeast-squares quadratic B-spline finite elementmethodfor the regularised long wave equationrdquo Computer Methods inApplied Mechanics and Engineering vol 182 no 1-2 pp 205ndash215 2000

[10] I Dag andM N Ozer ldquoApproximation of the RLW equation bythe least square cubic B-spline finite element methodrdquo AppliedMathematical Modelling vol 25 no 3 pp 221ndash231 2001

[11] H Gu and N Chen ldquoLeast-squares mixed finite elementmethods for the RLW equationsrdquoNumericalMethods for PartialDifferential Equations vol 24 no 3 pp 749ndash758 2008

[12] A A Soliman and K R Raslan ldquoCollocation method usingquadratic B-spline for the RLW equationrdquo International Journalof Computer Mathematics vol 78 no 3 pp 399ndash412 2001

[13] I Dag B Saka and D Irk ldquoApplication of cubic B-splines fornumerical solution of the RLW equationrdquo Applied Mathematicsand Computation vol 159 no 2 pp 373ndash389 2004

[14] A A Soliman and M H Hussien ldquoCollocation solution forRLW equation with septic splinerdquo Applied Mathematics andComputation vol 161 no 2 pp 623ndash636 2005

[15] S U Islam S Haq and A Ali ldquoA meshfree method for thenumerical solution of the RLW equationrdquo Journal of Compu-tational and Applied Mathematics vol 223 no 2 pp 997ndash10122009

[16] S Haq S U Islam andA Ali ldquoA numerical meshfree techniquefor the solution of the MEW equationrdquo Computer Modeling inEngineering amp Sciences vol 38 no 1 pp 1ndash23 2008

[17] L Mei and Y Chen ldquoExplicit multistep method for thenumerical solution of RLW equationrdquoAppliedMathematics andComputation vol 218 no 18 pp 9547ndash9554 2012

[18] L Zhang ldquoAfinite difference scheme for generalized regularizedlong-wave equationrdquo Applied Mathematics and Computationvol 168 no 2 pp 962ndash972 2005

[19] MMohammadi and RMokhtari ldquoSolving the generalized reg-ularized longwave equation on the basis of a reproducing kernelspacerdquo Journal of Computational and Applied Mathematics vol235 no 14 pp 4003ndash4014 2011

[20] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004

[21] T Roshan ldquoA Petrov-Galerkin method for solving the gener-alized regularized long wave (GRLW) equationrdquo Computers ampMathematics with Applications vol 63 no 5 pp 943ndash956 2012

[22] A A Soliman ldquoNumerical simulation of the generalizedregularized long wave equation by Hersquos variational iterationmethodrdquoMathematics and Computers in Simulation vol 70 no2 pp 119ndash124 2005

[23] R Mokhtari and M Mohammadi ldquoNumerical solution ofGRLW equation using Sinc-collocation methodrdquo ComputerPhysics Communications vol 181 no 7 pp 1266ndash1274 2010

[24] A H A Ali A A Soliman and K R Raslan ldquoSoliton solutionfor nonlinear partial differential equations by cosine-functionmethodrdquo Physics Letters A vol 368 no 3-4 pp 299ndash304 2007

[25] L R T Gardner G A Gardner F A Irk and N K AmeinldquoApproximations of solitary waves of the MRLW equation byB-spline finite elementsrdquo The Arabian Journal for Science andEngineering A vol 22 no 2 pp 183ndash193 1997

[26] A K Khalifa K R Raslan and H M Alzubaidi ldquoA finite dif-ference scheme for the MRLW and solitary wave interactionsrdquoApplied Mathematics and Computation vol 189 no 1 pp 346ndash354 2007

[27] A K Khalifa K R Raslan and H M Alzubaidi ldquoA collocationmethod with cubic B-splines for solving the MRLW equationrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 406ndash418 2008

[28] K R Raslan and S M Hassan ldquoSolitary waves for the MRLWequationrdquo Applied Mathematics Letters vol 22 no 7 pp 984ndash989 2009

[29] K R Raslan and T S EL-Danaf ldquoSolitary waves solutions of theMRLW equation using quintic B-splinesrdquo Journal of King SaudUniversity Science vol 22 no 3 pp 161ndash166 2010

[30] A K Khalifa K R Raslan and H M Alzubaidi ldquoNumericalstudy using ADM for the modified regularized long waveequationrdquo Applied Mathematical Modelling vol 32 no 12 pp2962ndash2972 2008

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

three invariants are given In Section 3 we propose thenumerical method including semidiscretization in space andfull discretization in space and time In Section 4 we presenta stability analysis In Section 5 some numerical experimentsare presented Finally we give a brief conclusion in Section 6

2 Governing Equation

We assumeΩ = (119886 119887) to be an open bounded subset of119877 and119879 gt 0 to be a final time We set Ω

119879= Ω times (0 119879) 119869 = (0 119879]

and consider the following MRLW equation

119906119905+ 119906119909+ 61199062119906119909minus 120583119906119909119909119905

= 0 in Ω119879

119906 = 0 on 120597Ω times [0 119879]

119906 = 1199060 on Ω times 119905 = 0

(3)

where 1199060 Ω rarr 119877 is the initial datum 120583 is a positive

constant and the subscripts 119909 and 119905 denote differentiation inspace and time respectively The physical boundary condi-tion requires 119906 rarr 0 as 119909 rarr plusmninfin

The MRLW equation (3) has the exact solution [25]119906 (119909 119905) = radic119888 sech (119901 (119909 minus (119888 + 1) 119905 minus 119909

0)) (4)

where 119901 = radic119888120583(119888 + 1) and 1199090and 119888 are arbitrary constants

Furthermore (3) possesses three invariants of motion corre-sponding to conservation of mass momentum and energy[25]

1198681= int

119887

119886

119906 119889119909

1198682= int

119887

119886

(1199062+ 120583119906119909

2) 119889119909

1198683= int

119887

119886

(1199064minus 120583119906119909

2) 119889119909

(5)

These invariants are used to check the conservative propertiesof a numerical method especially for problems without ananalytical solution and during collision of solitons

3 Numerical Methods

Set 119881 = 1198671

0(Ω) = V isin 1198671(Ω) V|

120597Ω= 0 Applying Greenrsquos

formula to problem (3) and using the boundary condition inthe definition of119881 we derive the variational form of problem(3) Find 119906 119869 rarr 119881 such that

(

120597119906

120597119905

V) + 119861 (119906 V) + 119862 (119906 V) + 119863 (119906 V) = 0 forallV isin 119881 119905 isin 119869

(119906 (119909 0) V) = (1199060(119909) V) forallV isin 119881 119909 isin Ω

(6)where

119861 (119906 V) = int119887

119886

120583119906119909119905sdot V119909119889119909

119862 (119906 V) = int119887

119886

119906119909sdot V 119889119909

119863 (119906 V) = int119887

119886

61199062119906119909sdot V 119889119909

(7)

31 Semidiscretization in Space Let us consider a uniform 1119863

meshwith themesh sizeℎ = 119909119894+1minus119909119894 119894 = 0 1 119873minus1 which

consists of119873 + 1 points 119886 = 1199090lt 1199091lt sdot sdot sdot lt 119909

119873minus1lt 119909119873= 119887

Then we obtain119873minus1

sum

119894=0

(int

119909119894+1

119909119894

120597119906

120597119905

sdot V 119889119909 + int119909119894+1

119909119894

120583119906119909119905sdot V119909119889119909

+int

119909119894+1

119909119894

119906119909sdot V 119889119909 + int

119909119894+1

119909119894

61199062119906119909sdot V 119889119909) = 0

(8)

We make the transformation 119909 = 119909119894+ 120578ℎ 0 le 120578 le 1 119894 =

0 1 119873minus1 in order to transfer an element into a standardinterval From (8) we derive the following equation

int

1

0

120597119906

120597119905

sdot V 119889120578 +120583

ℎ2int

1

0

119906120578119905sdot V120578119889120578 +

1

ℎ

int

1

0

119906120578sdot V 119889120578

+

6

ℎ

int

1

0

1199062119906120578sdot V 119889120578 = 0

(9)

Now we define the finite dimensional subspace 119881ℎsub 119881

119881ℎ= span119871

1 1198712 where

1198711= 120578 119871

2= 1 minus 120578 (10)

are the linear basis functions on each element Then thesemidiscrete scheme for problem (3) is formulated as followsFind 119906

ℎ 119869 rarr 119881

ℎsuch that

(

120597119906ℎ

120597119905

V) + 119861 (119906ℎ V) + 119862 (119906

ℎ V) + 119863 (119906

ℎ V) = 0

forallV isin 119881ℎ 119905 isin 119869

(119906ℎ(119909 0) V) = (119906ℎ

0(119909) V) forallV isin 119881

ℎ 119909 isin Ω

(11)

The variation of 119906 over the element [119909119894 119909119894+1] 119894 =

0 1 119873 minus 1 is expressed as

119906119890=

2

sum

119895=1

119871119895(119909) 119906119895(119905) (12)

For 119895 = 1 2 we choose V = 119871119895in problem (11) and

substitute (12) into (11) Then an elementrsquos contribution isobtained in the form of

2

sum

119894=1

(int

1

0

119871119894sdot 119871119895119889119909 +

120583

ℎ2int

1

0

1198711015840

119894sdot 1198711015840

119895119889119909)

120597119906119894

120597119905

+

1

ℎ

(int

1

0

1198711015840

119894sdot 119871119895119889119909)119906

119894

+(

6

ℎ

int

1

0

(

2

sum

119894=1

119906119894119871119894)

2

1198711015840

119894sdot 119871119895119889119909)119906

119894

= 0

(13)

where the symbol ( 1015840) denotes differentiation with respect to120578 which in matrix form is given by

(119860119890+ 119861119890)

120597119906119890

120597119905

+ 119862119890119906119890+ 119863119890(119906119890) 119906119890= 0 (14)

Abstract and Applied Analysis 3

where 119906119890 = (1199061 1199062)119879 are relevant nodal parameters The

element matrices are

119860119890

119895119896= int

1

0

119871119895119871119896119889120578

119861119890

119895119896=

120583

ℎ2int

1

0

1198711015840

1198951198711015840

119896119889120578

119862119890

119895119896=

1

ℎ

int

1

0

1198711015840

119895119871119896119889120578

119863119890

119895119896=

6

ℎ

int

1

0

(

2

sum

119898=1

119906119898119871119898)

2

1198711015840

119895119871119896119889120578

(15)

For the element [119909119894 119909119894+1] (119894 = 0 1 119873minus 1) the indices

119895 and 119896 take only the values 1 and 2 so that the matrices 119860119890119861119890 119862119890 and119863119890 are 2 times 2

119860119890= (

1

3

1

6

1

6

1

3

)

119861119890

119895119896=

120583

ℎ2(

1 minus1

minus1 1)

119862119890

119895119896=

1

ℎ

(

minus

1

2

1

2

minus

1

2

1

2

)

119863119890

119895119896

=

6

ℎ

times(

minus(

1

4

1199062

1+

1

6

11990611199062+

1

12

1199062

2)

1

4

1199062

1+

1

6

11990611199062+

1

12

1199062

2

minus(

1

12

1199062

1+

1

6

11990611199062+

1

4

1199062

2)

1

12

1199062

1+

1

6

11990611199062+

1

4

1199062

2

)

(16)

Assembling contributions from all elements leads to thefollowing matrix form of the coupled nonlinear ordinarydifferential equations

(119860 + 119861)

120597119906

120597119905

+ 119862119906 + 119863 (119906) 119906 = 0 (17)

where 119906 = (1199060 1199061 119906

119873)119879 contains all the nodal parame-

tersThe four assembledmatrices are tridiagonalThe generalrow for each matrix has the following form

119860 (

1

6

2

3

1

6

)

119861

120583

ℎ2(minus1 2 minus1)

119862

1

2ℎ

(minus1 0 1)

119863

6

ℎ

(minus

1

12

(1199062

119898minus1+ 2119906119898minus1

119906119898+ 31199062

119898)

1

12

(1199062

119898minus1minus 1199062

119898+1) +

1

6

119906119898(119906119898minus1

minus 119906119898+1

)

1

12

(31199062

119898+ 2119906119898119906119898+1

+ 1199062

119898+1))

(18)

32 Full Discretization We now consider a fully discretescheme Let 0 = 1199050 lt 1199051 lt sdot sdot sdot lt 119905119872 = 119879 be a uniformpartitionof 119869 into subintervals 119869119899 = (119905119899 119905119899+1) 119899 = 0 1 119872 minus 1 withlength Δ119905 = 119905

119899+1minus 119905119899= 119879119872 For a generic function V of

time set V119899 = V(119905119899)Weuse theCrank-Nicolson discretizationmethod in (17) with

120597119906

120597119905

=

119906119899minus 119906119899minus1

Δ119905

119899=

119906119899+ 119906119899minus1

2

(19)

Then (17) can be written as

(119860 + 119861)

120597119906

120597119905

+ 119862119899+ 119863 (

119899) 119899= 0 (20)

with 1199060 = 119906ℎ

0 This equation is symmetric about the point

119905 = 119905119899minus12

and one should therefore expect second orderaccuracy in time However the linearization decreases theorder of the time discretization error to119874(Δ119905)This drawbackcan be overcome by using an extrapolation technique in thelinearization of the nonlinear coefficient 119863 We choose 119906119899 =(32)119906

119899minus1minus (12)119906

119899minus2 for 119899 ge 2 and define

(119860 + 119861)

120597119906

120597119905

+ 119862119899+ 119863 (119906

119899) 119899= 0 (21)

Now the Crank-Nicolson method can be shown to producean error of order119874((Δ119905)2) Combining with the error of finiteelementmethod the Crank-Nicolson Galerkin finite elementmethod of (6) produces an error of order119874((Δ119905)2+ℎ2) overall[17]

We can obtain the following recurrence relation at point119909119898

[

1

6

minus 119887 minus

Δ119905

4ℎ

minus

6Δ119905

ℎ

(

1

24

V2119898minus1

+

1

12

V119898minus1

V119898+

1

8

V2119898)] 119906119899

119898minus1

+ [

2

3

+ 2119887 +

6Δ119905

ℎ

(

1

24

(V2119898minus1

minus V2119898+1

)

+

1

12

(V119898minus1

minus V119898+1

) V119898)] 119906119899

119898

+ [

1

6

minus 119887 +

Δ119905

4ℎ

+

6Δ119905

ℎ

(

1

8

V2119898+

1

12

V119898V119898+1

+

1

24

V2119898+1

)] 119906119899

119898+1

= [

1

6

minus 119887 +

Δ119905

4ℎ

+

6Δ119905

ℎ

(

1

24

V2119898minus1

+

1

12

V119898minus1

V119898+

1

8

V2119898)] 119906119899minus1

119898minus1

+ [

2

3

+ 2119887 minus

6Δ119905

ℎ

(

1

24

(V2119898minus1

minus V2119898+1

)

+

1

12

(V119898minus1

minus V119898+1

) V119898)] 119906119899minus1

119898

+ [

1

6

minus 119887 minus

Δ119905

4ℎ

minus

6Δ119905

ℎ

(

1

8

V2119898+

1

12

V119898V119898+1

+

1

24

V2119898+1

)] 119906119899minus1

119898+1

(22)

4 Abstract and Applied Analysis

where V119898and 119887 are given by

V119898=

3

2

119906119899minus1

119898minus

1

2

119906119899minus2

119898

119887 =

120583

ℎ2

(23)

As (22) is used only for 119899 ge 2 we must compute 1199061 byanother method For this we choose a predictor-correctormethod Let 119899 = 1 in (21) we can get the first approximation11990610 with 119906

1 replaced by 1199060 Then we use (12)(11990610 + 1199060)

to substitute 1199061 in (21) and regard the result as the finalapproximation 1199061 Or more precisely the concrete procedureis as follows First set

1199060= 1199060

ℎ (24)

then we calculate

(119860 + 119861)

11990610minus 1199060

Δ119905

+ 119862

11990610+ 1199060

2

+ 119863 (1199060)

11990610+ 1199060

2

= 0

(25)

finally we have

(119860 + 119861)

1199061minus 1199060

Δ119905

+ 119862

1199061+ 1199060

2

+ 119863(

11990610+ 1199060

2

)

1199061+ 1199060

2

= 0

(26)

Thus we can use theThomas algorithm to solve the linearalgebraic equations

In order to improve the accuracy in time we also useRunge-Kutta discretizationmethodThe fourth order Runge-Kutta discretization method in (17) can be described asfollows

119906119899+1

= 119906119899+

1198701+ 21198702+ 21198703+ 1198704

6

minus (119860 + 119861)1198701= [119862119906

119899+ 119863 (119906

119899) 119906119899] Δ119905

minus (119860 + 119861)1198702

= [119862(119906119899+

1

2

1198701) + 119863(119906

119899+

1

2

1198701)(119906119899+

1

2

1198701)]Δ119905

minus (119860 + 119861)1198703

= [119862(119906119899+

1

2

1198702) + 119863(119906

119899+

1

2

1198702)(119906119899+

1

2

1198702)]Δ119905

minus (119860 + 119861)1198704

= [119862 (119906119899+ 1198703) + 119863 (119906

119899+ 1198703) (119906119899+ 1198703)] Δ119905

(27)

4 Stability Analysis

A linear stability analysis is made of the growth factor 119892 of theerror 120576119899

119895in a typical Fourier mode of amplitude 120576119899

120576119899

119895= 119890119894119895119896ℎ

120576119899 (28)

where 119896 is a mode number and ℎ is the element size The vonNeumann stability method can only be applied to a linearscheme so we linearize the nonlinear term by assuming that119906 in this term is locally constant Substituting (28) into (22)gives

120576119899+1

= 119892 120576119899 (29)

119892 =

120572 + 120573119894

120572 minus 120573119894

(30)

where

120572 = (

1

3

minus 2119887) cos (119896ℎ) + 2

3

+ 2119887

120573 =

Δ119905

2ℎ

(1 + 120576119862) sin (119896ℎ) (31)

Taking the modulus of (30) gives

10038161003816100381610038161198921003816100381610038161003816= radic119892119892 = 1 (32)

Therefore our method is marginally stable

5 Numerical Experiments

In this section we present some numerical tests to checkthe efficiency and accuracy of our method which are thepropagation of single soliton and collision of two and threesolitons at different time levels Finally we investigate thedevelopment of theMaxwellian initial condition into solitarywaves

To illustrate the accuracy of the present method we use1198712- and 119871

infin-error norms

1198712=

10038171003817100381710038171003817119906exact119899

minus 119906119899100381710038171003817100381710038172= (ℎ

119873

sum

119894=1

10038161003816100381610038161003816119906exact119899119895

minus 119906119899

119895

10038161003816100381610038161003816

2

)

12

119871infin=

10038171003817100381710038171003817119906exact119899

minus 11990611989910038171003817100381710038171003817infin

= max119895

10038161003816100381610038161003816119906exact119899119895

minus 119906119899

119895

10038161003816100381610038161003816

(33)

to compare the numerical solution with the exact solutionwhich show the mean and maximum differences betweenthe numerical and analytical solutions The quantities 119868

1 1198682

and 1198683measure the conservation laws of our method during

propagation

51 Single Soliton The analytical values of the three variantsare

1198681=

120587radic119888

119901

1198682=

2119888

119901

+

2120583119901119888

3

1198683=

41198882

3119901

minus

2120583119901119888

3

(34)

For the purpose of comparison with the previous work[28] we choose the parameters 119909

0= 40 119888 = 005 ℎ =

119896 = 02 119879 = 10 and 0 le 119909 le 100 Then the amplitudeis 07071068 and the analytical values of three invariants are1198681= 3219174470 119868

2= 0465531499 and 119868

3= 0008001323

The relevant numerical results which applied the Crank-Nicolson scheme in (22) and the Runge-Kutta scheme in

Abstract and Applied Analysis 5

0 20 40 60 80 100

0

005

01

015

02

minus005

T = 0

T = 6

T = 10

Figure 1 Solitary wave with 119888 = 005 ℎ = 02 119896 = 02 and 1199090= 40

0 le 119909 le 100

0

50

100

24

68

10

0

05

1

15

2

Error-axis

minus1

minus05

times10minus4

t-axisx-ax

is

Figure 2 Error graph for 119888 = 005 ℎ = 02 119896 = 02 and1199090= 40 0 le

119909 le 100

(27) are presented in Tables 1 and 2 respectively The profilesof solitary waves at time = 5 and time = 10 are given inFigure 1 and the error distribution in the wave profile in thethree-dimensional view at different time level is illustratedin Figure 2 Table 3 presents the variants 119871

infinnorm and 119871

2

norm at time = 10 against the quadratic B-splines and thecubic B-splines in [28]

We also choose the parameters 119888 = 03 ℎ = 01 119896 = 001and 119909

0= 40 with the range [0 100] Then the amplitude

is 054772 and the analytical values of three invariants are1198681= 3581966678 119868

2= 1345076492 and 119868

3= 0153723028

Tables 4 and 5 show the results about the three invariants the1198712norm and the 119871

infinnorm at different times and compare

these results with those by the collocation method with

0 20 40 60 80 100

0

01

02

03

04

05

06

T = 2

T = 6

T = 10

minus01

Figure 3 Solitary wave with 119888 = 03 ℎ = 01 119896 = 001 and 1199090=

40 0 le 119909 le 100

0

2

4

0

50

100

x-axis

24

68

10

t-axis

minus4

minus2

Error-axis

times10minus4

Figure 4 Error graph for 119888 = 03 ℎ = 01 119896 = 001 and1199090= 40 0 le

119909 le 100

cubic B-splines in [27] using the Crank-Nicolson scheme in(22) and the Runge-Kutta scheme in (27) respectively Thechanges of the variants are satisfactorily small though ourresult for this case is notmore accurate than the result in [27]where the simulation is done up to 119905 = 10 The profile of theapproximate solution is given in Figures 3 and 4 shows theerror profile of the single solitary waves

The Runge-Kutta Galerkin method algorithm has beenrun for different space steps with a fixed time step 119896 = 001The error and convergence order of space for the 119871

2- and

119871infin-norm are recorded in Tables 6 and 7 respectively which

verify that the 1198712- and 119871

infin-error can achieve the order119874(ℎ2)

in space And the algorithm has also been run for differenttime steps with a fixed space step ℎ = 01 The error andconvergence order of time for the 119871

2- and 119871

infin-norm are

recorded in Tables 8 and 9 respectively

6 Abstract and Applied Analysis

Table 1 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007784E minus 03 0 02 3218839 04655233 8007697E minus 03 083581119864 minus 04 045779119864 minus 04

4 3218854 04655215 8007602E minus 03 136403119864 minus 04 045846119864 minus 04

6 3218842 04655196 8007507E minus 03 192290119864 minus 04 068890119864 minus 04

8 3218819 04655177 8007410E minus 03 247879119864 minus 04 091724119864 minus 04

10 3218794 04655159 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Table 2 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007778E minus 03 0 02 3218977 04655250 8007779E minus 03 2613735119864 minus 05 4577868119864 minus 05

4 3219083 04655250 8007780E minus 03 3432730119864 minus 05 2894973119864 minus 05

6 3219165 04655250 8007780E minus 03 4744950119864 minus 05 2214867119864 minus 05

8 3219229 04655250 8007780E minus 03 6052811119864 minus 05 2298029119864 minus 05

10 3219274 04655250 8007780E minus 03 7200630119864 minus 05 2346961119864 minus 05

Table 3 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 0 le 119909 le 100 and time = 10

Method 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

Our scheme in (22) 3218794 04655160 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Our scheme in (27) 3219274 04655250 8007780E minus 03 0720063119864 minus 04 0234696119864 minus 04

Quadratic [28] 3215653 04655665 8004883E minus 03 199288119864 minus 04 481156119864 minus 04

Cubic [28] 3215189 04655136 7999173E minus 03 453811119864 minus 04 625887119864 minus 04

Table 4 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson

time discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 142310119864 minus 04 045779119864 minus 04

4 3581966 1344973 01538262 265355119864 minus 04 153988119864 minus 04

6 3581966 1344973 01538261 374673119864 minus 04 200152119864 minus 04

8 3581966 1344972 01538260 477620119864 minus 04 242916119864 minus 04

10 3581966 1344972 01538260 577512119864 minus 04 284373119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Table 5 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 1396685119864 minus 04 0920780119864 minus 04

4 3581967 1344973 01538262 2592545119864 minus 04 1512530119864 minus 04

6 3581967 1344973 01538262 3647701119864 minus 04 1956752119864 minus 04

8 3581967 1344973 01538262 4638980119864 minus 04 2368004119864 minus 04

10 3581967 1344973 01538262 5600205119864 minus 04 2766940119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Abstract and Applied Analysis 7

Table 6 The order of convergence in space for 1198712error norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δℎ = 04

Order Order(0102) (0204)

2 1396685 times 10minus4 5598770 times 10minus4 2258793 times 10minus3 20031 201244 2592545 times 10minus4 1038054 times 10minus3 4168527 times 10minus3 20014 200576 3647701 times 10minus4 1459612 times 10minus3 5846531 times 10minus3 20005 200208 4638980 times 10minus4 1855582 times 10minus3 7421648 times 10minus3 20000 1999910 5600205 times 10minus4 2239531 times 10minus3 8948683 times 10minus3 19996 19985

Table 7 The order of convergence in space for 119871infinerror norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δ119905 = 04

Order Order(0102) (0204)

2 0920780 times 10minus4 3689931 times 10minus4 1480253 times 10minus3 20027 200424 1512530 times 10minus4 6055462 times 10minus4 2430139 times 10minus3 20012 200476 1956752 times 10minus4 7811765 times 10minus4 3126531 times 10minus3 19972 200088 2368004 times 10minus4 9454389 times 10minus4 3768796 times 10minus3 19973 1995010 2766940 times 10minus4 1104758 times 10minus3 4416067 times 10minus3 19974 19990

Table 8 The order of convergence in time for 1198712error norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00511878 000418182 4219760 times 10minus4 3613597 33088994 00835505 000806197 8133115 times 10minus4 3309253 33092536 01027137 001176588 1180928 times 10minus3 3125948 33166178 01141693 001561415 1547231 times 10minus3 2870249 333509410 01220153 001982141 1925846 times 10minus3 2621930 3363496

Table 9 The order of convergence in time for 119871infinerror norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00318115 2528470 times 10minus3 2505105 times 10minus4 3653213 33353224 00498808 4159059 times 10minus3 4456332 times 10minus4 3584155 32223286 00594582 5587773 times 10minus3 5905399 times 10minus4 3411531 32421678 00641225 7106547 times 10minus3 7270440 times 10minus4 3173611 328903410 00661177 8861734 times 10minus3 8671150 times 10minus4 2899376 3353294

52 Collision of Two Solitons In this section we study theinteraction of two solitary waves with the initial conditionsgiven by a linear sum of two separate solitary waves of variousamplitudes

119906 (119909 0) =

2

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (35)

where 119860119894= radic119888119894

119901119894= radic119888

119894120583(119888119894+ 1) 119894 = 1 2 and 119909

119894and

119888119894(119894 = 1 2) are arbitrary constants The analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

1198682=

21198881

1199011

+

21198882

1199012

+

212058311988811199011

3

+

212058311990121198882

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

minus

212058311990111198881

3

minus

212058311990121198882

3

(36)

In our numerical scheme we choose ℎ = 01 119896 = 001 1198881= 1

1198882= 05 119909

1= 10 and 119909

2= 40 with interval [0 100] Then

the amplitudes are in ratio 2 1 where 21198602= 1198601and the

analytical values for the conservation are 1198681= 82905324294

1198682= 5224332548 and 119868

3= 1799113742 The changes of

the invariants are satisfactorily small since the changes ofthe invariants 119868

1 1198682 and 119868

3are 13295 times 10minus4 22259 times 10minus5

and 68989 times 10minus5 respectively The results are recorded inTable 10 A graph of the two solitary waves collision plottedin a three-dimensional view at some discrete times is shownin Figure 5

53 Collision of Three Solitons In this section we study theinteraction of three solitary waves with the initial conditionsgiven by a linear sum of three separate solitary waves ofvarious amplitudes

119906 (119909 0) =

3

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (37)

8 Abstract and Applied Analysis

Table 10 Invariants and error norms for two solitary waves 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 ℎ = 01 119896 = 001 and 0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 8288129 8288129 5222934 5222933 1800512 18005132 8289522 8293010 5222910 5222951 1800479 18005074 8289462 8295348 5222887 5222954 1800458 18225076 8289304 8297822 5222864 5222956 1800433 18005078 8289233 8300322 5222840 5222958 1800411 180050810 8289231 8302630 5222817 5222960 1800388 1800509

Table 11 Invariants and error norms for three solitary waves 1198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 119909

3= 40 ℎ = 01 119896 = 001 and

0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 11800513 11800513 6490318 6490318 1997082 19970824 11799364 11807800 6485747 6490319 1992012 19966238 11797850 11812825 6483120 6490298 1988853 199607612 11797570 11817192 6482677 6490279 1988441 199636016 11791837 11820521 6474939 6490305 1982852 199687520 11788577 11823320 6469884 6490341 1977926 1996758

0

02

04

06

08

1

u-axis

0

50

100

x-axis

24

68

10

t-axis

minus02

Figure 5 Interaction of two solitons at times from 119905 = 2 4 6 8 10

for 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 and 0 le 119909 le 100

where 119860119894= radic119888119894

119901119894= radic119888119894120583(119888119894+ 1) 119894 = 1 2 3 and 119909

119894and

119888119894(119894 = 1 2 3) are arbitrary constantsThe analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

+

120587radic1198883

1199013

1198682=

21198881

1199011

+

21198882

1199012

+

21198883

1199013

+

212058311988811199011

3

+

212058311990121198882

3

+

212058311990131198883

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

+

41198883

2

31199013

minus

212058311990111198881

3

minus

212058311990121198882

3

minus

212058311990131198883

3

(38)

In this case we choose paraments as ℎ = 01 119896 = 0051198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 and 119909

3= 40

with interval [0 100]Then the amplitudes are in ratio 4 2 1

where 1198601= 2119860

2= 4119860

3and the analytical values for the

conservation are 1198681= 11802939794 119868

2= 6416902131

and 1198683= 1910917141 The changes of the invariants from

the initial variants approach zero throughout and agree withthe analytical values for the three invariants as presentedin Table 11 which indicates that our scheme is satisfactorilyconservativeThe geometry of the initial state and the profilesat time 119905 = 10 119905 = 20 and 119905 = 35 is shown graphically inFigure 6 respectively

54 The Maxwellian Initial Condition Finally we considerthe development of the Maxwellian initial condition

119906 (119909 0) = 119890minus(119909minus40)

2 (39)

into a train of solitary waves We choose different values 120583 =10 120583 = 05 and 120583 = 01 in our numerical scheme Thecomparison of the three variants 119868

1 1198682 and 119868

3with the earlier

result in [29] is presented in Table 12 here Crank-Nicolsonmethod is used for time discretization For 120583 = 10 thechanges of the variants 119868

1 1198682 and 119868

3with respect to the initial

values are 39492 times 10minus6 69575 times 10minus4 and 31509 times 10minus3respectively whereas they are 28210 times 10minus6 260000 times 10minus4and 70771times 10minus4 in [29] For 120583 = 05 the variants are changedby 78982 times 10minus6 68214 times 10minus5 and 61410 times 10minus4 in this caseThedevelopment of theMaxwellian initial condition is shownin Figure 7 with different parameter values respectively Thesmaller 120583 there is the more the number of solitary waves willform The simulations are done up to time = 10 in this caseFrom the previous work [25] the total number of solitarywaves and the values 120583 have the approximate relation

119873 cong [

1

5radic120583

] (40)

Abstract and Applied Analysis 9

Table 12 Values of 1198681 1198682 and 119868

3for Maxwellian initial condition when ℎ = 01 and 119896 = 001 and the space interval [0 100]

120583 Time 1198681

1198682

1198683

1198681[29] 119868

2[29] 119868

3[29]

1

2 1772528 2499467 minus03633570 1772449 2506352 minus036681494 1772544 2499870 minus03643380 1772446 2506235 minus036669746 1772532 2500822 minus03657774 1772447 2506171 minus036663268 1772524 2500977 minus03661331 1772446 2506123 minus0366586010 1772521 2501206 minus03665079 1772444 22060926 minus03665553

05

2 1772534 1876446 02597450 1772451 1879888 025964944 1772534 1876534 02587968 1772446 1879855 025968286 1772528 1876600 02583802 1772449 1879841 025969738 1772523 1876581 02582386 1772450 1879834 0259705010 1772520 1876574 02581499 1772449 1879828 02597092

01

2 1770991 1373320 07687222 1772452 1378607 076087774 1769863 1371236 07658462 1772451 1378577 076083646 1768742 1369139 07630269 1772451 1378546 076079378 1767630 1367058 07602389 1772451 1378515 0760752910 1766526 1364993 07574795 1772453 1378483 07607117

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(a) Time = 0

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(b) Time = 10

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(c) Time = 20

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(d) Time = 35

Figure 6 Interaction of three solitary waves

10 Abstract and Applied Analysis

0 20 40 60 80 100

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

(a) 120583 = 10

0 20 40 60 80 100minus04

minus02

0

02

04

06

08

1

(b) 120583 = 05

0 20 40 60 80 100minus02

0

02

04

06

08

1

(c) 120583 = 01

Figure 7 Maxwellian initial condition with different values 120583

6 Conclusion

In this paper we have developed a Galerkin linear finiteelementmethod to investigate the propagation of solitons andtheir interactions governed by the nonlinear MRLW equa-tion An extrapolation technique has been used to improvethe accuracy in time for this numerical method A linearstability analysis shows that this method is marginally stableThe high efficiency and accuracy of our method is testedby the numerical experiments propagation of single solitoncollision of two and three solitons and development of theMaxwellian initial condition into solitary waves Moreoverthis method numerically satisfies the conservation laws ofmass momentum and energy

Conflict of Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFund of China (11371289) and the China Scholarship Council(201206285005)

References

[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966

[2] T B Benjamin J L Bona and J J Mahony ldquoModel equationsfor long waves in nonlinear dispersive systemsrdquo PhilosophicalTransactions of the Royal Society of London A vol 272 no 1220pp 47ndash78 1972

[3] S Kutluay and A Esen ldquoA finite difference solution of theregularized long-wave equationrdquo Mathematical Problems inEngineering vol 2006 Article ID 85743 14 pages 2006

Abstract and Applied Analysis 11

[4] J Lin Z Xie and J Zhou ldquoHigh-order compact differencescheme for the regularized long wave equationrdquo Communica-tions in Numerical Methods in Engineering vol 23 no 2 pp135ndash156 2007

[5] P Avilez-Valente and F J Seabra-Santos ldquoA Petrov-Galerkinfinite element scheme for the regularized long wave equationrdquoComputational Mechanics vol 34 no 4 pp 256ndash270 2004

[6] A Esen and S Kutluay ldquoApplication of a lumped Galerkinmethod to the regularized long wave equationrdquo Applied Mathe-matics and Computation vol 174 no 2 pp 833ndash845 2006

[7] Y Liu H Li Y Du and J Wang ldquoExplicit multistep mixedfinite element method for RLW equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 768976 12 pages 2013

[8] B Saka and I Dag ldquoA numerical solution of the RLW equationby Galerkin method using quartic B-splinesrdquo Communicationsin Numerical Methods in Engineering with Biomedical Applica-tions vol 24 no 11 pp 1339ndash1361 2008

[9] I Dag ldquoLeast-squares quadratic B-spline finite elementmethodfor the regularised long wave equationrdquo Computer Methods inApplied Mechanics and Engineering vol 182 no 1-2 pp 205ndash215 2000

[10] I Dag andM N Ozer ldquoApproximation of the RLW equation bythe least square cubic B-spline finite element methodrdquo AppliedMathematical Modelling vol 25 no 3 pp 221ndash231 2001

[11] H Gu and N Chen ldquoLeast-squares mixed finite elementmethods for the RLW equationsrdquoNumericalMethods for PartialDifferential Equations vol 24 no 3 pp 749ndash758 2008

[12] A A Soliman and K R Raslan ldquoCollocation method usingquadratic B-spline for the RLW equationrdquo International Journalof Computer Mathematics vol 78 no 3 pp 399ndash412 2001

[13] I Dag B Saka and D Irk ldquoApplication of cubic B-splines fornumerical solution of the RLW equationrdquo Applied Mathematicsand Computation vol 159 no 2 pp 373ndash389 2004

[14] A A Soliman and M H Hussien ldquoCollocation solution forRLW equation with septic splinerdquo Applied Mathematics andComputation vol 161 no 2 pp 623ndash636 2005

[15] S U Islam S Haq and A Ali ldquoA meshfree method for thenumerical solution of the RLW equationrdquo Journal of Compu-tational and Applied Mathematics vol 223 no 2 pp 997ndash10122009

[16] S Haq S U Islam andA Ali ldquoA numerical meshfree techniquefor the solution of the MEW equationrdquo Computer Modeling inEngineering amp Sciences vol 38 no 1 pp 1ndash23 2008

[17] L Mei and Y Chen ldquoExplicit multistep method for thenumerical solution of RLW equationrdquoAppliedMathematics andComputation vol 218 no 18 pp 9547ndash9554 2012

[18] L Zhang ldquoAfinite difference scheme for generalized regularizedlong-wave equationrdquo Applied Mathematics and Computationvol 168 no 2 pp 962ndash972 2005

[19] MMohammadi and RMokhtari ldquoSolving the generalized reg-ularized longwave equation on the basis of a reproducing kernelspacerdquo Journal of Computational and Applied Mathematics vol235 no 14 pp 4003ndash4014 2011

[20] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004

[21] T Roshan ldquoA Petrov-Galerkin method for solving the gener-alized regularized long wave (GRLW) equationrdquo Computers ampMathematics with Applications vol 63 no 5 pp 943ndash956 2012

[22] A A Soliman ldquoNumerical simulation of the generalizedregularized long wave equation by Hersquos variational iterationmethodrdquoMathematics and Computers in Simulation vol 70 no2 pp 119ndash124 2005

[23] R Mokhtari and M Mohammadi ldquoNumerical solution ofGRLW equation using Sinc-collocation methodrdquo ComputerPhysics Communications vol 181 no 7 pp 1266ndash1274 2010

[24] A H A Ali A A Soliman and K R Raslan ldquoSoliton solutionfor nonlinear partial differential equations by cosine-functionmethodrdquo Physics Letters A vol 368 no 3-4 pp 299ndash304 2007

[25] L R T Gardner G A Gardner F A Irk and N K AmeinldquoApproximations of solitary waves of the MRLW equation byB-spline finite elementsrdquo The Arabian Journal for Science andEngineering A vol 22 no 2 pp 183ndash193 1997

[26] A K Khalifa K R Raslan and H M Alzubaidi ldquoA finite dif-ference scheme for the MRLW and solitary wave interactionsrdquoApplied Mathematics and Computation vol 189 no 1 pp 346ndash354 2007

[27] A K Khalifa K R Raslan and H M Alzubaidi ldquoA collocationmethod with cubic B-splines for solving the MRLW equationrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 406ndash418 2008

[28] K R Raslan and S M Hassan ldquoSolitary waves for the MRLWequationrdquo Applied Mathematics Letters vol 22 no 7 pp 984ndash989 2009

[29] K R Raslan and T S EL-Danaf ldquoSolitary waves solutions of theMRLW equation using quintic B-splinesrdquo Journal of King SaudUniversity Science vol 22 no 3 pp 161ndash166 2010

[30] A K Khalifa K R Raslan and H M Alzubaidi ldquoNumericalstudy using ADM for the modified regularized long waveequationrdquo Applied Mathematical Modelling vol 32 no 12 pp2962ndash2972 2008

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 3

where 119906119890 = (1199061 1199062)119879 are relevant nodal parameters The

element matrices are

119860119890

119895119896= int

1

0

119871119895119871119896119889120578

119861119890

119895119896=

120583

ℎ2int

1

0

1198711015840

1198951198711015840

119896119889120578

119862119890

119895119896=

1

ℎ

int

1

0

1198711015840

119895119871119896119889120578

119863119890

119895119896=

6

ℎ

int

1

0

(

2

sum

119898=1

119906119898119871119898)

2

1198711015840

119895119871119896119889120578

(15)

For the element [119909119894 119909119894+1] (119894 = 0 1 119873minus 1) the indices

119895 and 119896 take only the values 1 and 2 so that the matrices 119860119890119861119890 119862119890 and119863119890 are 2 times 2

119860119890= (

1

3

1

6

1

6

1

3

)

119861119890

119895119896=

120583

ℎ2(

1 minus1

minus1 1)

119862119890

119895119896=

1

ℎ

(

minus

1

2

1

2

minus

1

2

1

2

)

119863119890

119895119896

=

6

ℎ

times(

minus(

1

4

1199062

1+

1

6

11990611199062+

1

12

1199062

2)

1

4

1199062

1+

1

6

11990611199062+

1

12

1199062

2

minus(

1

12

1199062

1+

1

6

11990611199062+

1

4

1199062

2)

1

12

1199062

1+

1

6

11990611199062+

1

4

1199062

2

)

(16)

Assembling contributions from all elements leads to thefollowing matrix form of the coupled nonlinear ordinarydifferential equations

(119860 + 119861)

120597119906

120597119905

+ 119862119906 + 119863 (119906) 119906 = 0 (17)

where 119906 = (1199060 1199061 119906

119873)119879 contains all the nodal parame-

tersThe four assembledmatrices are tridiagonalThe generalrow for each matrix has the following form

119860 (

1

6

2

3

1

6

)

119861

120583

ℎ2(minus1 2 minus1)

119862

1

2ℎ

(minus1 0 1)

119863

6

ℎ

(minus

1

12

(1199062

119898minus1+ 2119906119898minus1

119906119898+ 31199062

119898)

1

12

(1199062

119898minus1minus 1199062

119898+1) +

1

6

119906119898(119906119898minus1

minus 119906119898+1

)

1

12

(31199062

119898+ 2119906119898119906119898+1

+ 1199062

119898+1))

(18)

32 Full Discretization We now consider a fully discretescheme Let 0 = 1199050 lt 1199051 lt sdot sdot sdot lt 119905119872 = 119879 be a uniformpartitionof 119869 into subintervals 119869119899 = (119905119899 119905119899+1) 119899 = 0 1 119872 minus 1 withlength Δ119905 = 119905

119899+1minus 119905119899= 119879119872 For a generic function V of

time set V119899 = V(119905119899)Weuse theCrank-Nicolson discretizationmethod in (17) with

120597119906

120597119905

=

119906119899minus 119906119899minus1

Δ119905

119899=

119906119899+ 119906119899minus1

2

(19)

Then (17) can be written as

(119860 + 119861)

120597119906

120597119905

+ 119862119899+ 119863 (

119899) 119899= 0 (20)

with 1199060 = 119906ℎ

0 This equation is symmetric about the point

119905 = 119905119899minus12

and one should therefore expect second orderaccuracy in time However the linearization decreases theorder of the time discretization error to119874(Δ119905)This drawbackcan be overcome by using an extrapolation technique in thelinearization of the nonlinear coefficient 119863 We choose 119906119899 =(32)119906

119899minus1minus (12)119906

119899minus2 for 119899 ge 2 and define

(119860 + 119861)

120597119906

120597119905

+ 119862119899+ 119863 (119906

119899) 119899= 0 (21)

Now the Crank-Nicolson method can be shown to producean error of order119874((Δ119905)2) Combining with the error of finiteelementmethod the Crank-Nicolson Galerkin finite elementmethod of (6) produces an error of order119874((Δ119905)2+ℎ2) overall[17]

We can obtain the following recurrence relation at point119909119898

[

1

6

minus 119887 minus

Δ119905

4ℎ

minus

6Δ119905

ℎ

(

1

24

V2119898minus1

+

1

12

V119898minus1

V119898+

1

8

V2119898)] 119906119899

119898minus1

+ [

2

3

+ 2119887 +

6Δ119905

ℎ

(

1

24

(V2119898minus1

minus V2119898+1

)

+

1

12

(V119898minus1

minus V119898+1

) V119898)] 119906119899

119898

+ [

1

6

minus 119887 +

Δ119905

4ℎ

+

6Δ119905

ℎ

(

1

8

V2119898+

1

12

V119898V119898+1

+

1

24

V2119898+1

)] 119906119899

119898+1

= [

1

6

minus 119887 +

Δ119905

4ℎ

+

6Δ119905

ℎ

(

1

24

V2119898minus1

+

1

12

V119898minus1

V119898+

1

8

V2119898)] 119906119899minus1

119898minus1

+ [

2

3

+ 2119887 minus

6Δ119905

ℎ

(

1

24

(V2119898minus1

minus V2119898+1

)

+

1

12

(V119898minus1

minus V119898+1

) V119898)] 119906119899minus1

119898

+ [

1

6

minus 119887 minus

Δ119905

4ℎ

minus

6Δ119905

ℎ

(

1

8

V2119898+

1

12

V119898V119898+1

+

1

24

V2119898+1

)] 119906119899minus1

119898+1

(22)

4 Abstract and Applied Analysis

where V119898and 119887 are given by

V119898=

3

2

119906119899minus1

119898minus

1

2

119906119899minus2

119898

119887 =

120583

ℎ2

(23)

As (22) is used only for 119899 ge 2 we must compute 1199061 byanother method For this we choose a predictor-correctormethod Let 119899 = 1 in (21) we can get the first approximation11990610 with 119906

1 replaced by 1199060 Then we use (12)(11990610 + 1199060)

to substitute 1199061 in (21) and regard the result as the finalapproximation 1199061 Or more precisely the concrete procedureis as follows First set

1199060= 1199060

ℎ (24)

then we calculate

(119860 + 119861)

11990610minus 1199060

Δ119905

+ 119862

11990610+ 1199060

2

+ 119863 (1199060)

11990610+ 1199060

2

= 0

(25)

finally we have

(119860 + 119861)

1199061minus 1199060

Δ119905

+ 119862

1199061+ 1199060

2

+ 119863(

11990610+ 1199060

2

)

1199061+ 1199060

2

= 0

(26)

Thus we can use theThomas algorithm to solve the linearalgebraic equations

In order to improve the accuracy in time we also useRunge-Kutta discretizationmethodThe fourth order Runge-Kutta discretization method in (17) can be described asfollows

119906119899+1

= 119906119899+

1198701+ 21198702+ 21198703+ 1198704

6

minus (119860 + 119861)1198701= [119862119906

119899+ 119863 (119906

119899) 119906119899] Δ119905

minus (119860 + 119861)1198702

= [119862(119906119899+

1

2

1198701) + 119863(119906

119899+

1

2

1198701)(119906119899+

1

2

1198701)]Δ119905

minus (119860 + 119861)1198703

= [119862(119906119899+

1

2

1198702) + 119863(119906

119899+

1

2

1198702)(119906119899+

1

2

1198702)]Δ119905

minus (119860 + 119861)1198704

= [119862 (119906119899+ 1198703) + 119863 (119906

119899+ 1198703) (119906119899+ 1198703)] Δ119905

(27)

4 Stability Analysis

A linear stability analysis is made of the growth factor 119892 of theerror 120576119899

119895in a typical Fourier mode of amplitude 120576119899

120576119899

119895= 119890119894119895119896ℎ

120576119899 (28)

where 119896 is a mode number and ℎ is the element size The vonNeumann stability method can only be applied to a linearscheme so we linearize the nonlinear term by assuming that119906 in this term is locally constant Substituting (28) into (22)gives

120576119899+1

= 119892 120576119899 (29)

119892 =

120572 + 120573119894

120572 minus 120573119894

(30)

where

120572 = (

1

3

minus 2119887) cos (119896ℎ) + 2

3

+ 2119887

120573 =

Δ119905

2ℎ

(1 + 120576119862) sin (119896ℎ) (31)

Taking the modulus of (30) gives

10038161003816100381610038161198921003816100381610038161003816= radic119892119892 = 1 (32)

Therefore our method is marginally stable

5 Numerical Experiments

In this section we present some numerical tests to checkthe efficiency and accuracy of our method which are thepropagation of single soliton and collision of two and threesolitons at different time levels Finally we investigate thedevelopment of theMaxwellian initial condition into solitarywaves

To illustrate the accuracy of the present method we use1198712- and 119871

infin-error norms

1198712=

10038171003817100381710038171003817119906exact119899

minus 119906119899100381710038171003817100381710038172= (ℎ

119873

sum

119894=1

10038161003816100381610038161003816119906exact119899119895

minus 119906119899

119895

10038161003816100381610038161003816

2

)

12

119871infin=

10038171003817100381710038171003817119906exact119899

minus 11990611989910038171003817100381710038171003817infin

= max119895

10038161003816100381610038161003816119906exact119899119895

minus 119906119899

119895

10038161003816100381610038161003816

(33)

to compare the numerical solution with the exact solutionwhich show the mean and maximum differences betweenthe numerical and analytical solutions The quantities 119868

1 1198682

and 1198683measure the conservation laws of our method during

propagation

51 Single Soliton The analytical values of the three variantsare

1198681=

120587radic119888

119901

1198682=

2119888

119901

+

2120583119901119888

3

1198683=

41198882

3119901

minus

2120583119901119888

3

(34)

For the purpose of comparison with the previous work[28] we choose the parameters 119909

0= 40 119888 = 005 ℎ =

119896 = 02 119879 = 10 and 0 le 119909 le 100 Then the amplitudeis 07071068 and the analytical values of three invariants are1198681= 3219174470 119868

2= 0465531499 and 119868

3= 0008001323

The relevant numerical results which applied the Crank-Nicolson scheme in (22) and the Runge-Kutta scheme in

Abstract and Applied Analysis 5

0 20 40 60 80 100

0

005

01

015

02

minus005

T = 0

T = 6

T = 10

Figure 1 Solitary wave with 119888 = 005 ℎ = 02 119896 = 02 and 1199090= 40

0 le 119909 le 100

0

50

100

24

68

10

0

05

1

15

2

Error-axis

minus1

minus05

times10minus4

t-axisx-ax

is

Figure 2 Error graph for 119888 = 005 ℎ = 02 119896 = 02 and1199090= 40 0 le

119909 le 100

(27) are presented in Tables 1 and 2 respectively The profilesof solitary waves at time = 5 and time = 10 are given inFigure 1 and the error distribution in the wave profile in thethree-dimensional view at different time level is illustratedin Figure 2 Table 3 presents the variants 119871

infinnorm and 119871

2

norm at time = 10 against the quadratic B-splines and thecubic B-splines in [28]

We also choose the parameters 119888 = 03 ℎ = 01 119896 = 001and 119909

0= 40 with the range [0 100] Then the amplitude

is 054772 and the analytical values of three invariants are1198681= 3581966678 119868

2= 1345076492 and 119868

3= 0153723028

Tables 4 and 5 show the results about the three invariants the1198712norm and the 119871

infinnorm at different times and compare

these results with those by the collocation method with

0 20 40 60 80 100

0

01

02

03

04

05

06

T = 2

T = 6

T = 10

minus01

Figure 3 Solitary wave with 119888 = 03 ℎ = 01 119896 = 001 and 1199090=

40 0 le 119909 le 100

0

2

4

0

50

100

x-axis

24

68

10

t-axis

minus4

minus2

Error-axis

times10minus4

Figure 4 Error graph for 119888 = 03 ℎ = 01 119896 = 001 and1199090= 40 0 le

119909 le 100

cubic B-splines in [27] using the Crank-Nicolson scheme in(22) and the Runge-Kutta scheme in (27) respectively Thechanges of the variants are satisfactorily small though ourresult for this case is notmore accurate than the result in [27]where the simulation is done up to 119905 = 10 The profile of theapproximate solution is given in Figures 3 and 4 shows theerror profile of the single solitary waves

The Runge-Kutta Galerkin method algorithm has beenrun for different space steps with a fixed time step 119896 = 001The error and convergence order of space for the 119871

2- and

119871infin-norm are recorded in Tables 6 and 7 respectively which

verify that the 1198712- and 119871

infin-error can achieve the order119874(ℎ2)

in space And the algorithm has also been run for differenttime steps with a fixed space step ℎ = 01 The error andconvergence order of time for the 119871

2- and 119871

infin-norm are

recorded in Tables 8 and 9 respectively

6 Abstract and Applied Analysis

Table 1 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007784E minus 03 0 02 3218839 04655233 8007697E minus 03 083581119864 minus 04 045779119864 minus 04

4 3218854 04655215 8007602E minus 03 136403119864 minus 04 045846119864 minus 04

6 3218842 04655196 8007507E minus 03 192290119864 minus 04 068890119864 minus 04

8 3218819 04655177 8007410E minus 03 247879119864 minus 04 091724119864 minus 04

10 3218794 04655159 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Table 2 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007778E minus 03 0 02 3218977 04655250 8007779E minus 03 2613735119864 minus 05 4577868119864 minus 05

4 3219083 04655250 8007780E minus 03 3432730119864 minus 05 2894973119864 minus 05

6 3219165 04655250 8007780E minus 03 4744950119864 minus 05 2214867119864 minus 05

8 3219229 04655250 8007780E minus 03 6052811119864 minus 05 2298029119864 minus 05

10 3219274 04655250 8007780E minus 03 7200630119864 minus 05 2346961119864 minus 05

Table 3 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 0 le 119909 le 100 and time = 10

Method 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

Our scheme in (22) 3218794 04655160 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Our scheme in (27) 3219274 04655250 8007780E minus 03 0720063119864 minus 04 0234696119864 minus 04

Quadratic [28] 3215653 04655665 8004883E minus 03 199288119864 minus 04 481156119864 minus 04

Cubic [28] 3215189 04655136 7999173E minus 03 453811119864 minus 04 625887119864 minus 04

Table 4 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson

time discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 142310119864 minus 04 045779119864 minus 04

4 3581966 1344973 01538262 265355119864 minus 04 153988119864 minus 04

6 3581966 1344973 01538261 374673119864 minus 04 200152119864 minus 04

8 3581966 1344972 01538260 477620119864 minus 04 242916119864 minus 04

10 3581966 1344972 01538260 577512119864 minus 04 284373119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Table 5 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 1396685119864 minus 04 0920780119864 minus 04

4 3581967 1344973 01538262 2592545119864 minus 04 1512530119864 minus 04

6 3581967 1344973 01538262 3647701119864 minus 04 1956752119864 minus 04

8 3581967 1344973 01538262 4638980119864 minus 04 2368004119864 minus 04

10 3581967 1344973 01538262 5600205119864 minus 04 2766940119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Abstract and Applied Analysis 7

Table 6 The order of convergence in space for 1198712error norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δℎ = 04

Order Order(0102) (0204)

2 1396685 times 10minus4 5598770 times 10minus4 2258793 times 10minus3 20031 201244 2592545 times 10minus4 1038054 times 10minus3 4168527 times 10minus3 20014 200576 3647701 times 10minus4 1459612 times 10minus3 5846531 times 10minus3 20005 200208 4638980 times 10minus4 1855582 times 10minus3 7421648 times 10minus3 20000 1999910 5600205 times 10minus4 2239531 times 10minus3 8948683 times 10minus3 19996 19985

Table 7 The order of convergence in space for 119871infinerror norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δ119905 = 04

Order Order(0102) (0204)

2 0920780 times 10minus4 3689931 times 10minus4 1480253 times 10minus3 20027 200424 1512530 times 10minus4 6055462 times 10minus4 2430139 times 10minus3 20012 200476 1956752 times 10minus4 7811765 times 10minus4 3126531 times 10minus3 19972 200088 2368004 times 10minus4 9454389 times 10minus4 3768796 times 10minus3 19973 1995010 2766940 times 10minus4 1104758 times 10minus3 4416067 times 10minus3 19974 19990

Table 8 The order of convergence in time for 1198712error norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00511878 000418182 4219760 times 10minus4 3613597 33088994 00835505 000806197 8133115 times 10minus4 3309253 33092536 01027137 001176588 1180928 times 10minus3 3125948 33166178 01141693 001561415 1547231 times 10minus3 2870249 333509410 01220153 001982141 1925846 times 10minus3 2621930 3363496

Table 9 The order of convergence in time for 119871infinerror norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00318115 2528470 times 10minus3 2505105 times 10minus4 3653213 33353224 00498808 4159059 times 10minus3 4456332 times 10minus4 3584155 32223286 00594582 5587773 times 10minus3 5905399 times 10minus4 3411531 32421678 00641225 7106547 times 10minus3 7270440 times 10minus4 3173611 328903410 00661177 8861734 times 10minus3 8671150 times 10minus4 2899376 3353294

52 Collision of Two Solitons In this section we study theinteraction of two solitary waves with the initial conditionsgiven by a linear sum of two separate solitary waves of variousamplitudes

119906 (119909 0) =

2

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (35)

where 119860119894= radic119888119894

119901119894= radic119888

119894120583(119888119894+ 1) 119894 = 1 2 and 119909

119894and

119888119894(119894 = 1 2) are arbitrary constants The analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

1198682=

21198881

1199011

+

21198882

1199012

+

212058311988811199011

3

+

212058311990121198882

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

minus

212058311990111198881

3

minus

212058311990121198882

3

(36)

In our numerical scheme we choose ℎ = 01 119896 = 001 1198881= 1

1198882= 05 119909

1= 10 and 119909

2= 40 with interval [0 100] Then

the amplitudes are in ratio 2 1 where 21198602= 1198601and the

analytical values for the conservation are 1198681= 82905324294

1198682= 5224332548 and 119868

3= 1799113742 The changes of

the invariants are satisfactorily small since the changes ofthe invariants 119868

1 1198682 and 119868

3are 13295 times 10minus4 22259 times 10minus5

and 68989 times 10minus5 respectively The results are recorded inTable 10 A graph of the two solitary waves collision plottedin a three-dimensional view at some discrete times is shownin Figure 5

53 Collision of Three Solitons In this section we study theinteraction of three solitary waves with the initial conditionsgiven by a linear sum of three separate solitary waves ofvarious amplitudes

119906 (119909 0) =

3

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (37)

8 Abstract and Applied Analysis

Table 10 Invariants and error norms for two solitary waves 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 ℎ = 01 119896 = 001 and 0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 8288129 8288129 5222934 5222933 1800512 18005132 8289522 8293010 5222910 5222951 1800479 18005074 8289462 8295348 5222887 5222954 1800458 18225076 8289304 8297822 5222864 5222956 1800433 18005078 8289233 8300322 5222840 5222958 1800411 180050810 8289231 8302630 5222817 5222960 1800388 1800509

Table 11 Invariants and error norms for three solitary waves 1198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 119909

3= 40 ℎ = 01 119896 = 001 and

0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 11800513 11800513 6490318 6490318 1997082 19970824 11799364 11807800 6485747 6490319 1992012 19966238 11797850 11812825 6483120 6490298 1988853 199607612 11797570 11817192 6482677 6490279 1988441 199636016 11791837 11820521 6474939 6490305 1982852 199687520 11788577 11823320 6469884 6490341 1977926 1996758

0

02

04

06

08

1

u-axis

0

50

100

x-axis

24

68

10

t-axis

minus02

Figure 5 Interaction of two solitons at times from 119905 = 2 4 6 8 10

for 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 and 0 le 119909 le 100

where 119860119894= radic119888119894

119901119894= radic119888119894120583(119888119894+ 1) 119894 = 1 2 3 and 119909

119894and

119888119894(119894 = 1 2 3) are arbitrary constantsThe analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

+

120587radic1198883

1199013

1198682=

21198881

1199011

+

21198882

1199012

+

21198883

1199013

+

212058311988811199011

3

+

212058311990121198882

3

+

212058311990131198883

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

+

41198883

2

31199013

minus

212058311990111198881

3

minus

212058311990121198882

3

minus

212058311990131198883

3

(38)

In this case we choose paraments as ℎ = 01 119896 = 0051198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 and 119909

3= 40

with interval [0 100]Then the amplitudes are in ratio 4 2 1

where 1198601= 2119860

2= 4119860

3and the analytical values for the

conservation are 1198681= 11802939794 119868

2= 6416902131

and 1198683= 1910917141 The changes of the invariants from

the initial variants approach zero throughout and agree withthe analytical values for the three invariants as presentedin Table 11 which indicates that our scheme is satisfactorilyconservativeThe geometry of the initial state and the profilesat time 119905 = 10 119905 = 20 and 119905 = 35 is shown graphically inFigure 6 respectively

54 The Maxwellian Initial Condition Finally we considerthe development of the Maxwellian initial condition

119906 (119909 0) = 119890minus(119909minus40)

2 (39)

into a train of solitary waves We choose different values 120583 =10 120583 = 05 and 120583 = 01 in our numerical scheme Thecomparison of the three variants 119868

1 1198682 and 119868

3with the earlier

result in [29] is presented in Table 12 here Crank-Nicolsonmethod is used for time discretization For 120583 = 10 thechanges of the variants 119868

1 1198682 and 119868

3with respect to the initial

values are 39492 times 10minus6 69575 times 10minus4 and 31509 times 10minus3respectively whereas they are 28210 times 10minus6 260000 times 10minus4and 70771times 10minus4 in [29] For 120583 = 05 the variants are changedby 78982 times 10minus6 68214 times 10minus5 and 61410 times 10minus4 in this caseThedevelopment of theMaxwellian initial condition is shownin Figure 7 with different parameter values respectively Thesmaller 120583 there is the more the number of solitary waves willform The simulations are done up to time = 10 in this caseFrom the previous work [25] the total number of solitarywaves and the values 120583 have the approximate relation

119873 cong [

1

5radic120583

] (40)

Abstract and Applied Analysis 9

Table 12 Values of 1198681 1198682 and 119868

3for Maxwellian initial condition when ℎ = 01 and 119896 = 001 and the space interval [0 100]

120583 Time 1198681

1198682

1198683

1198681[29] 119868

2[29] 119868

3[29]

1

2 1772528 2499467 minus03633570 1772449 2506352 minus036681494 1772544 2499870 minus03643380 1772446 2506235 minus036669746 1772532 2500822 minus03657774 1772447 2506171 minus036663268 1772524 2500977 minus03661331 1772446 2506123 minus0366586010 1772521 2501206 minus03665079 1772444 22060926 minus03665553

05

2 1772534 1876446 02597450 1772451 1879888 025964944 1772534 1876534 02587968 1772446 1879855 025968286 1772528 1876600 02583802 1772449 1879841 025969738 1772523 1876581 02582386 1772450 1879834 0259705010 1772520 1876574 02581499 1772449 1879828 02597092

01

2 1770991 1373320 07687222 1772452 1378607 076087774 1769863 1371236 07658462 1772451 1378577 076083646 1768742 1369139 07630269 1772451 1378546 076079378 1767630 1367058 07602389 1772451 1378515 0760752910 1766526 1364993 07574795 1772453 1378483 07607117

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(a) Time = 0

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(b) Time = 10

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(c) Time = 20

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(d) Time = 35

Figure 6 Interaction of three solitary waves

10 Abstract and Applied Analysis

0 20 40 60 80 100

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

(a) 120583 = 10

0 20 40 60 80 100minus04

minus02

0

02

04

06

08

1

(b) 120583 = 05

0 20 40 60 80 100minus02

0

02

04

06

08

1

(c) 120583 = 01

Figure 7 Maxwellian initial condition with different values 120583

6 Conclusion

In this paper we have developed a Galerkin linear finiteelementmethod to investigate the propagation of solitons andtheir interactions governed by the nonlinear MRLW equa-tion An extrapolation technique has been used to improvethe accuracy in time for this numerical method A linearstability analysis shows that this method is marginally stableThe high efficiency and accuracy of our method is testedby the numerical experiments propagation of single solitoncollision of two and three solitons and development of theMaxwellian initial condition into solitary waves Moreoverthis method numerically satisfies the conservation laws ofmass momentum and energy

Conflict of Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFund of China (11371289) and the China Scholarship Council(201206285005)

References

[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966

[2] T B Benjamin J L Bona and J J Mahony ldquoModel equationsfor long waves in nonlinear dispersive systemsrdquo PhilosophicalTransactions of the Royal Society of London A vol 272 no 1220pp 47ndash78 1972

[3] S Kutluay and A Esen ldquoA finite difference solution of theregularized long-wave equationrdquo Mathematical Problems inEngineering vol 2006 Article ID 85743 14 pages 2006

Abstract and Applied Analysis 11

[4] J Lin Z Xie and J Zhou ldquoHigh-order compact differencescheme for the regularized long wave equationrdquo Communica-tions in Numerical Methods in Engineering vol 23 no 2 pp135ndash156 2007

[5] P Avilez-Valente and F J Seabra-Santos ldquoA Petrov-Galerkinfinite element scheme for the regularized long wave equationrdquoComputational Mechanics vol 34 no 4 pp 256ndash270 2004

[6] A Esen and S Kutluay ldquoApplication of a lumped Galerkinmethod to the regularized long wave equationrdquo Applied Mathe-matics and Computation vol 174 no 2 pp 833ndash845 2006

[7] Y Liu H Li Y Du and J Wang ldquoExplicit multistep mixedfinite element method for RLW equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 768976 12 pages 2013

[8] B Saka and I Dag ldquoA numerical solution of the RLW equationby Galerkin method using quartic B-splinesrdquo Communicationsin Numerical Methods in Engineering with Biomedical Applica-tions vol 24 no 11 pp 1339ndash1361 2008

[9] I Dag ldquoLeast-squares quadratic B-spline finite elementmethodfor the regularised long wave equationrdquo Computer Methods inApplied Mechanics and Engineering vol 182 no 1-2 pp 205ndash215 2000

[10] I Dag andM N Ozer ldquoApproximation of the RLW equation bythe least square cubic B-spline finite element methodrdquo AppliedMathematical Modelling vol 25 no 3 pp 221ndash231 2001

[11] H Gu and N Chen ldquoLeast-squares mixed finite elementmethods for the RLW equationsrdquoNumericalMethods for PartialDifferential Equations vol 24 no 3 pp 749ndash758 2008

[12] A A Soliman and K R Raslan ldquoCollocation method usingquadratic B-spline for the RLW equationrdquo International Journalof Computer Mathematics vol 78 no 3 pp 399ndash412 2001

[13] I Dag B Saka and D Irk ldquoApplication of cubic B-splines fornumerical solution of the RLW equationrdquo Applied Mathematicsand Computation vol 159 no 2 pp 373ndash389 2004

[14] A A Soliman and M H Hussien ldquoCollocation solution forRLW equation with septic splinerdquo Applied Mathematics andComputation vol 161 no 2 pp 623ndash636 2005

[15] S U Islam S Haq and A Ali ldquoA meshfree method for thenumerical solution of the RLW equationrdquo Journal of Compu-tational and Applied Mathematics vol 223 no 2 pp 997ndash10122009

[16] S Haq S U Islam andA Ali ldquoA numerical meshfree techniquefor the solution of the MEW equationrdquo Computer Modeling inEngineering amp Sciences vol 38 no 1 pp 1ndash23 2008

[17] L Mei and Y Chen ldquoExplicit multistep method for thenumerical solution of RLW equationrdquoAppliedMathematics andComputation vol 218 no 18 pp 9547ndash9554 2012

[18] L Zhang ldquoAfinite difference scheme for generalized regularizedlong-wave equationrdquo Applied Mathematics and Computationvol 168 no 2 pp 962ndash972 2005

[19] MMohammadi and RMokhtari ldquoSolving the generalized reg-ularized longwave equation on the basis of a reproducing kernelspacerdquo Journal of Computational and Applied Mathematics vol235 no 14 pp 4003ndash4014 2011

[20] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004

[21] T Roshan ldquoA Petrov-Galerkin method for solving the gener-alized regularized long wave (GRLW) equationrdquo Computers ampMathematics with Applications vol 63 no 5 pp 943ndash956 2012

[22] A A Soliman ldquoNumerical simulation of the generalizedregularized long wave equation by Hersquos variational iterationmethodrdquoMathematics and Computers in Simulation vol 70 no2 pp 119ndash124 2005

[23] R Mokhtari and M Mohammadi ldquoNumerical solution ofGRLW equation using Sinc-collocation methodrdquo ComputerPhysics Communications vol 181 no 7 pp 1266ndash1274 2010

[24] A H A Ali A A Soliman and K R Raslan ldquoSoliton solutionfor nonlinear partial differential equations by cosine-functionmethodrdquo Physics Letters A vol 368 no 3-4 pp 299ndash304 2007

[25] L R T Gardner G A Gardner F A Irk and N K AmeinldquoApproximations of solitary waves of the MRLW equation byB-spline finite elementsrdquo The Arabian Journal for Science andEngineering A vol 22 no 2 pp 183ndash193 1997

[26] A K Khalifa K R Raslan and H M Alzubaidi ldquoA finite dif-ference scheme for the MRLW and solitary wave interactionsrdquoApplied Mathematics and Computation vol 189 no 1 pp 346ndash354 2007

[27] A K Khalifa K R Raslan and H M Alzubaidi ldquoA collocationmethod with cubic B-splines for solving the MRLW equationrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 406ndash418 2008

[28] K R Raslan and S M Hassan ldquoSolitary waves for the MRLWequationrdquo Applied Mathematics Letters vol 22 no 7 pp 984ndash989 2009

[29] K R Raslan and T S EL-Danaf ldquoSolitary waves solutions of theMRLW equation using quintic B-splinesrdquo Journal of King SaudUniversity Science vol 22 no 3 pp 161ndash166 2010

[30] A K Khalifa K R Raslan and H M Alzubaidi ldquoNumericalstudy using ADM for the modified regularized long waveequationrdquo Applied Mathematical Modelling vol 32 no 12 pp2962ndash2972 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

where V119898and 119887 are given by

V119898=

3

2

119906119899minus1

119898minus

1

2

119906119899minus2

119898

119887 =

120583

ℎ2

(23)

As (22) is used only for 119899 ge 2 we must compute 1199061 byanother method For this we choose a predictor-correctormethod Let 119899 = 1 in (21) we can get the first approximation11990610 with 119906

1 replaced by 1199060 Then we use (12)(11990610 + 1199060)

to substitute 1199061 in (21) and regard the result as the finalapproximation 1199061 Or more precisely the concrete procedureis as follows First set

1199060= 1199060

ℎ (24)

then we calculate

(119860 + 119861)

11990610minus 1199060

Δ119905

+ 119862

11990610+ 1199060

2

+ 119863 (1199060)

11990610+ 1199060

2

= 0

(25)

finally we have

(119860 + 119861)

1199061minus 1199060

Δ119905

+ 119862

1199061+ 1199060

2

+ 119863(

11990610+ 1199060

2

)

1199061+ 1199060

2

= 0

(26)

Thus we can use theThomas algorithm to solve the linearalgebraic equations

In order to improve the accuracy in time we also useRunge-Kutta discretizationmethodThe fourth order Runge-Kutta discretization method in (17) can be described asfollows

119906119899+1

= 119906119899+

1198701+ 21198702+ 21198703+ 1198704

6

minus (119860 + 119861)1198701= [119862119906

119899+ 119863 (119906

119899) 119906119899] Δ119905

minus (119860 + 119861)1198702

= [119862(119906119899+

1

2

1198701) + 119863(119906

119899+

1

2

1198701)(119906119899+

1

2

1198701)]Δ119905

minus (119860 + 119861)1198703

= [119862(119906119899+

1

2

1198702) + 119863(119906

119899+

1

2

1198702)(119906119899+

1

2

1198702)]Δ119905

minus (119860 + 119861)1198704

= [119862 (119906119899+ 1198703) + 119863 (119906

119899+ 1198703) (119906119899+ 1198703)] Δ119905

(27)

4 Stability Analysis

A linear stability analysis is made of the growth factor 119892 of theerror 120576119899

119895in a typical Fourier mode of amplitude 120576119899

120576119899

119895= 119890119894119895119896ℎ

120576119899 (28)

where 119896 is a mode number and ℎ is the element size The vonNeumann stability method can only be applied to a linearscheme so we linearize the nonlinear term by assuming that119906 in this term is locally constant Substituting (28) into (22)gives

120576119899+1

= 119892 120576119899 (29)

119892 =

120572 + 120573119894

120572 minus 120573119894

(30)

where

120572 = (

1

3

minus 2119887) cos (119896ℎ) + 2

3

+ 2119887

120573 =

Δ119905

2ℎ

(1 + 120576119862) sin (119896ℎ) (31)

Taking the modulus of (30) gives

10038161003816100381610038161198921003816100381610038161003816= radic119892119892 = 1 (32)

Therefore our method is marginally stable

5 Numerical Experiments

In this section we present some numerical tests to checkthe efficiency and accuracy of our method which are thepropagation of single soliton and collision of two and threesolitons at different time levels Finally we investigate thedevelopment of theMaxwellian initial condition into solitarywaves

To illustrate the accuracy of the present method we use1198712- and 119871

infin-error norms

1198712=

10038171003817100381710038171003817119906exact119899

minus 119906119899100381710038171003817100381710038172= (ℎ

119873

sum

119894=1

10038161003816100381610038161003816119906exact119899119895

minus 119906119899

119895

10038161003816100381610038161003816

2

)

12

119871infin=

10038171003817100381710038171003817119906exact119899

minus 11990611989910038171003817100381710038171003817infin

= max119895

10038161003816100381610038161003816119906exact119899119895

minus 119906119899

119895

10038161003816100381610038161003816

(33)

to compare the numerical solution with the exact solutionwhich show the mean and maximum differences betweenthe numerical and analytical solutions The quantities 119868

1 1198682

and 1198683measure the conservation laws of our method during

propagation

51 Single Soliton The analytical values of the three variantsare

1198681=

120587radic119888

119901

1198682=

2119888

119901

+

2120583119901119888

3

1198683=

41198882

3119901

minus

2120583119901119888

3

(34)

For the purpose of comparison with the previous work[28] we choose the parameters 119909

0= 40 119888 = 005 ℎ =

119896 = 02 119879 = 10 and 0 le 119909 le 100 Then the amplitudeis 07071068 and the analytical values of three invariants are1198681= 3219174470 119868

2= 0465531499 and 119868

3= 0008001323

The relevant numerical results which applied the Crank-Nicolson scheme in (22) and the Runge-Kutta scheme in

Abstract and Applied Analysis 5

0 20 40 60 80 100

0

005

01

015

02

minus005

T = 0

T = 6

T = 10

Figure 1 Solitary wave with 119888 = 005 ℎ = 02 119896 = 02 and 1199090= 40

0 le 119909 le 100

0

50

100

24

68

10

0

05

1

15

2

Error-axis

minus1

minus05

times10minus4

t-axisx-ax

is

Figure 2 Error graph for 119888 = 005 ℎ = 02 119896 = 02 and1199090= 40 0 le

119909 le 100

(27) are presented in Tables 1 and 2 respectively The profilesof solitary waves at time = 5 and time = 10 are given inFigure 1 and the error distribution in the wave profile in thethree-dimensional view at different time level is illustratedin Figure 2 Table 3 presents the variants 119871

infinnorm and 119871

2

norm at time = 10 against the quadratic B-splines and thecubic B-splines in [28]

We also choose the parameters 119888 = 03 ℎ = 01 119896 = 001and 119909

0= 40 with the range [0 100] Then the amplitude

is 054772 and the analytical values of three invariants are1198681= 3581966678 119868

2= 1345076492 and 119868

3= 0153723028

Tables 4 and 5 show the results about the three invariants the1198712norm and the 119871

infinnorm at different times and compare

these results with those by the collocation method with

0 20 40 60 80 100

0

01

02

03

04

05

06

T = 2

T = 6

T = 10

minus01

Figure 3 Solitary wave with 119888 = 03 ℎ = 01 119896 = 001 and 1199090=

40 0 le 119909 le 100

0

2

4

0

50

100

x-axis

24

68

10

t-axis

minus4

minus2

Error-axis

times10minus4

Figure 4 Error graph for 119888 = 03 ℎ = 01 119896 = 001 and1199090= 40 0 le

119909 le 100

cubic B-splines in [27] using the Crank-Nicolson scheme in(22) and the Runge-Kutta scheme in (27) respectively Thechanges of the variants are satisfactorily small though ourresult for this case is notmore accurate than the result in [27]where the simulation is done up to 119905 = 10 The profile of theapproximate solution is given in Figures 3 and 4 shows theerror profile of the single solitary waves

The Runge-Kutta Galerkin method algorithm has beenrun for different space steps with a fixed time step 119896 = 001The error and convergence order of space for the 119871

2- and

119871infin-norm are recorded in Tables 6 and 7 respectively which

verify that the 1198712- and 119871

infin-error can achieve the order119874(ℎ2)

in space And the algorithm has also been run for differenttime steps with a fixed space step ℎ = 01 The error andconvergence order of time for the 119871

2- and 119871

infin-norm are

recorded in Tables 8 and 9 respectively

6 Abstract and Applied Analysis

Table 1 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007784E minus 03 0 02 3218839 04655233 8007697E minus 03 083581119864 minus 04 045779119864 minus 04

4 3218854 04655215 8007602E minus 03 136403119864 minus 04 045846119864 minus 04

6 3218842 04655196 8007507E minus 03 192290119864 minus 04 068890119864 minus 04

8 3218819 04655177 8007410E minus 03 247879119864 minus 04 091724119864 minus 04

10 3218794 04655159 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Table 2 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007778E minus 03 0 02 3218977 04655250 8007779E minus 03 2613735119864 minus 05 4577868119864 minus 05

4 3219083 04655250 8007780E minus 03 3432730119864 minus 05 2894973119864 minus 05

6 3219165 04655250 8007780E minus 03 4744950119864 minus 05 2214867119864 minus 05

8 3219229 04655250 8007780E minus 03 6052811119864 minus 05 2298029119864 minus 05

10 3219274 04655250 8007780E minus 03 7200630119864 minus 05 2346961119864 minus 05

Table 3 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 0 le 119909 le 100 and time = 10

Method 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

Our scheme in (22) 3218794 04655160 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Our scheme in (27) 3219274 04655250 8007780E minus 03 0720063119864 minus 04 0234696119864 minus 04

Quadratic [28] 3215653 04655665 8004883E minus 03 199288119864 minus 04 481156119864 minus 04

Cubic [28] 3215189 04655136 7999173E minus 03 453811119864 minus 04 625887119864 minus 04

Table 4 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson

time discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 142310119864 minus 04 045779119864 minus 04

4 3581966 1344973 01538262 265355119864 minus 04 153988119864 minus 04

6 3581966 1344973 01538261 374673119864 minus 04 200152119864 minus 04

8 3581966 1344972 01538260 477620119864 minus 04 242916119864 minus 04

10 3581966 1344972 01538260 577512119864 minus 04 284373119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Table 5 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 1396685119864 minus 04 0920780119864 minus 04

4 3581967 1344973 01538262 2592545119864 minus 04 1512530119864 minus 04

6 3581967 1344973 01538262 3647701119864 minus 04 1956752119864 minus 04

8 3581967 1344973 01538262 4638980119864 minus 04 2368004119864 minus 04

10 3581967 1344973 01538262 5600205119864 minus 04 2766940119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Abstract and Applied Analysis 7

Table 6 The order of convergence in space for 1198712error norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δℎ = 04

Order Order(0102) (0204)

2 1396685 times 10minus4 5598770 times 10minus4 2258793 times 10minus3 20031 201244 2592545 times 10minus4 1038054 times 10minus3 4168527 times 10minus3 20014 200576 3647701 times 10minus4 1459612 times 10minus3 5846531 times 10minus3 20005 200208 4638980 times 10minus4 1855582 times 10minus3 7421648 times 10minus3 20000 1999910 5600205 times 10minus4 2239531 times 10minus3 8948683 times 10minus3 19996 19985

Table 7 The order of convergence in space for 119871infinerror norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δ119905 = 04

Order Order(0102) (0204)

2 0920780 times 10minus4 3689931 times 10minus4 1480253 times 10minus3 20027 200424 1512530 times 10minus4 6055462 times 10minus4 2430139 times 10minus3 20012 200476 1956752 times 10minus4 7811765 times 10minus4 3126531 times 10minus3 19972 200088 2368004 times 10minus4 9454389 times 10minus4 3768796 times 10minus3 19973 1995010 2766940 times 10minus4 1104758 times 10minus3 4416067 times 10minus3 19974 19990

Table 8 The order of convergence in time for 1198712error norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00511878 000418182 4219760 times 10minus4 3613597 33088994 00835505 000806197 8133115 times 10minus4 3309253 33092536 01027137 001176588 1180928 times 10minus3 3125948 33166178 01141693 001561415 1547231 times 10minus3 2870249 333509410 01220153 001982141 1925846 times 10minus3 2621930 3363496

Table 9 The order of convergence in time for 119871infinerror norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00318115 2528470 times 10minus3 2505105 times 10minus4 3653213 33353224 00498808 4159059 times 10minus3 4456332 times 10minus4 3584155 32223286 00594582 5587773 times 10minus3 5905399 times 10minus4 3411531 32421678 00641225 7106547 times 10minus3 7270440 times 10minus4 3173611 328903410 00661177 8861734 times 10minus3 8671150 times 10minus4 2899376 3353294

52 Collision of Two Solitons In this section we study theinteraction of two solitary waves with the initial conditionsgiven by a linear sum of two separate solitary waves of variousamplitudes

119906 (119909 0) =

2

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (35)

where 119860119894= radic119888119894

119901119894= radic119888

119894120583(119888119894+ 1) 119894 = 1 2 and 119909

119894and

119888119894(119894 = 1 2) are arbitrary constants The analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

1198682=

21198881

1199011

+

21198882

1199012

+

212058311988811199011

3

+

212058311990121198882

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

minus

212058311990111198881

3

minus

212058311990121198882

3

(36)

In our numerical scheme we choose ℎ = 01 119896 = 001 1198881= 1

1198882= 05 119909

1= 10 and 119909

2= 40 with interval [0 100] Then

the amplitudes are in ratio 2 1 where 21198602= 1198601and the

analytical values for the conservation are 1198681= 82905324294

1198682= 5224332548 and 119868

3= 1799113742 The changes of

the invariants are satisfactorily small since the changes ofthe invariants 119868

1 1198682 and 119868

3are 13295 times 10minus4 22259 times 10minus5

and 68989 times 10minus5 respectively The results are recorded inTable 10 A graph of the two solitary waves collision plottedin a three-dimensional view at some discrete times is shownin Figure 5

53 Collision of Three Solitons In this section we study theinteraction of three solitary waves with the initial conditionsgiven by a linear sum of three separate solitary waves ofvarious amplitudes

119906 (119909 0) =

3

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (37)

8 Abstract and Applied Analysis

Table 10 Invariants and error norms for two solitary waves 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 ℎ = 01 119896 = 001 and 0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 8288129 8288129 5222934 5222933 1800512 18005132 8289522 8293010 5222910 5222951 1800479 18005074 8289462 8295348 5222887 5222954 1800458 18225076 8289304 8297822 5222864 5222956 1800433 18005078 8289233 8300322 5222840 5222958 1800411 180050810 8289231 8302630 5222817 5222960 1800388 1800509

Table 11 Invariants and error norms for three solitary waves 1198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 119909

3= 40 ℎ = 01 119896 = 001 and

0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 11800513 11800513 6490318 6490318 1997082 19970824 11799364 11807800 6485747 6490319 1992012 19966238 11797850 11812825 6483120 6490298 1988853 199607612 11797570 11817192 6482677 6490279 1988441 199636016 11791837 11820521 6474939 6490305 1982852 199687520 11788577 11823320 6469884 6490341 1977926 1996758

0

02

04

06

08

1

u-axis

0

50

100

x-axis

24

68

10

t-axis

minus02

Figure 5 Interaction of two solitons at times from 119905 = 2 4 6 8 10

for 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 and 0 le 119909 le 100

where 119860119894= radic119888119894

119901119894= radic119888119894120583(119888119894+ 1) 119894 = 1 2 3 and 119909

119894and

119888119894(119894 = 1 2 3) are arbitrary constantsThe analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

+

120587radic1198883

1199013

1198682=

21198881

1199011

+

21198882

1199012

+

21198883

1199013

+

212058311988811199011

3

+

212058311990121198882

3

+

212058311990131198883

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

+

41198883

2

31199013

minus

212058311990111198881

3

minus

212058311990121198882

3

minus

212058311990131198883

3

(38)

In this case we choose paraments as ℎ = 01 119896 = 0051198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 and 119909

3= 40

with interval [0 100]Then the amplitudes are in ratio 4 2 1

where 1198601= 2119860

2= 4119860

3and the analytical values for the

conservation are 1198681= 11802939794 119868

2= 6416902131

and 1198683= 1910917141 The changes of the invariants from

the initial variants approach zero throughout and agree withthe analytical values for the three invariants as presentedin Table 11 which indicates that our scheme is satisfactorilyconservativeThe geometry of the initial state and the profilesat time 119905 = 10 119905 = 20 and 119905 = 35 is shown graphically inFigure 6 respectively

54 The Maxwellian Initial Condition Finally we considerthe development of the Maxwellian initial condition

119906 (119909 0) = 119890minus(119909minus40)

2 (39)

into a train of solitary waves We choose different values 120583 =10 120583 = 05 and 120583 = 01 in our numerical scheme Thecomparison of the three variants 119868

1 1198682 and 119868

3with the earlier

result in [29] is presented in Table 12 here Crank-Nicolsonmethod is used for time discretization For 120583 = 10 thechanges of the variants 119868

1 1198682 and 119868

3with respect to the initial

values are 39492 times 10minus6 69575 times 10minus4 and 31509 times 10minus3respectively whereas they are 28210 times 10minus6 260000 times 10minus4and 70771times 10minus4 in [29] For 120583 = 05 the variants are changedby 78982 times 10minus6 68214 times 10minus5 and 61410 times 10minus4 in this caseThedevelopment of theMaxwellian initial condition is shownin Figure 7 with different parameter values respectively Thesmaller 120583 there is the more the number of solitary waves willform The simulations are done up to time = 10 in this caseFrom the previous work [25] the total number of solitarywaves and the values 120583 have the approximate relation

119873 cong [

1

5radic120583

] (40)

Abstract and Applied Analysis 9

Table 12 Values of 1198681 1198682 and 119868

3for Maxwellian initial condition when ℎ = 01 and 119896 = 001 and the space interval [0 100]

120583 Time 1198681

1198682

1198683

1198681[29] 119868

2[29] 119868

3[29]

1

2 1772528 2499467 minus03633570 1772449 2506352 minus036681494 1772544 2499870 minus03643380 1772446 2506235 minus036669746 1772532 2500822 minus03657774 1772447 2506171 minus036663268 1772524 2500977 minus03661331 1772446 2506123 minus0366586010 1772521 2501206 minus03665079 1772444 22060926 minus03665553

05

2 1772534 1876446 02597450 1772451 1879888 025964944 1772534 1876534 02587968 1772446 1879855 025968286 1772528 1876600 02583802 1772449 1879841 025969738 1772523 1876581 02582386 1772450 1879834 0259705010 1772520 1876574 02581499 1772449 1879828 02597092

01

2 1770991 1373320 07687222 1772452 1378607 076087774 1769863 1371236 07658462 1772451 1378577 076083646 1768742 1369139 07630269 1772451 1378546 076079378 1767630 1367058 07602389 1772451 1378515 0760752910 1766526 1364993 07574795 1772453 1378483 07607117

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(a) Time = 0

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(b) Time = 10

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(c) Time = 20

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(d) Time = 35

Figure 6 Interaction of three solitary waves

10 Abstract and Applied Analysis

0 20 40 60 80 100

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

(a) 120583 = 10

0 20 40 60 80 100minus04

minus02

0

02

04

06

08

1

(b) 120583 = 05

0 20 40 60 80 100minus02

0

02

04

06

08

1

(c) 120583 = 01

Figure 7 Maxwellian initial condition with different values 120583

6 Conclusion

In this paper we have developed a Galerkin linear finiteelementmethod to investigate the propagation of solitons andtheir interactions governed by the nonlinear MRLW equa-tion An extrapolation technique has been used to improvethe accuracy in time for this numerical method A linearstability analysis shows that this method is marginally stableThe high efficiency and accuracy of our method is testedby the numerical experiments propagation of single solitoncollision of two and three solitons and development of theMaxwellian initial condition into solitary waves Moreoverthis method numerically satisfies the conservation laws ofmass momentum and energy

Conflict of Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFund of China (11371289) and the China Scholarship Council(201206285005)

References

[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966

[2] T B Benjamin J L Bona and J J Mahony ldquoModel equationsfor long waves in nonlinear dispersive systemsrdquo PhilosophicalTransactions of the Royal Society of London A vol 272 no 1220pp 47ndash78 1972

[3] S Kutluay and A Esen ldquoA finite difference solution of theregularized long-wave equationrdquo Mathematical Problems inEngineering vol 2006 Article ID 85743 14 pages 2006

Abstract and Applied Analysis 11

[4] J Lin Z Xie and J Zhou ldquoHigh-order compact differencescheme for the regularized long wave equationrdquo Communica-tions in Numerical Methods in Engineering vol 23 no 2 pp135ndash156 2007

[5] P Avilez-Valente and F J Seabra-Santos ldquoA Petrov-Galerkinfinite element scheme for the regularized long wave equationrdquoComputational Mechanics vol 34 no 4 pp 256ndash270 2004

[6] A Esen and S Kutluay ldquoApplication of a lumped Galerkinmethod to the regularized long wave equationrdquo Applied Mathe-matics and Computation vol 174 no 2 pp 833ndash845 2006

[7] Y Liu H Li Y Du and J Wang ldquoExplicit multistep mixedfinite element method for RLW equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 768976 12 pages 2013

[8] B Saka and I Dag ldquoA numerical solution of the RLW equationby Galerkin method using quartic B-splinesrdquo Communicationsin Numerical Methods in Engineering with Biomedical Applica-tions vol 24 no 11 pp 1339ndash1361 2008

[9] I Dag ldquoLeast-squares quadratic B-spline finite elementmethodfor the regularised long wave equationrdquo Computer Methods inApplied Mechanics and Engineering vol 182 no 1-2 pp 205ndash215 2000

[10] I Dag andM N Ozer ldquoApproximation of the RLW equation bythe least square cubic B-spline finite element methodrdquo AppliedMathematical Modelling vol 25 no 3 pp 221ndash231 2001

[11] H Gu and N Chen ldquoLeast-squares mixed finite elementmethods for the RLW equationsrdquoNumericalMethods for PartialDifferential Equations vol 24 no 3 pp 749ndash758 2008

[12] A A Soliman and K R Raslan ldquoCollocation method usingquadratic B-spline for the RLW equationrdquo International Journalof Computer Mathematics vol 78 no 3 pp 399ndash412 2001

[13] I Dag B Saka and D Irk ldquoApplication of cubic B-splines fornumerical solution of the RLW equationrdquo Applied Mathematicsand Computation vol 159 no 2 pp 373ndash389 2004

[14] A A Soliman and M H Hussien ldquoCollocation solution forRLW equation with septic splinerdquo Applied Mathematics andComputation vol 161 no 2 pp 623ndash636 2005

[15] S U Islam S Haq and A Ali ldquoA meshfree method for thenumerical solution of the RLW equationrdquo Journal of Compu-tational and Applied Mathematics vol 223 no 2 pp 997ndash10122009

[16] S Haq S U Islam andA Ali ldquoA numerical meshfree techniquefor the solution of the MEW equationrdquo Computer Modeling inEngineering amp Sciences vol 38 no 1 pp 1ndash23 2008

[17] L Mei and Y Chen ldquoExplicit multistep method for thenumerical solution of RLW equationrdquoAppliedMathematics andComputation vol 218 no 18 pp 9547ndash9554 2012

[18] L Zhang ldquoAfinite difference scheme for generalized regularizedlong-wave equationrdquo Applied Mathematics and Computationvol 168 no 2 pp 962ndash972 2005

[19] MMohammadi and RMokhtari ldquoSolving the generalized reg-ularized longwave equation on the basis of a reproducing kernelspacerdquo Journal of Computational and Applied Mathematics vol235 no 14 pp 4003ndash4014 2011

[20] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004

[21] T Roshan ldquoA Petrov-Galerkin method for solving the gener-alized regularized long wave (GRLW) equationrdquo Computers ampMathematics with Applications vol 63 no 5 pp 943ndash956 2012

[22] A A Soliman ldquoNumerical simulation of the generalizedregularized long wave equation by Hersquos variational iterationmethodrdquoMathematics and Computers in Simulation vol 70 no2 pp 119ndash124 2005

[23] R Mokhtari and M Mohammadi ldquoNumerical solution ofGRLW equation using Sinc-collocation methodrdquo ComputerPhysics Communications vol 181 no 7 pp 1266ndash1274 2010

[24] A H A Ali A A Soliman and K R Raslan ldquoSoliton solutionfor nonlinear partial differential equations by cosine-functionmethodrdquo Physics Letters A vol 368 no 3-4 pp 299ndash304 2007

[25] L R T Gardner G A Gardner F A Irk and N K AmeinldquoApproximations of solitary waves of the MRLW equation byB-spline finite elementsrdquo The Arabian Journal for Science andEngineering A vol 22 no 2 pp 183ndash193 1997

[26] A K Khalifa K R Raslan and H M Alzubaidi ldquoA finite dif-ference scheme for the MRLW and solitary wave interactionsrdquoApplied Mathematics and Computation vol 189 no 1 pp 346ndash354 2007

[27] A K Khalifa K R Raslan and H M Alzubaidi ldquoA collocationmethod with cubic B-splines for solving the MRLW equationrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 406ndash418 2008

[28] K R Raslan and S M Hassan ldquoSolitary waves for the MRLWequationrdquo Applied Mathematics Letters vol 22 no 7 pp 984ndash989 2009

[29] K R Raslan and T S EL-Danaf ldquoSolitary waves solutions of theMRLW equation using quintic B-splinesrdquo Journal of King SaudUniversity Science vol 22 no 3 pp 161ndash166 2010

[30] A K Khalifa K R Raslan and H M Alzubaidi ldquoNumericalstudy using ADM for the modified regularized long waveequationrdquo Applied Mathematical Modelling vol 32 no 12 pp2962ndash2972 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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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

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

0 20 40 60 80 100

0

005

01

015

02

minus005

T = 0

T = 6

T = 10

Figure 1 Solitary wave with 119888 = 005 ℎ = 02 119896 = 02 and 1199090= 40

0 le 119909 le 100

0

50

100

24

68

10

0

05

1

15

2

Error-axis

minus1

minus05

times10minus4

t-axisx-ax

is

Figure 2 Error graph for 119888 = 005 ℎ = 02 119896 = 02 and1199090= 40 0 le

119909 le 100

(27) are presented in Tables 1 and 2 respectively The profilesof solitary waves at time = 5 and time = 10 are given inFigure 1 and the error distribution in the wave profile in thethree-dimensional view at different time level is illustratedin Figure 2 Table 3 presents the variants 119871

infinnorm and 119871

2

norm at time = 10 against the quadratic B-splines and thecubic B-splines in [28]

We also choose the parameters 119888 = 03 ℎ = 01 119896 = 001and 119909

0= 40 with the range [0 100] Then the amplitude

is 054772 and the analytical values of three invariants are1198681= 3581966678 119868

2= 1345076492 and 119868

3= 0153723028

Tables 4 and 5 show the results about the three invariants the1198712norm and the 119871

infinnorm at different times and compare

these results with those by the collocation method with

0 20 40 60 80 100

0

01

02

03

04

05

06

T = 2

T = 6

T = 10

minus01

Figure 3 Solitary wave with 119888 = 03 ℎ = 01 119896 = 001 and 1199090=

40 0 le 119909 le 100

0

2

4

0

50

100

x-axis

24

68

10

t-axis

minus4

minus2

Error-axis

times10minus4

Figure 4 Error graph for 119888 = 03 ℎ = 01 119896 = 001 and1199090= 40 0 le

119909 le 100

cubic B-splines in [27] using the Crank-Nicolson scheme in(22) and the Runge-Kutta scheme in (27) respectively Thechanges of the variants are satisfactorily small though ourresult for this case is notmore accurate than the result in [27]where the simulation is done up to 119905 = 10 The profile of theapproximate solution is given in Figures 3 and 4 shows theerror profile of the single solitary waves

The Runge-Kutta Galerkin method algorithm has beenrun for different space steps with a fixed time step 119896 = 001The error and convergence order of space for the 119871

2- and

119871infin-norm are recorded in Tables 6 and 7 respectively which

verify that the 1198712- and 119871

infin-error can achieve the order119874(ℎ2)

in space And the algorithm has also been run for differenttime steps with a fixed space step ℎ = 01 The error andconvergence order of time for the 119871

2- and 119871

infin-norm are

recorded in Tables 8 and 9 respectively

6 Abstract and Applied Analysis

Table 1 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007784E minus 03 0 02 3218839 04655233 8007697E minus 03 083581119864 minus 04 045779119864 minus 04

4 3218854 04655215 8007602E minus 03 136403119864 minus 04 045846119864 minus 04

6 3218842 04655196 8007507E minus 03 192290119864 minus 04 068890119864 minus 04

8 3218819 04655177 8007410E minus 03 247879119864 minus 04 091724119864 minus 04

10 3218794 04655159 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Table 2 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007778E minus 03 0 02 3218977 04655250 8007779E minus 03 2613735119864 minus 05 4577868119864 minus 05

4 3219083 04655250 8007780E minus 03 3432730119864 minus 05 2894973119864 minus 05

6 3219165 04655250 8007780E minus 03 4744950119864 minus 05 2214867119864 minus 05

8 3219229 04655250 8007780E minus 03 6052811119864 minus 05 2298029119864 minus 05

10 3219274 04655250 8007780E minus 03 7200630119864 minus 05 2346961119864 minus 05

Table 3 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 0 le 119909 le 100 and time = 10

Method 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

Our scheme in (22) 3218794 04655160 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Our scheme in (27) 3219274 04655250 8007780E minus 03 0720063119864 minus 04 0234696119864 minus 04

Quadratic [28] 3215653 04655665 8004883E minus 03 199288119864 minus 04 481156119864 minus 04

Cubic [28] 3215189 04655136 7999173E minus 03 453811119864 minus 04 625887119864 minus 04

Table 4 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson

time discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 142310119864 minus 04 045779119864 minus 04

4 3581966 1344973 01538262 265355119864 minus 04 153988119864 minus 04

6 3581966 1344973 01538261 374673119864 minus 04 200152119864 minus 04

8 3581966 1344972 01538260 477620119864 minus 04 242916119864 minus 04

10 3581966 1344972 01538260 577512119864 minus 04 284373119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Table 5 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 1396685119864 minus 04 0920780119864 minus 04

4 3581967 1344973 01538262 2592545119864 minus 04 1512530119864 minus 04

6 3581967 1344973 01538262 3647701119864 minus 04 1956752119864 minus 04

8 3581967 1344973 01538262 4638980119864 minus 04 2368004119864 minus 04

10 3581967 1344973 01538262 5600205119864 minus 04 2766940119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Abstract and Applied Analysis 7

Table 6 The order of convergence in space for 1198712error norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δℎ = 04

Order Order(0102) (0204)

2 1396685 times 10minus4 5598770 times 10minus4 2258793 times 10minus3 20031 201244 2592545 times 10minus4 1038054 times 10minus3 4168527 times 10minus3 20014 200576 3647701 times 10minus4 1459612 times 10minus3 5846531 times 10minus3 20005 200208 4638980 times 10minus4 1855582 times 10minus3 7421648 times 10minus3 20000 1999910 5600205 times 10minus4 2239531 times 10minus3 8948683 times 10minus3 19996 19985

Table 7 The order of convergence in space for 119871infinerror norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δ119905 = 04

Order Order(0102) (0204)

2 0920780 times 10minus4 3689931 times 10minus4 1480253 times 10minus3 20027 200424 1512530 times 10minus4 6055462 times 10minus4 2430139 times 10minus3 20012 200476 1956752 times 10minus4 7811765 times 10minus4 3126531 times 10minus3 19972 200088 2368004 times 10minus4 9454389 times 10minus4 3768796 times 10minus3 19973 1995010 2766940 times 10minus4 1104758 times 10minus3 4416067 times 10minus3 19974 19990

Table 8 The order of convergence in time for 1198712error norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00511878 000418182 4219760 times 10minus4 3613597 33088994 00835505 000806197 8133115 times 10minus4 3309253 33092536 01027137 001176588 1180928 times 10minus3 3125948 33166178 01141693 001561415 1547231 times 10minus3 2870249 333509410 01220153 001982141 1925846 times 10minus3 2621930 3363496

Table 9 The order of convergence in time for 119871infinerror norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00318115 2528470 times 10minus3 2505105 times 10minus4 3653213 33353224 00498808 4159059 times 10minus3 4456332 times 10minus4 3584155 32223286 00594582 5587773 times 10minus3 5905399 times 10minus4 3411531 32421678 00641225 7106547 times 10minus3 7270440 times 10minus4 3173611 328903410 00661177 8861734 times 10minus3 8671150 times 10minus4 2899376 3353294

52 Collision of Two Solitons In this section we study theinteraction of two solitary waves with the initial conditionsgiven by a linear sum of two separate solitary waves of variousamplitudes

119906 (119909 0) =

2

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (35)

where 119860119894= radic119888119894

119901119894= radic119888

119894120583(119888119894+ 1) 119894 = 1 2 and 119909

119894and

119888119894(119894 = 1 2) are arbitrary constants The analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

1198682=

21198881

1199011

+

21198882

1199012

+

212058311988811199011

3

+

212058311990121198882

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

minus

212058311990111198881

3

minus

212058311990121198882

3

(36)

In our numerical scheme we choose ℎ = 01 119896 = 001 1198881= 1

1198882= 05 119909

1= 10 and 119909

2= 40 with interval [0 100] Then

the amplitudes are in ratio 2 1 where 21198602= 1198601and the

analytical values for the conservation are 1198681= 82905324294

1198682= 5224332548 and 119868

3= 1799113742 The changes of

the invariants are satisfactorily small since the changes ofthe invariants 119868

1 1198682 and 119868

3are 13295 times 10minus4 22259 times 10minus5

and 68989 times 10minus5 respectively The results are recorded inTable 10 A graph of the two solitary waves collision plottedin a three-dimensional view at some discrete times is shownin Figure 5

53 Collision of Three Solitons In this section we study theinteraction of three solitary waves with the initial conditionsgiven by a linear sum of three separate solitary waves ofvarious amplitudes

119906 (119909 0) =

3

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (37)

8 Abstract and Applied Analysis

Table 10 Invariants and error norms for two solitary waves 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 ℎ = 01 119896 = 001 and 0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 8288129 8288129 5222934 5222933 1800512 18005132 8289522 8293010 5222910 5222951 1800479 18005074 8289462 8295348 5222887 5222954 1800458 18225076 8289304 8297822 5222864 5222956 1800433 18005078 8289233 8300322 5222840 5222958 1800411 180050810 8289231 8302630 5222817 5222960 1800388 1800509

Table 11 Invariants and error norms for three solitary waves 1198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 119909

3= 40 ℎ = 01 119896 = 001 and

0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 11800513 11800513 6490318 6490318 1997082 19970824 11799364 11807800 6485747 6490319 1992012 19966238 11797850 11812825 6483120 6490298 1988853 199607612 11797570 11817192 6482677 6490279 1988441 199636016 11791837 11820521 6474939 6490305 1982852 199687520 11788577 11823320 6469884 6490341 1977926 1996758

0

02

04

06

08

1

u-axis

0

50

100

x-axis

24

68

10

t-axis

minus02

Figure 5 Interaction of two solitons at times from 119905 = 2 4 6 8 10

for 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 and 0 le 119909 le 100

where 119860119894= radic119888119894

119901119894= radic119888119894120583(119888119894+ 1) 119894 = 1 2 3 and 119909

119894and

119888119894(119894 = 1 2 3) are arbitrary constantsThe analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

+

120587radic1198883

1199013

1198682=

21198881

1199011

+

21198882

1199012

+

21198883

1199013

+

212058311988811199011

3

+

212058311990121198882

3

+

212058311990131198883

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

+

41198883

2

31199013

minus

212058311990111198881

3

minus

212058311990121198882

3

minus

212058311990131198883

3

(38)

In this case we choose paraments as ℎ = 01 119896 = 0051198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 and 119909

3= 40

with interval [0 100]Then the amplitudes are in ratio 4 2 1

where 1198601= 2119860

2= 4119860

3and the analytical values for the

conservation are 1198681= 11802939794 119868

2= 6416902131

and 1198683= 1910917141 The changes of the invariants from

the initial variants approach zero throughout and agree withthe analytical values for the three invariants as presentedin Table 11 which indicates that our scheme is satisfactorilyconservativeThe geometry of the initial state and the profilesat time 119905 = 10 119905 = 20 and 119905 = 35 is shown graphically inFigure 6 respectively

54 The Maxwellian Initial Condition Finally we considerthe development of the Maxwellian initial condition

119906 (119909 0) = 119890minus(119909minus40)

2 (39)

into a train of solitary waves We choose different values 120583 =10 120583 = 05 and 120583 = 01 in our numerical scheme Thecomparison of the three variants 119868

1 1198682 and 119868

3with the earlier

result in [29] is presented in Table 12 here Crank-Nicolsonmethod is used for time discretization For 120583 = 10 thechanges of the variants 119868

1 1198682 and 119868

3with respect to the initial

values are 39492 times 10minus6 69575 times 10minus4 and 31509 times 10minus3respectively whereas they are 28210 times 10minus6 260000 times 10minus4and 70771times 10minus4 in [29] For 120583 = 05 the variants are changedby 78982 times 10minus6 68214 times 10minus5 and 61410 times 10minus4 in this caseThedevelopment of theMaxwellian initial condition is shownin Figure 7 with different parameter values respectively Thesmaller 120583 there is the more the number of solitary waves willform The simulations are done up to time = 10 in this caseFrom the previous work [25] the total number of solitarywaves and the values 120583 have the approximate relation

119873 cong [

1

5radic120583

] (40)

Abstract and Applied Analysis 9

Table 12 Values of 1198681 1198682 and 119868

3for Maxwellian initial condition when ℎ = 01 and 119896 = 001 and the space interval [0 100]

120583 Time 1198681

1198682

1198683

1198681[29] 119868

2[29] 119868

3[29]

1

2 1772528 2499467 minus03633570 1772449 2506352 minus036681494 1772544 2499870 minus03643380 1772446 2506235 minus036669746 1772532 2500822 minus03657774 1772447 2506171 minus036663268 1772524 2500977 minus03661331 1772446 2506123 minus0366586010 1772521 2501206 minus03665079 1772444 22060926 minus03665553

05

2 1772534 1876446 02597450 1772451 1879888 025964944 1772534 1876534 02587968 1772446 1879855 025968286 1772528 1876600 02583802 1772449 1879841 025969738 1772523 1876581 02582386 1772450 1879834 0259705010 1772520 1876574 02581499 1772449 1879828 02597092

01

2 1770991 1373320 07687222 1772452 1378607 076087774 1769863 1371236 07658462 1772451 1378577 076083646 1768742 1369139 07630269 1772451 1378546 076079378 1767630 1367058 07602389 1772451 1378515 0760752910 1766526 1364993 07574795 1772453 1378483 07607117

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(a) Time = 0

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(b) Time = 10

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(c) Time = 20

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(d) Time = 35

Figure 6 Interaction of three solitary waves

10 Abstract and Applied Analysis

0 20 40 60 80 100

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

(a) 120583 = 10

0 20 40 60 80 100minus04

minus02

0

02

04

06

08

1

(b) 120583 = 05

0 20 40 60 80 100minus02

0

02

04

06

08

1

(c) 120583 = 01

Figure 7 Maxwellian initial condition with different values 120583

6 Conclusion

In this paper we have developed a Galerkin linear finiteelementmethod to investigate the propagation of solitons andtheir interactions governed by the nonlinear MRLW equa-tion An extrapolation technique has been used to improvethe accuracy in time for this numerical method A linearstability analysis shows that this method is marginally stableThe high efficiency and accuracy of our method is testedby the numerical experiments propagation of single solitoncollision of two and three solitons and development of theMaxwellian initial condition into solitary waves Moreoverthis method numerically satisfies the conservation laws ofmass momentum and energy

Conflict of Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFund of China (11371289) and the China Scholarship Council(201206285005)

References

[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966

[2] T B Benjamin J L Bona and J J Mahony ldquoModel equationsfor long waves in nonlinear dispersive systemsrdquo PhilosophicalTransactions of the Royal Society of London A vol 272 no 1220pp 47ndash78 1972

[3] S Kutluay and A Esen ldquoA finite difference solution of theregularized long-wave equationrdquo Mathematical Problems inEngineering vol 2006 Article ID 85743 14 pages 2006

Abstract and Applied Analysis 11

[4] J Lin Z Xie and J Zhou ldquoHigh-order compact differencescheme for the regularized long wave equationrdquo Communica-tions in Numerical Methods in Engineering vol 23 no 2 pp135ndash156 2007

[5] P Avilez-Valente and F J Seabra-Santos ldquoA Petrov-Galerkinfinite element scheme for the regularized long wave equationrdquoComputational Mechanics vol 34 no 4 pp 256ndash270 2004

[6] A Esen and S Kutluay ldquoApplication of a lumped Galerkinmethod to the regularized long wave equationrdquo Applied Mathe-matics and Computation vol 174 no 2 pp 833ndash845 2006

[7] Y Liu H Li Y Du and J Wang ldquoExplicit multistep mixedfinite element method for RLW equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 768976 12 pages 2013

[8] B Saka and I Dag ldquoA numerical solution of the RLW equationby Galerkin method using quartic B-splinesrdquo Communicationsin Numerical Methods in Engineering with Biomedical Applica-tions vol 24 no 11 pp 1339ndash1361 2008

[9] I Dag ldquoLeast-squares quadratic B-spline finite elementmethodfor the regularised long wave equationrdquo Computer Methods inApplied Mechanics and Engineering vol 182 no 1-2 pp 205ndash215 2000

[10] I Dag andM N Ozer ldquoApproximation of the RLW equation bythe least square cubic B-spline finite element methodrdquo AppliedMathematical Modelling vol 25 no 3 pp 221ndash231 2001

[11] H Gu and N Chen ldquoLeast-squares mixed finite elementmethods for the RLW equationsrdquoNumericalMethods for PartialDifferential Equations vol 24 no 3 pp 749ndash758 2008

[12] A A Soliman and K R Raslan ldquoCollocation method usingquadratic B-spline for the RLW equationrdquo International Journalof Computer Mathematics vol 78 no 3 pp 399ndash412 2001

[13] I Dag B Saka and D Irk ldquoApplication of cubic B-splines fornumerical solution of the RLW equationrdquo Applied Mathematicsand Computation vol 159 no 2 pp 373ndash389 2004

[14] A A Soliman and M H Hussien ldquoCollocation solution forRLW equation with septic splinerdquo Applied Mathematics andComputation vol 161 no 2 pp 623ndash636 2005

[15] S U Islam S Haq and A Ali ldquoA meshfree method for thenumerical solution of the RLW equationrdquo Journal of Compu-tational and Applied Mathematics vol 223 no 2 pp 997ndash10122009

[16] S Haq S U Islam andA Ali ldquoA numerical meshfree techniquefor the solution of the MEW equationrdquo Computer Modeling inEngineering amp Sciences vol 38 no 1 pp 1ndash23 2008

[17] L Mei and Y Chen ldquoExplicit multistep method for thenumerical solution of RLW equationrdquoAppliedMathematics andComputation vol 218 no 18 pp 9547ndash9554 2012

[18] L Zhang ldquoAfinite difference scheme for generalized regularizedlong-wave equationrdquo Applied Mathematics and Computationvol 168 no 2 pp 962ndash972 2005

[19] MMohammadi and RMokhtari ldquoSolving the generalized reg-ularized longwave equation on the basis of a reproducing kernelspacerdquo Journal of Computational and Applied Mathematics vol235 no 14 pp 4003ndash4014 2011

[20] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004

[21] T Roshan ldquoA Petrov-Galerkin method for solving the gener-alized regularized long wave (GRLW) equationrdquo Computers ampMathematics with Applications vol 63 no 5 pp 943ndash956 2012

[22] A A Soliman ldquoNumerical simulation of the generalizedregularized long wave equation by Hersquos variational iterationmethodrdquoMathematics and Computers in Simulation vol 70 no2 pp 119ndash124 2005

[23] R Mokhtari and M Mohammadi ldquoNumerical solution ofGRLW equation using Sinc-collocation methodrdquo ComputerPhysics Communications vol 181 no 7 pp 1266ndash1274 2010

[24] A H A Ali A A Soliman and K R Raslan ldquoSoliton solutionfor nonlinear partial differential equations by cosine-functionmethodrdquo Physics Letters A vol 368 no 3-4 pp 299ndash304 2007

[25] L R T Gardner G A Gardner F A Irk and N K AmeinldquoApproximations of solitary waves of the MRLW equation byB-spline finite elementsrdquo The Arabian Journal for Science andEngineering A vol 22 no 2 pp 183ndash193 1997

[26] A K Khalifa K R Raslan and H M Alzubaidi ldquoA finite dif-ference scheme for the MRLW and solitary wave interactionsrdquoApplied Mathematics and Computation vol 189 no 1 pp 346ndash354 2007

[27] A K Khalifa K R Raslan and H M Alzubaidi ldquoA collocationmethod with cubic B-splines for solving the MRLW equationrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 406ndash418 2008

[28] K R Raslan and S M Hassan ldquoSolitary waves for the MRLWequationrdquo Applied Mathematics Letters vol 22 no 7 pp 984ndash989 2009

[29] K R Raslan and T S EL-Danaf ldquoSolitary waves solutions of theMRLW equation using quintic B-splinesrdquo Journal of King SaudUniversity Science vol 22 no 3 pp 161ndash166 2010

[30] A K Khalifa K R Raslan and H M Alzubaidi ldquoNumericalstudy using ADM for the modified regularized long waveequationrdquo Applied Mathematical Modelling vol 32 no 12 pp2962ndash2972 2008

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 1 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007784E minus 03 0 02 3218839 04655233 8007697E minus 03 083581119864 minus 04 045779119864 minus 04

4 3218854 04655215 8007602E minus 03 136403119864 minus 04 045846119864 minus 04

6 3218842 04655196 8007507E minus 03 192290119864 minus 04 068890119864 minus 04

8 3218819 04655177 8007410E minus 03 247879119864 minus 04 091724119864 minus 04

10 3218794 04655159 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Table 2 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

0 3218839 04655250 8007778E minus 03 0 02 3218977 04655250 8007779E minus 03 2613735119864 minus 05 4577868119864 minus 05

4 3219083 04655250 8007780E minus 03 3432730119864 minus 05 2894973119864 minus 05

6 3219165 04655250 8007780E minus 03 4744950119864 minus 05 2214867119864 minus 05

8 3219229 04655250 8007780E minus 03 6052811119864 minus 05 2298029119864 minus 05

10 3219274 04655250 8007780E minus 03 7200630119864 minus 05 2346961119864 minus 05

Table 3 Invariants and error norms for single solitary wave 119888 = 005 ℎ = 02 119896 = 02 1199090= 40 0 le 119909 le 100 and time = 10

Method 1198681

1198682

1198683

1198712

119871infinAnalytical 3219174 0465531 8001323E minus 03

Our scheme in (22) 3218794 04655160 8007313E minus 03 302940119864 minus 04 113929119864 minus 04

Our scheme in (27) 3219274 04655250 8007780E minus 03 0720063119864 minus 04 0234696119864 minus 04

Quadratic [28] 3215653 04655665 8004883E minus 03 199288119864 minus 04 481156119864 minus 04

Cubic [28] 3215189 04655136 7999173E minus 03 453811119864 minus 04 625887119864 minus 04

Table 4 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Crank-Nicolson

time discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 142310119864 minus 04 045779119864 minus 04

4 3581966 1344973 01538262 265355119864 minus 04 153988119864 minus 04

6 3581966 1344973 01538261 374673119864 minus 04 200152119864 minus 04

8 3581966 1344972 01538260 477620119864 minus 04 242916119864 minus 04

10 3581966 1344972 01538260 577512119864 minus 04 284373119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Table 5 Invariants and error norms for single solitary wave 119888 = 03 ℎ = 01 119896 = 001 1199090= 40 and 0 le 119909 le 100 (using Runge-Kutta time

discretization)

Time 1198681

1198682

1198683

1198712

119871infinAnalytical 3581967 1345076 01537230

0 3581967 1344973 01538264 0 02 3581967 1344973 01538263 1396685119864 minus 04 0920780119864 minus 04

4 3581967 1344973 01538262 2592545119864 minus 04 1512530119864 minus 04

6 3581967 1344973 01538262 3647701119864 minus 04 1956752119864 minus 04

8 3581967 1344973 01538262 4638980119864 minus 04 2368004119864 minus 04

10 3581967 1344973 01538262 5600205119864 minus 04 2766940119864 minus 04

[27] 358197 134508 0153723 402927119864 minus 04 206732119864 minus 04

Abstract and Applied Analysis 7

Table 6 The order of convergence in space for 1198712error norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δℎ = 04

Order Order(0102) (0204)

2 1396685 times 10minus4 5598770 times 10minus4 2258793 times 10minus3 20031 201244 2592545 times 10minus4 1038054 times 10minus3 4168527 times 10minus3 20014 200576 3647701 times 10minus4 1459612 times 10minus3 5846531 times 10minus3 20005 200208 4638980 times 10minus4 1855582 times 10minus3 7421648 times 10minus3 20000 1999910 5600205 times 10minus4 2239531 times 10minus3 8948683 times 10minus3 19996 19985

Table 7 The order of convergence in space for 119871infinerror norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δ119905 = 04

Order Order(0102) (0204)

2 0920780 times 10minus4 3689931 times 10minus4 1480253 times 10minus3 20027 200424 1512530 times 10minus4 6055462 times 10minus4 2430139 times 10minus3 20012 200476 1956752 times 10minus4 7811765 times 10minus4 3126531 times 10minus3 19972 200088 2368004 times 10minus4 9454389 times 10minus4 3768796 times 10minus3 19973 1995010 2766940 times 10minus4 1104758 times 10minus3 4416067 times 10minus3 19974 19990

Table 8 The order of convergence in time for 1198712error norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00511878 000418182 4219760 times 10minus4 3613597 33088994 00835505 000806197 8133115 times 10minus4 3309253 33092536 01027137 001176588 1180928 times 10minus3 3125948 33166178 01141693 001561415 1547231 times 10minus3 2870249 333509410 01220153 001982141 1925846 times 10minus3 2621930 3363496

Table 9 The order of convergence in time for 119871infinerror norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00318115 2528470 times 10minus3 2505105 times 10minus4 3653213 33353224 00498808 4159059 times 10minus3 4456332 times 10minus4 3584155 32223286 00594582 5587773 times 10minus3 5905399 times 10minus4 3411531 32421678 00641225 7106547 times 10minus3 7270440 times 10minus4 3173611 328903410 00661177 8861734 times 10minus3 8671150 times 10minus4 2899376 3353294

52 Collision of Two Solitons In this section we study theinteraction of two solitary waves with the initial conditionsgiven by a linear sum of two separate solitary waves of variousamplitudes

119906 (119909 0) =

2

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (35)

where 119860119894= radic119888119894

119901119894= radic119888

119894120583(119888119894+ 1) 119894 = 1 2 and 119909

119894and

119888119894(119894 = 1 2) are arbitrary constants The analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

1198682=

21198881

1199011

+

21198882

1199012

+

212058311988811199011

3

+

212058311990121198882

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

minus

212058311990111198881

3

minus

212058311990121198882

3

(36)

In our numerical scheme we choose ℎ = 01 119896 = 001 1198881= 1

1198882= 05 119909

1= 10 and 119909

2= 40 with interval [0 100] Then

the amplitudes are in ratio 2 1 where 21198602= 1198601and the

analytical values for the conservation are 1198681= 82905324294

1198682= 5224332548 and 119868

3= 1799113742 The changes of

the invariants are satisfactorily small since the changes ofthe invariants 119868

1 1198682 and 119868

3are 13295 times 10minus4 22259 times 10minus5

and 68989 times 10minus5 respectively The results are recorded inTable 10 A graph of the two solitary waves collision plottedin a three-dimensional view at some discrete times is shownin Figure 5

53 Collision of Three Solitons In this section we study theinteraction of three solitary waves with the initial conditionsgiven by a linear sum of three separate solitary waves ofvarious amplitudes

119906 (119909 0) =

3

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (37)

8 Abstract and Applied Analysis

Table 10 Invariants and error norms for two solitary waves 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 ℎ = 01 119896 = 001 and 0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 8288129 8288129 5222934 5222933 1800512 18005132 8289522 8293010 5222910 5222951 1800479 18005074 8289462 8295348 5222887 5222954 1800458 18225076 8289304 8297822 5222864 5222956 1800433 18005078 8289233 8300322 5222840 5222958 1800411 180050810 8289231 8302630 5222817 5222960 1800388 1800509

Table 11 Invariants and error norms for three solitary waves 1198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 119909

3= 40 ℎ = 01 119896 = 001 and

0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 11800513 11800513 6490318 6490318 1997082 19970824 11799364 11807800 6485747 6490319 1992012 19966238 11797850 11812825 6483120 6490298 1988853 199607612 11797570 11817192 6482677 6490279 1988441 199636016 11791837 11820521 6474939 6490305 1982852 199687520 11788577 11823320 6469884 6490341 1977926 1996758

0

02

04

06

08

1

u-axis

0

50

100

x-axis

24

68

10

t-axis

minus02

Figure 5 Interaction of two solitons at times from 119905 = 2 4 6 8 10

for 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 and 0 le 119909 le 100

where 119860119894= radic119888119894

119901119894= radic119888119894120583(119888119894+ 1) 119894 = 1 2 3 and 119909

119894and

119888119894(119894 = 1 2 3) are arbitrary constantsThe analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

+

120587radic1198883

1199013

1198682=

21198881

1199011

+

21198882

1199012

+

21198883

1199013

+

212058311988811199011

3

+

212058311990121198882

3

+

212058311990131198883

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

+

41198883

2

31199013

minus

212058311990111198881

3

minus

212058311990121198882

3

minus

212058311990131198883

3

(38)

In this case we choose paraments as ℎ = 01 119896 = 0051198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 and 119909

3= 40

with interval [0 100]Then the amplitudes are in ratio 4 2 1

where 1198601= 2119860

2= 4119860

3and the analytical values for the

conservation are 1198681= 11802939794 119868

2= 6416902131

and 1198683= 1910917141 The changes of the invariants from

the initial variants approach zero throughout and agree withthe analytical values for the three invariants as presentedin Table 11 which indicates that our scheme is satisfactorilyconservativeThe geometry of the initial state and the profilesat time 119905 = 10 119905 = 20 and 119905 = 35 is shown graphically inFigure 6 respectively

54 The Maxwellian Initial Condition Finally we considerthe development of the Maxwellian initial condition

119906 (119909 0) = 119890minus(119909minus40)

2 (39)

into a train of solitary waves We choose different values 120583 =10 120583 = 05 and 120583 = 01 in our numerical scheme Thecomparison of the three variants 119868

1 1198682 and 119868

3with the earlier

result in [29] is presented in Table 12 here Crank-Nicolsonmethod is used for time discretization For 120583 = 10 thechanges of the variants 119868

1 1198682 and 119868

3with respect to the initial

values are 39492 times 10minus6 69575 times 10minus4 and 31509 times 10minus3respectively whereas they are 28210 times 10minus6 260000 times 10minus4and 70771times 10minus4 in [29] For 120583 = 05 the variants are changedby 78982 times 10minus6 68214 times 10minus5 and 61410 times 10minus4 in this caseThedevelopment of theMaxwellian initial condition is shownin Figure 7 with different parameter values respectively Thesmaller 120583 there is the more the number of solitary waves willform The simulations are done up to time = 10 in this caseFrom the previous work [25] the total number of solitarywaves and the values 120583 have the approximate relation

119873 cong [

1

5radic120583

] (40)

Abstract and Applied Analysis 9

Table 12 Values of 1198681 1198682 and 119868

3for Maxwellian initial condition when ℎ = 01 and 119896 = 001 and the space interval [0 100]

120583 Time 1198681

1198682

1198683

1198681[29] 119868

2[29] 119868

3[29]

1

2 1772528 2499467 minus03633570 1772449 2506352 minus036681494 1772544 2499870 minus03643380 1772446 2506235 minus036669746 1772532 2500822 minus03657774 1772447 2506171 minus036663268 1772524 2500977 minus03661331 1772446 2506123 minus0366586010 1772521 2501206 minus03665079 1772444 22060926 minus03665553

05

2 1772534 1876446 02597450 1772451 1879888 025964944 1772534 1876534 02587968 1772446 1879855 025968286 1772528 1876600 02583802 1772449 1879841 025969738 1772523 1876581 02582386 1772450 1879834 0259705010 1772520 1876574 02581499 1772449 1879828 02597092

01

2 1770991 1373320 07687222 1772452 1378607 076087774 1769863 1371236 07658462 1772451 1378577 076083646 1768742 1369139 07630269 1772451 1378546 076079378 1767630 1367058 07602389 1772451 1378515 0760752910 1766526 1364993 07574795 1772453 1378483 07607117

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(a) Time = 0

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(b) Time = 10

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(c) Time = 20

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(d) Time = 35

Figure 6 Interaction of three solitary waves

10 Abstract and Applied Analysis

0 20 40 60 80 100

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

(a) 120583 = 10

0 20 40 60 80 100minus04

minus02

0

02

04

06

08

1

(b) 120583 = 05

0 20 40 60 80 100minus02

0

02

04

06

08

1

(c) 120583 = 01

Figure 7 Maxwellian initial condition with different values 120583

6 Conclusion

In this paper we have developed a Galerkin linear finiteelementmethod to investigate the propagation of solitons andtheir interactions governed by the nonlinear MRLW equa-tion An extrapolation technique has been used to improvethe accuracy in time for this numerical method A linearstability analysis shows that this method is marginally stableThe high efficiency and accuracy of our method is testedby the numerical experiments propagation of single solitoncollision of two and three solitons and development of theMaxwellian initial condition into solitary waves Moreoverthis method numerically satisfies the conservation laws ofmass momentum and energy

Conflict of Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFund of China (11371289) and the China Scholarship Council(201206285005)

References

[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966

[2] T B Benjamin J L Bona and J J Mahony ldquoModel equationsfor long waves in nonlinear dispersive systemsrdquo PhilosophicalTransactions of the Royal Society of London A vol 272 no 1220pp 47ndash78 1972

[3] S Kutluay and A Esen ldquoA finite difference solution of theregularized long-wave equationrdquo Mathematical Problems inEngineering vol 2006 Article ID 85743 14 pages 2006

Abstract and Applied Analysis 11

[4] J Lin Z Xie and J Zhou ldquoHigh-order compact differencescheme for the regularized long wave equationrdquo Communica-tions in Numerical Methods in Engineering vol 23 no 2 pp135ndash156 2007

[5] P Avilez-Valente and F J Seabra-Santos ldquoA Petrov-Galerkinfinite element scheme for the regularized long wave equationrdquoComputational Mechanics vol 34 no 4 pp 256ndash270 2004

[6] A Esen and S Kutluay ldquoApplication of a lumped Galerkinmethod to the regularized long wave equationrdquo Applied Mathe-matics and Computation vol 174 no 2 pp 833ndash845 2006

[7] Y Liu H Li Y Du and J Wang ldquoExplicit multistep mixedfinite element method for RLW equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 768976 12 pages 2013

[8] B Saka and I Dag ldquoA numerical solution of the RLW equationby Galerkin method using quartic B-splinesrdquo Communicationsin Numerical Methods in Engineering with Biomedical Applica-tions vol 24 no 11 pp 1339ndash1361 2008

[9] I Dag ldquoLeast-squares quadratic B-spline finite elementmethodfor the regularised long wave equationrdquo Computer Methods inApplied Mechanics and Engineering vol 182 no 1-2 pp 205ndash215 2000

[10] I Dag andM N Ozer ldquoApproximation of the RLW equation bythe least square cubic B-spline finite element methodrdquo AppliedMathematical Modelling vol 25 no 3 pp 221ndash231 2001

[11] H Gu and N Chen ldquoLeast-squares mixed finite elementmethods for the RLW equationsrdquoNumericalMethods for PartialDifferential Equations vol 24 no 3 pp 749ndash758 2008

[12] A A Soliman and K R Raslan ldquoCollocation method usingquadratic B-spline for the RLW equationrdquo International Journalof Computer Mathematics vol 78 no 3 pp 399ndash412 2001

[13] I Dag B Saka and D Irk ldquoApplication of cubic B-splines fornumerical solution of the RLW equationrdquo Applied Mathematicsand Computation vol 159 no 2 pp 373ndash389 2004

[14] A A Soliman and M H Hussien ldquoCollocation solution forRLW equation with septic splinerdquo Applied Mathematics andComputation vol 161 no 2 pp 623ndash636 2005

[15] S U Islam S Haq and A Ali ldquoA meshfree method for thenumerical solution of the RLW equationrdquo Journal of Compu-tational and Applied Mathematics vol 223 no 2 pp 997ndash10122009

[16] S Haq S U Islam andA Ali ldquoA numerical meshfree techniquefor the solution of the MEW equationrdquo Computer Modeling inEngineering amp Sciences vol 38 no 1 pp 1ndash23 2008

[17] L Mei and Y Chen ldquoExplicit multistep method for thenumerical solution of RLW equationrdquoAppliedMathematics andComputation vol 218 no 18 pp 9547ndash9554 2012

[18] L Zhang ldquoAfinite difference scheme for generalized regularizedlong-wave equationrdquo Applied Mathematics and Computationvol 168 no 2 pp 962ndash972 2005

[19] MMohammadi and RMokhtari ldquoSolving the generalized reg-ularized longwave equation on the basis of a reproducing kernelspacerdquo Journal of Computational and Applied Mathematics vol235 no 14 pp 4003ndash4014 2011

[20] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004

[21] T Roshan ldquoA Petrov-Galerkin method for solving the gener-alized regularized long wave (GRLW) equationrdquo Computers ampMathematics with Applications vol 63 no 5 pp 943ndash956 2012

[22] A A Soliman ldquoNumerical simulation of the generalizedregularized long wave equation by Hersquos variational iterationmethodrdquoMathematics and Computers in Simulation vol 70 no2 pp 119ndash124 2005

[23] R Mokhtari and M Mohammadi ldquoNumerical solution ofGRLW equation using Sinc-collocation methodrdquo ComputerPhysics Communications vol 181 no 7 pp 1266ndash1274 2010

[24] A H A Ali A A Soliman and K R Raslan ldquoSoliton solutionfor nonlinear partial differential equations by cosine-functionmethodrdquo Physics Letters A vol 368 no 3-4 pp 299ndash304 2007

[25] L R T Gardner G A Gardner F A Irk and N K AmeinldquoApproximations of solitary waves of the MRLW equation byB-spline finite elementsrdquo The Arabian Journal for Science andEngineering A vol 22 no 2 pp 183ndash193 1997

[26] A K Khalifa K R Raslan and H M Alzubaidi ldquoA finite dif-ference scheme for the MRLW and solitary wave interactionsrdquoApplied Mathematics and Computation vol 189 no 1 pp 346ndash354 2007

[27] A K Khalifa K R Raslan and H M Alzubaidi ldquoA collocationmethod with cubic B-splines for solving the MRLW equationrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 406ndash418 2008

[28] K R Raslan and S M Hassan ldquoSolitary waves for the MRLWequationrdquo Applied Mathematics Letters vol 22 no 7 pp 984ndash989 2009

[29] K R Raslan and T S EL-Danaf ldquoSolitary waves solutions of theMRLW equation using quintic B-splinesrdquo Journal of King SaudUniversity Science vol 22 no 3 pp 161ndash166 2010

[30] A K Khalifa K R Raslan and H M Alzubaidi ldquoNumericalstudy using ADM for the modified regularized long waveequationrdquo Applied Mathematical Modelling vol 32 no 12 pp2962ndash2972 2008

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

Table 6 The order of convergence in space for 1198712error norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δℎ = 04

Order Order(0102) (0204)

2 1396685 times 10minus4 5598770 times 10minus4 2258793 times 10minus3 20031 201244 2592545 times 10minus4 1038054 times 10minus3 4168527 times 10minus3 20014 200576 3647701 times 10minus4 1459612 times 10minus3 5846531 times 10minus3 20005 200208 4638980 times 10minus4 1855582 times 10minus3 7421648 times 10minus3 20000 1999910 5600205 times 10minus4 2239531 times 10minus3 8948683 times 10minus3 19996 19985

Table 7 The order of convergence in space for 119871infinerror norm with 119896 = 001 and 119888 = 03

Space Δℎ = 01 Δℎ = 02 Δ119905 = 04

Order Order(0102) (0204)

2 0920780 times 10minus4 3689931 times 10minus4 1480253 times 10minus3 20027 200424 1512530 times 10minus4 6055462 times 10minus4 2430139 times 10minus3 20012 200476 1956752 times 10minus4 7811765 times 10minus4 3126531 times 10minus3 19972 200088 2368004 times 10minus4 9454389 times 10minus4 3768796 times 10minus3 19973 1995010 2766940 times 10minus4 1104758 times 10minus3 4416067 times 10minus3 19974 19990

Table 8 The order of convergence in time for 1198712error norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00511878 000418182 4219760 times 10minus4 3613597 33088994 00835505 000806197 8133115 times 10minus4 3309253 33092536 01027137 001176588 1180928 times 10minus3 3125948 33166178 01141693 001561415 1547231 times 10minus3 2870249 333509410 01220153 001982141 1925846 times 10minus3 2621930 3363496

Table 9 The order of convergence in time for 119871infinerror norm with ℎ = 01 and 119888 = 03

Time Δ119905 = 20 Δ119905 = 10 Δ119905 = 05

Order Order(2010) (1005)

2 00318115 2528470 times 10minus3 2505105 times 10minus4 3653213 33353224 00498808 4159059 times 10minus3 4456332 times 10minus4 3584155 32223286 00594582 5587773 times 10minus3 5905399 times 10minus4 3411531 32421678 00641225 7106547 times 10minus3 7270440 times 10minus4 3173611 328903410 00661177 8861734 times 10minus3 8671150 times 10minus4 2899376 3353294

52 Collision of Two Solitons In this section we study theinteraction of two solitary waves with the initial conditionsgiven by a linear sum of two separate solitary waves of variousamplitudes

119906 (119909 0) =

2

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (35)

where 119860119894= radic119888119894

119901119894= radic119888

119894120583(119888119894+ 1) 119894 = 1 2 and 119909

119894and

119888119894(119894 = 1 2) are arbitrary constants The analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

1198682=

21198881

1199011

+

21198882

1199012

+

212058311988811199011

3

+

212058311990121198882

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

minus

212058311990111198881

3

minus

212058311990121198882

3

(36)

In our numerical scheme we choose ℎ = 01 119896 = 001 1198881= 1

1198882= 05 119909

1= 10 and 119909

2= 40 with interval [0 100] Then

the amplitudes are in ratio 2 1 where 21198602= 1198601and the

analytical values for the conservation are 1198681= 82905324294

1198682= 5224332548 and 119868

3= 1799113742 The changes of

the invariants are satisfactorily small since the changes ofthe invariants 119868

1 1198682 and 119868

3are 13295 times 10minus4 22259 times 10minus5

and 68989 times 10minus5 respectively The results are recorded inTable 10 A graph of the two solitary waves collision plottedin a three-dimensional view at some discrete times is shownin Figure 5

53 Collision of Three Solitons In this section we study theinteraction of three solitary waves with the initial conditionsgiven by a linear sum of three separate solitary waves ofvarious amplitudes

119906 (119909 0) =

3

sum

119894=1

119860119894sech (119901

119894(119909 minus 119909

119894)) (37)

8 Abstract and Applied Analysis

Table 10 Invariants and error norms for two solitary waves 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 ℎ = 01 119896 = 001 and 0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 8288129 8288129 5222934 5222933 1800512 18005132 8289522 8293010 5222910 5222951 1800479 18005074 8289462 8295348 5222887 5222954 1800458 18225076 8289304 8297822 5222864 5222956 1800433 18005078 8289233 8300322 5222840 5222958 1800411 180050810 8289231 8302630 5222817 5222960 1800388 1800509

Table 11 Invariants and error norms for three solitary waves 1198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 119909

3= 40 ℎ = 01 119896 = 001 and

0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 11800513 11800513 6490318 6490318 1997082 19970824 11799364 11807800 6485747 6490319 1992012 19966238 11797850 11812825 6483120 6490298 1988853 199607612 11797570 11817192 6482677 6490279 1988441 199636016 11791837 11820521 6474939 6490305 1982852 199687520 11788577 11823320 6469884 6490341 1977926 1996758

0

02

04

06

08

1

u-axis

0

50

100

x-axis

24

68

10

t-axis

minus02

Figure 5 Interaction of two solitons at times from 119905 = 2 4 6 8 10

for 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 and 0 le 119909 le 100

where 119860119894= radic119888119894

119901119894= radic119888119894120583(119888119894+ 1) 119894 = 1 2 3 and 119909

119894and

119888119894(119894 = 1 2 3) are arbitrary constantsThe analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

+

120587radic1198883

1199013

1198682=

21198881

1199011

+

21198882

1199012

+

21198883

1199013

+

212058311988811199011

3

+

212058311990121198882

3

+

212058311990131198883

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

+

41198883

2

31199013

minus

212058311990111198881

3

minus

212058311990121198882

3

minus

212058311990131198883

3

(38)

In this case we choose paraments as ℎ = 01 119896 = 0051198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 and 119909

3= 40

with interval [0 100]Then the amplitudes are in ratio 4 2 1

where 1198601= 2119860

2= 4119860

3and the analytical values for the

conservation are 1198681= 11802939794 119868

2= 6416902131

and 1198683= 1910917141 The changes of the invariants from

the initial variants approach zero throughout and agree withthe analytical values for the three invariants as presentedin Table 11 which indicates that our scheme is satisfactorilyconservativeThe geometry of the initial state and the profilesat time 119905 = 10 119905 = 20 and 119905 = 35 is shown graphically inFigure 6 respectively

54 The Maxwellian Initial Condition Finally we considerthe development of the Maxwellian initial condition

119906 (119909 0) = 119890minus(119909minus40)

2 (39)

into a train of solitary waves We choose different values 120583 =10 120583 = 05 and 120583 = 01 in our numerical scheme Thecomparison of the three variants 119868

1 1198682 and 119868

3with the earlier

result in [29] is presented in Table 12 here Crank-Nicolsonmethod is used for time discretization For 120583 = 10 thechanges of the variants 119868

1 1198682 and 119868

3with respect to the initial

values are 39492 times 10minus6 69575 times 10minus4 and 31509 times 10minus3respectively whereas they are 28210 times 10minus6 260000 times 10minus4and 70771times 10minus4 in [29] For 120583 = 05 the variants are changedby 78982 times 10minus6 68214 times 10minus5 and 61410 times 10minus4 in this caseThedevelopment of theMaxwellian initial condition is shownin Figure 7 with different parameter values respectively Thesmaller 120583 there is the more the number of solitary waves willform The simulations are done up to time = 10 in this caseFrom the previous work [25] the total number of solitarywaves and the values 120583 have the approximate relation

119873 cong [

1

5radic120583

] (40)

Abstract and Applied Analysis 9

Table 12 Values of 1198681 1198682 and 119868

3for Maxwellian initial condition when ℎ = 01 and 119896 = 001 and the space interval [0 100]

120583 Time 1198681

1198682

1198683

1198681[29] 119868

2[29] 119868

3[29]

1

2 1772528 2499467 minus03633570 1772449 2506352 minus036681494 1772544 2499870 minus03643380 1772446 2506235 minus036669746 1772532 2500822 minus03657774 1772447 2506171 minus036663268 1772524 2500977 minus03661331 1772446 2506123 minus0366586010 1772521 2501206 minus03665079 1772444 22060926 minus03665553

05

2 1772534 1876446 02597450 1772451 1879888 025964944 1772534 1876534 02587968 1772446 1879855 025968286 1772528 1876600 02583802 1772449 1879841 025969738 1772523 1876581 02582386 1772450 1879834 0259705010 1772520 1876574 02581499 1772449 1879828 02597092

01

2 1770991 1373320 07687222 1772452 1378607 076087774 1769863 1371236 07658462 1772451 1378577 076083646 1768742 1369139 07630269 1772451 1378546 076079378 1767630 1367058 07602389 1772451 1378515 0760752910 1766526 1364993 07574795 1772453 1378483 07607117

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(a) Time = 0

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(b) Time = 10

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(c) Time = 20

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(d) Time = 35

Figure 6 Interaction of three solitary waves

10 Abstract and Applied Analysis

0 20 40 60 80 100

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

(a) 120583 = 10

0 20 40 60 80 100minus04

minus02

0

02

04

06

08

1

(b) 120583 = 05

0 20 40 60 80 100minus02

0

02

04

06

08

1

(c) 120583 = 01

Figure 7 Maxwellian initial condition with different values 120583

6 Conclusion

In this paper we have developed a Galerkin linear finiteelementmethod to investigate the propagation of solitons andtheir interactions governed by the nonlinear MRLW equa-tion An extrapolation technique has been used to improvethe accuracy in time for this numerical method A linearstability analysis shows that this method is marginally stableThe high efficiency and accuracy of our method is testedby the numerical experiments propagation of single solitoncollision of two and three solitons and development of theMaxwellian initial condition into solitary waves Moreoverthis method numerically satisfies the conservation laws ofmass momentum and energy

Conflict of Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFund of China (11371289) and the China Scholarship Council(201206285005)

References

[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966

[2] T B Benjamin J L Bona and J J Mahony ldquoModel equationsfor long waves in nonlinear dispersive systemsrdquo PhilosophicalTransactions of the Royal Society of London A vol 272 no 1220pp 47ndash78 1972

[3] S Kutluay and A Esen ldquoA finite difference solution of theregularized long-wave equationrdquo Mathematical Problems inEngineering vol 2006 Article ID 85743 14 pages 2006

Abstract and Applied Analysis 11

[4] J Lin Z Xie and J Zhou ldquoHigh-order compact differencescheme for the regularized long wave equationrdquo Communica-tions in Numerical Methods in Engineering vol 23 no 2 pp135ndash156 2007

[5] P Avilez-Valente and F J Seabra-Santos ldquoA Petrov-Galerkinfinite element scheme for the regularized long wave equationrdquoComputational Mechanics vol 34 no 4 pp 256ndash270 2004

[6] A Esen and S Kutluay ldquoApplication of a lumped Galerkinmethod to the regularized long wave equationrdquo Applied Mathe-matics and Computation vol 174 no 2 pp 833ndash845 2006

[7] Y Liu H Li Y Du and J Wang ldquoExplicit multistep mixedfinite element method for RLW equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 768976 12 pages 2013

[8] B Saka and I Dag ldquoA numerical solution of the RLW equationby Galerkin method using quartic B-splinesrdquo Communicationsin Numerical Methods in Engineering with Biomedical Applica-tions vol 24 no 11 pp 1339ndash1361 2008

[9] I Dag ldquoLeast-squares quadratic B-spline finite elementmethodfor the regularised long wave equationrdquo Computer Methods inApplied Mechanics and Engineering vol 182 no 1-2 pp 205ndash215 2000

[10] I Dag andM N Ozer ldquoApproximation of the RLW equation bythe least square cubic B-spline finite element methodrdquo AppliedMathematical Modelling vol 25 no 3 pp 221ndash231 2001

[11] H Gu and N Chen ldquoLeast-squares mixed finite elementmethods for the RLW equationsrdquoNumericalMethods for PartialDifferential Equations vol 24 no 3 pp 749ndash758 2008

[12] A A Soliman and K R Raslan ldquoCollocation method usingquadratic B-spline for the RLW equationrdquo International Journalof Computer Mathematics vol 78 no 3 pp 399ndash412 2001

[13] I Dag B Saka and D Irk ldquoApplication of cubic B-splines fornumerical solution of the RLW equationrdquo Applied Mathematicsand Computation vol 159 no 2 pp 373ndash389 2004

[14] A A Soliman and M H Hussien ldquoCollocation solution forRLW equation with septic splinerdquo Applied Mathematics andComputation vol 161 no 2 pp 623ndash636 2005

[15] S U Islam S Haq and A Ali ldquoA meshfree method for thenumerical solution of the RLW equationrdquo Journal of Compu-tational and Applied Mathematics vol 223 no 2 pp 997ndash10122009

[16] S Haq S U Islam andA Ali ldquoA numerical meshfree techniquefor the solution of the MEW equationrdquo Computer Modeling inEngineering amp Sciences vol 38 no 1 pp 1ndash23 2008

[17] L Mei and Y Chen ldquoExplicit multistep method for thenumerical solution of RLW equationrdquoAppliedMathematics andComputation vol 218 no 18 pp 9547ndash9554 2012

[18] L Zhang ldquoAfinite difference scheme for generalized regularizedlong-wave equationrdquo Applied Mathematics and Computationvol 168 no 2 pp 962ndash972 2005

[19] MMohammadi and RMokhtari ldquoSolving the generalized reg-ularized longwave equation on the basis of a reproducing kernelspacerdquo Journal of Computational and Applied Mathematics vol235 no 14 pp 4003ndash4014 2011

[20] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004

[21] T Roshan ldquoA Petrov-Galerkin method for solving the gener-alized regularized long wave (GRLW) equationrdquo Computers ampMathematics with Applications vol 63 no 5 pp 943ndash956 2012

[22] A A Soliman ldquoNumerical simulation of the generalizedregularized long wave equation by Hersquos variational iterationmethodrdquoMathematics and Computers in Simulation vol 70 no2 pp 119ndash124 2005

[23] R Mokhtari and M Mohammadi ldquoNumerical solution ofGRLW equation using Sinc-collocation methodrdquo ComputerPhysics Communications vol 181 no 7 pp 1266ndash1274 2010

[24] A H A Ali A A Soliman and K R Raslan ldquoSoliton solutionfor nonlinear partial differential equations by cosine-functionmethodrdquo Physics Letters A vol 368 no 3-4 pp 299ndash304 2007

[25] L R T Gardner G A Gardner F A Irk and N K AmeinldquoApproximations of solitary waves of the MRLW equation byB-spline finite elementsrdquo The Arabian Journal for Science andEngineering A vol 22 no 2 pp 183ndash193 1997

[26] A K Khalifa K R Raslan and H M Alzubaidi ldquoA finite dif-ference scheme for the MRLW and solitary wave interactionsrdquoApplied Mathematics and Computation vol 189 no 1 pp 346ndash354 2007

[27] A K Khalifa K R Raslan and H M Alzubaidi ldquoA collocationmethod with cubic B-splines for solving the MRLW equationrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 406ndash418 2008

[28] K R Raslan and S M Hassan ldquoSolitary waves for the MRLWequationrdquo Applied Mathematics Letters vol 22 no 7 pp 984ndash989 2009

[29] K R Raslan and T S EL-Danaf ldquoSolitary waves solutions of theMRLW equation using quintic B-splinesrdquo Journal of King SaudUniversity Science vol 22 no 3 pp 161ndash166 2010

[30] A K Khalifa K R Raslan and H M Alzubaidi ldquoNumericalstudy using ADM for the modified regularized long waveequationrdquo Applied Mathematical Modelling vol 32 no 12 pp2962ndash2972 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

Table 10 Invariants and error norms for two solitary waves 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 ℎ = 01 119896 = 001 and 0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 8288129 8288129 5222934 5222933 1800512 18005132 8289522 8293010 5222910 5222951 1800479 18005074 8289462 8295348 5222887 5222954 1800458 18225076 8289304 8297822 5222864 5222956 1800433 18005078 8289233 8300322 5222840 5222958 1800411 180050810 8289231 8302630 5222817 5222960 1800388 1800509

Table 11 Invariants and error norms for three solitary waves 1198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 119909

3= 40 ℎ = 01 119896 = 001 and

0 le 119909 le 100

Time 1198681

1198682

1198683

C-N R-K C-N R-K C-N R-K0 11800513 11800513 6490318 6490318 1997082 19970824 11799364 11807800 6485747 6490319 1992012 19966238 11797850 11812825 6483120 6490298 1988853 199607612 11797570 11817192 6482677 6490279 1988441 199636016 11791837 11820521 6474939 6490305 1982852 199687520 11788577 11823320 6469884 6490341 1977926 1996758

0

02

04

06

08

1

u-axis

0

50

100

x-axis

24

68

10

t-axis

minus02

Figure 5 Interaction of two solitons at times from 119905 = 2 4 6 8 10

for 1198881= 1 1198882= 05 119909

1= 10 119909

2= 40 and 0 le 119909 le 100

where 119860119894= radic119888119894

119901119894= radic119888119894120583(119888119894+ 1) 119894 = 1 2 3 and 119909

119894and

119888119894(119894 = 1 2 3) are arbitrary constantsThe analytical values for

the conservation laws in this case have the following form

1198681=

120587radic1198881

1199011

+

120587radic1198882

1199012

+

120587radic1198883

1199013

1198682=

21198881

1199011

+

21198882

1199012

+

21198883

1199013

+

212058311988811199011

3

+

212058311990121198882

3

+

212058311990131198883

3

1198683=

41198881

2

31199011

+

41198882

2

31199012

+

41198883

2

31199013

minus

212058311990111198881

3

minus

212058311990121198882

3

minus

212058311990131198883

3

(38)

In this case we choose paraments as ℎ = 01 119896 = 0051198881= 1 119888

2= 05 119888

3= 025 119909

1= 10 119909

2= 20 and 119909

3= 40

with interval [0 100]Then the amplitudes are in ratio 4 2 1

where 1198601= 2119860

2= 4119860

3and the analytical values for the

conservation are 1198681= 11802939794 119868

2= 6416902131

and 1198683= 1910917141 The changes of the invariants from

the initial variants approach zero throughout and agree withthe analytical values for the three invariants as presentedin Table 11 which indicates that our scheme is satisfactorilyconservativeThe geometry of the initial state and the profilesat time 119905 = 10 119905 = 20 and 119905 = 35 is shown graphically inFigure 6 respectively

54 The Maxwellian Initial Condition Finally we considerthe development of the Maxwellian initial condition

119906 (119909 0) = 119890minus(119909minus40)

2 (39)

into a train of solitary waves We choose different values 120583 =10 120583 = 05 and 120583 = 01 in our numerical scheme Thecomparison of the three variants 119868

1 1198682 and 119868

3with the earlier

result in [29] is presented in Table 12 here Crank-Nicolsonmethod is used for time discretization For 120583 = 10 thechanges of the variants 119868

1 1198682 and 119868

3with respect to the initial

values are 39492 times 10minus6 69575 times 10minus4 and 31509 times 10minus3respectively whereas they are 28210 times 10minus6 260000 times 10minus4and 70771times 10minus4 in [29] For 120583 = 05 the variants are changedby 78982 times 10minus6 68214 times 10minus5 and 61410 times 10minus4 in this caseThedevelopment of theMaxwellian initial condition is shownin Figure 7 with different parameter values respectively Thesmaller 120583 there is the more the number of solitary waves willform The simulations are done up to time = 10 in this caseFrom the previous work [25] the total number of solitarywaves and the values 120583 have the approximate relation

119873 cong [

1

5radic120583

] (40)

Abstract and Applied Analysis 9

Table 12 Values of 1198681 1198682 and 119868

3for Maxwellian initial condition when ℎ = 01 and 119896 = 001 and the space interval [0 100]

120583 Time 1198681

1198682

1198683

1198681[29] 119868

2[29] 119868

3[29]

1

2 1772528 2499467 minus03633570 1772449 2506352 minus036681494 1772544 2499870 minus03643380 1772446 2506235 minus036669746 1772532 2500822 minus03657774 1772447 2506171 minus036663268 1772524 2500977 minus03661331 1772446 2506123 minus0366586010 1772521 2501206 minus03665079 1772444 22060926 minus03665553

05

2 1772534 1876446 02597450 1772451 1879888 025964944 1772534 1876534 02587968 1772446 1879855 025968286 1772528 1876600 02583802 1772449 1879841 025969738 1772523 1876581 02582386 1772450 1879834 0259705010 1772520 1876574 02581499 1772449 1879828 02597092

01

2 1770991 1373320 07687222 1772452 1378607 076087774 1769863 1371236 07658462 1772451 1378577 076083646 1768742 1369139 07630269 1772451 1378546 076079378 1767630 1367058 07602389 1772451 1378515 0760752910 1766526 1364993 07574795 1772453 1378483 07607117

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(a) Time = 0

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(b) Time = 10

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(c) Time = 20

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(d) Time = 35

Figure 6 Interaction of three solitary waves

10 Abstract and Applied Analysis

0 20 40 60 80 100

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

(a) 120583 = 10

0 20 40 60 80 100minus04

minus02

0

02

04

06

08

1

(b) 120583 = 05

0 20 40 60 80 100minus02

0

02

04

06

08

1

(c) 120583 = 01

Figure 7 Maxwellian initial condition with different values 120583

6 Conclusion

In this paper we have developed a Galerkin linear finiteelementmethod to investigate the propagation of solitons andtheir interactions governed by the nonlinear MRLW equa-tion An extrapolation technique has been used to improvethe accuracy in time for this numerical method A linearstability analysis shows that this method is marginally stableThe high efficiency and accuracy of our method is testedby the numerical experiments propagation of single solitoncollision of two and three solitons and development of theMaxwellian initial condition into solitary waves Moreoverthis method numerically satisfies the conservation laws ofmass momentum and energy

Conflict of Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFund of China (11371289) and the China Scholarship Council(201206285005)

References

[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966

[2] T B Benjamin J L Bona and J J Mahony ldquoModel equationsfor long waves in nonlinear dispersive systemsrdquo PhilosophicalTransactions of the Royal Society of London A vol 272 no 1220pp 47ndash78 1972

[3] S Kutluay and A Esen ldquoA finite difference solution of theregularized long-wave equationrdquo Mathematical Problems inEngineering vol 2006 Article ID 85743 14 pages 2006

Abstract and Applied Analysis 11

[4] J Lin Z Xie and J Zhou ldquoHigh-order compact differencescheme for the regularized long wave equationrdquo Communica-tions in Numerical Methods in Engineering vol 23 no 2 pp135ndash156 2007

[5] P Avilez-Valente and F J Seabra-Santos ldquoA Petrov-Galerkinfinite element scheme for the regularized long wave equationrdquoComputational Mechanics vol 34 no 4 pp 256ndash270 2004

[6] A Esen and S Kutluay ldquoApplication of a lumped Galerkinmethod to the regularized long wave equationrdquo Applied Mathe-matics and Computation vol 174 no 2 pp 833ndash845 2006

[7] Y Liu H Li Y Du and J Wang ldquoExplicit multistep mixedfinite element method for RLW equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 768976 12 pages 2013

[8] B Saka and I Dag ldquoA numerical solution of the RLW equationby Galerkin method using quartic B-splinesrdquo Communicationsin Numerical Methods in Engineering with Biomedical Applica-tions vol 24 no 11 pp 1339ndash1361 2008

[9] I Dag ldquoLeast-squares quadratic B-spline finite elementmethodfor the regularised long wave equationrdquo Computer Methods inApplied Mechanics and Engineering vol 182 no 1-2 pp 205ndash215 2000

[10] I Dag andM N Ozer ldquoApproximation of the RLW equation bythe least square cubic B-spline finite element methodrdquo AppliedMathematical Modelling vol 25 no 3 pp 221ndash231 2001

[11] H Gu and N Chen ldquoLeast-squares mixed finite elementmethods for the RLW equationsrdquoNumericalMethods for PartialDifferential Equations vol 24 no 3 pp 749ndash758 2008

[12] A A Soliman and K R Raslan ldquoCollocation method usingquadratic B-spline for the RLW equationrdquo International Journalof Computer Mathematics vol 78 no 3 pp 399ndash412 2001

[13] I Dag B Saka and D Irk ldquoApplication of cubic B-splines fornumerical solution of the RLW equationrdquo Applied Mathematicsand Computation vol 159 no 2 pp 373ndash389 2004

[14] A A Soliman and M H Hussien ldquoCollocation solution forRLW equation with septic splinerdquo Applied Mathematics andComputation vol 161 no 2 pp 623ndash636 2005

[15] S U Islam S Haq and A Ali ldquoA meshfree method for thenumerical solution of the RLW equationrdquo Journal of Compu-tational and Applied Mathematics vol 223 no 2 pp 997ndash10122009

[16] S Haq S U Islam andA Ali ldquoA numerical meshfree techniquefor the solution of the MEW equationrdquo Computer Modeling inEngineering amp Sciences vol 38 no 1 pp 1ndash23 2008

[17] L Mei and Y Chen ldquoExplicit multistep method for thenumerical solution of RLW equationrdquoAppliedMathematics andComputation vol 218 no 18 pp 9547ndash9554 2012

[18] L Zhang ldquoAfinite difference scheme for generalized regularizedlong-wave equationrdquo Applied Mathematics and Computationvol 168 no 2 pp 962ndash972 2005

[19] MMohammadi and RMokhtari ldquoSolving the generalized reg-ularized longwave equation on the basis of a reproducing kernelspacerdquo Journal of Computational and Applied Mathematics vol235 no 14 pp 4003ndash4014 2011

[20] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004

[21] T Roshan ldquoA Petrov-Galerkin method for solving the gener-alized regularized long wave (GRLW) equationrdquo Computers ampMathematics with Applications vol 63 no 5 pp 943ndash956 2012

[22] A A Soliman ldquoNumerical simulation of the generalizedregularized long wave equation by Hersquos variational iterationmethodrdquoMathematics and Computers in Simulation vol 70 no2 pp 119ndash124 2005

[23] R Mokhtari and M Mohammadi ldquoNumerical solution ofGRLW equation using Sinc-collocation methodrdquo ComputerPhysics Communications vol 181 no 7 pp 1266ndash1274 2010

[24] A H A Ali A A Soliman and K R Raslan ldquoSoliton solutionfor nonlinear partial differential equations by cosine-functionmethodrdquo Physics Letters A vol 368 no 3-4 pp 299ndash304 2007

[25] L R T Gardner G A Gardner F A Irk and N K AmeinldquoApproximations of solitary waves of the MRLW equation byB-spline finite elementsrdquo The Arabian Journal for Science andEngineering A vol 22 no 2 pp 183ndash193 1997

[26] A K Khalifa K R Raslan and H M Alzubaidi ldquoA finite dif-ference scheme for the MRLW and solitary wave interactionsrdquoApplied Mathematics and Computation vol 189 no 1 pp 346ndash354 2007

[27] A K Khalifa K R Raslan and H M Alzubaidi ldquoA collocationmethod with cubic B-splines for solving the MRLW equationrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 406ndash418 2008

[28] K R Raslan and S M Hassan ldquoSolitary waves for the MRLWequationrdquo Applied Mathematics Letters vol 22 no 7 pp 984ndash989 2009

[29] K R Raslan and T S EL-Danaf ldquoSolitary waves solutions of theMRLW equation using quintic B-splinesrdquo Journal of King SaudUniversity Science vol 22 no 3 pp 161ndash166 2010

[30] A K Khalifa K R Raslan and H M Alzubaidi ldquoNumericalstudy using ADM for the modified regularized long waveequationrdquo Applied Mathematical Modelling vol 32 no 12 pp2962ndash2972 2008

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

Table 12 Values of 1198681 1198682 and 119868

3for Maxwellian initial condition when ℎ = 01 and 119896 = 001 and the space interval [0 100]

120583 Time 1198681

1198682

1198683

1198681[29] 119868

2[29] 119868

3[29]

1

2 1772528 2499467 minus03633570 1772449 2506352 minus036681494 1772544 2499870 minus03643380 1772446 2506235 minus036669746 1772532 2500822 minus03657774 1772447 2506171 minus036663268 1772524 2500977 minus03661331 1772446 2506123 minus0366586010 1772521 2501206 minus03665079 1772444 22060926 minus03665553

05

2 1772534 1876446 02597450 1772451 1879888 025964944 1772534 1876534 02587968 1772446 1879855 025968286 1772528 1876600 02583802 1772449 1879841 025969738 1772523 1876581 02582386 1772450 1879834 0259705010 1772520 1876574 02581499 1772449 1879828 02597092

01

2 1770991 1373320 07687222 1772452 1378607 076087774 1769863 1371236 07658462 1772451 1378577 076083646 1768742 1369139 07630269 1772451 1378546 076079378 1767630 1367058 07602389 1772451 1378515 0760752910 1766526 1364993 07574795 1772453 1378483 07607117

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(a) Time = 0

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(b) Time = 10

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(c) Time = 20

0 20 40 60 80 100

0

02

04

06

08

1

minus02

(d) Time = 35

Figure 6 Interaction of three solitary waves

10 Abstract and Applied Analysis

0 20 40 60 80 100

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

(a) 120583 = 10

0 20 40 60 80 100minus04

minus02

0

02

04

06

08

1

(b) 120583 = 05

0 20 40 60 80 100minus02

0

02

04

06

08

1

(c) 120583 = 01

Figure 7 Maxwellian initial condition with different values 120583

6 Conclusion

In this paper we have developed a Galerkin linear finiteelementmethod to investigate the propagation of solitons andtheir interactions governed by the nonlinear MRLW equa-tion An extrapolation technique has been used to improvethe accuracy in time for this numerical method A linearstability analysis shows that this method is marginally stableThe high efficiency and accuracy of our method is testedby the numerical experiments propagation of single solitoncollision of two and three solitons and development of theMaxwellian initial condition into solitary waves Moreoverthis method numerically satisfies the conservation laws ofmass momentum and energy

Conflict of Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFund of China (11371289) and the China Scholarship Council(201206285005)

References

[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966

[2] T B Benjamin J L Bona and J J Mahony ldquoModel equationsfor long waves in nonlinear dispersive systemsrdquo PhilosophicalTransactions of the Royal Society of London A vol 272 no 1220pp 47ndash78 1972

[3] S Kutluay and A Esen ldquoA finite difference solution of theregularized long-wave equationrdquo Mathematical Problems inEngineering vol 2006 Article ID 85743 14 pages 2006

Abstract and Applied Analysis 11

[4] J Lin Z Xie and J Zhou ldquoHigh-order compact differencescheme for the regularized long wave equationrdquo Communica-tions in Numerical Methods in Engineering vol 23 no 2 pp135ndash156 2007

[5] P Avilez-Valente and F J Seabra-Santos ldquoA Petrov-Galerkinfinite element scheme for the regularized long wave equationrdquoComputational Mechanics vol 34 no 4 pp 256ndash270 2004

[6] A Esen and S Kutluay ldquoApplication of a lumped Galerkinmethod to the regularized long wave equationrdquo Applied Mathe-matics and Computation vol 174 no 2 pp 833ndash845 2006

[7] Y Liu H Li Y Du and J Wang ldquoExplicit multistep mixedfinite element method for RLW equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 768976 12 pages 2013

[8] B Saka and I Dag ldquoA numerical solution of the RLW equationby Galerkin method using quartic B-splinesrdquo Communicationsin Numerical Methods in Engineering with Biomedical Applica-tions vol 24 no 11 pp 1339ndash1361 2008

[9] I Dag ldquoLeast-squares quadratic B-spline finite elementmethodfor the regularised long wave equationrdquo Computer Methods inApplied Mechanics and Engineering vol 182 no 1-2 pp 205ndash215 2000

[10] I Dag andM N Ozer ldquoApproximation of the RLW equation bythe least square cubic B-spline finite element methodrdquo AppliedMathematical Modelling vol 25 no 3 pp 221ndash231 2001

[11] H Gu and N Chen ldquoLeast-squares mixed finite elementmethods for the RLW equationsrdquoNumericalMethods for PartialDifferential Equations vol 24 no 3 pp 749ndash758 2008

[12] A A Soliman and K R Raslan ldquoCollocation method usingquadratic B-spline for the RLW equationrdquo International Journalof Computer Mathematics vol 78 no 3 pp 399ndash412 2001

[13] I Dag B Saka and D Irk ldquoApplication of cubic B-splines fornumerical solution of the RLW equationrdquo Applied Mathematicsand Computation vol 159 no 2 pp 373ndash389 2004

[14] A A Soliman and M H Hussien ldquoCollocation solution forRLW equation with septic splinerdquo Applied Mathematics andComputation vol 161 no 2 pp 623ndash636 2005

[15] S U Islam S Haq and A Ali ldquoA meshfree method for thenumerical solution of the RLW equationrdquo Journal of Compu-tational and Applied Mathematics vol 223 no 2 pp 997ndash10122009

[16] S Haq S U Islam andA Ali ldquoA numerical meshfree techniquefor the solution of the MEW equationrdquo Computer Modeling inEngineering amp Sciences vol 38 no 1 pp 1ndash23 2008

[17] L Mei and Y Chen ldquoExplicit multistep method for thenumerical solution of RLW equationrdquoAppliedMathematics andComputation vol 218 no 18 pp 9547ndash9554 2012

[18] L Zhang ldquoAfinite difference scheme for generalized regularizedlong-wave equationrdquo Applied Mathematics and Computationvol 168 no 2 pp 962ndash972 2005

[19] MMohammadi and RMokhtari ldquoSolving the generalized reg-ularized longwave equation on the basis of a reproducing kernelspacerdquo Journal of Computational and Applied Mathematics vol235 no 14 pp 4003ndash4014 2011

[20] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004

[21] T Roshan ldquoA Petrov-Galerkin method for solving the gener-alized regularized long wave (GRLW) equationrdquo Computers ampMathematics with Applications vol 63 no 5 pp 943ndash956 2012

[22] A A Soliman ldquoNumerical simulation of the generalizedregularized long wave equation by Hersquos variational iterationmethodrdquoMathematics and Computers in Simulation vol 70 no2 pp 119ndash124 2005

[23] R Mokhtari and M Mohammadi ldquoNumerical solution ofGRLW equation using Sinc-collocation methodrdquo ComputerPhysics Communications vol 181 no 7 pp 1266ndash1274 2010

[24] A H A Ali A A Soliman and K R Raslan ldquoSoliton solutionfor nonlinear partial differential equations by cosine-functionmethodrdquo Physics Letters A vol 368 no 3-4 pp 299ndash304 2007

[25] L R T Gardner G A Gardner F A Irk and N K AmeinldquoApproximations of solitary waves of the MRLW equation byB-spline finite elementsrdquo The Arabian Journal for Science andEngineering A vol 22 no 2 pp 183ndash193 1997

[26] A K Khalifa K R Raslan and H M Alzubaidi ldquoA finite dif-ference scheme for the MRLW and solitary wave interactionsrdquoApplied Mathematics and Computation vol 189 no 1 pp 346ndash354 2007

[27] A K Khalifa K R Raslan and H M Alzubaidi ldquoA collocationmethod with cubic B-splines for solving the MRLW equationrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 406ndash418 2008

[28] K R Raslan and S M Hassan ldquoSolitary waves for the MRLWequationrdquo Applied Mathematics Letters vol 22 no 7 pp 984ndash989 2009

[29] K R Raslan and T S EL-Danaf ldquoSolitary waves solutions of theMRLW equation using quintic B-splinesrdquo Journal of King SaudUniversity Science vol 22 no 3 pp 161ndash166 2010

[30] A K Khalifa K R Raslan and H M Alzubaidi ldquoNumericalstudy using ADM for the modified regularized long waveequationrdquo Applied Mathematical Modelling vol 32 no 12 pp2962ndash2972 2008

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

10 Abstract and Applied Analysis

0 20 40 60 80 100

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

(a) 120583 = 10

0 20 40 60 80 100minus04

minus02

0

02

04

06

08

1

(b) 120583 = 05

0 20 40 60 80 100minus02

0

02

04

06

08

1

(c) 120583 = 01

Figure 7 Maxwellian initial condition with different values 120583

6 Conclusion

In this paper we have developed a Galerkin linear finiteelementmethod to investigate the propagation of solitons andtheir interactions governed by the nonlinear MRLW equa-tion An extrapolation technique has been used to improvethe accuracy in time for this numerical method A linearstability analysis shows that this method is marginally stableThe high efficiency and accuracy of our method is testedby the numerical experiments propagation of single solitoncollision of two and three solitons and development of theMaxwellian initial condition into solitary waves Moreoverthis method numerically satisfies the conservation laws ofmass momentum and energy

Conflict of Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFund of China (11371289) and the China Scholarship Council(201206285005)

References

[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966

[2] T B Benjamin J L Bona and J J Mahony ldquoModel equationsfor long waves in nonlinear dispersive systemsrdquo PhilosophicalTransactions of the Royal Society of London A vol 272 no 1220pp 47ndash78 1972

[3] S Kutluay and A Esen ldquoA finite difference solution of theregularized long-wave equationrdquo Mathematical Problems inEngineering vol 2006 Article ID 85743 14 pages 2006

Abstract and Applied Analysis 11

[4] J Lin Z Xie and J Zhou ldquoHigh-order compact differencescheme for the regularized long wave equationrdquo Communica-tions in Numerical Methods in Engineering vol 23 no 2 pp135ndash156 2007

[5] P Avilez-Valente and F J Seabra-Santos ldquoA Petrov-Galerkinfinite element scheme for the regularized long wave equationrdquoComputational Mechanics vol 34 no 4 pp 256ndash270 2004

[6] A Esen and S Kutluay ldquoApplication of a lumped Galerkinmethod to the regularized long wave equationrdquo Applied Mathe-matics and Computation vol 174 no 2 pp 833ndash845 2006

[7] Y Liu H Li Y Du and J Wang ldquoExplicit multistep mixedfinite element method for RLW equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 768976 12 pages 2013

[8] B Saka and I Dag ldquoA numerical solution of the RLW equationby Galerkin method using quartic B-splinesrdquo Communicationsin Numerical Methods in Engineering with Biomedical Applica-tions vol 24 no 11 pp 1339ndash1361 2008

[9] I Dag ldquoLeast-squares quadratic B-spline finite elementmethodfor the regularised long wave equationrdquo Computer Methods inApplied Mechanics and Engineering vol 182 no 1-2 pp 205ndash215 2000

[10] I Dag andM N Ozer ldquoApproximation of the RLW equation bythe least square cubic B-spline finite element methodrdquo AppliedMathematical Modelling vol 25 no 3 pp 221ndash231 2001

[11] H Gu and N Chen ldquoLeast-squares mixed finite elementmethods for the RLW equationsrdquoNumericalMethods for PartialDifferential Equations vol 24 no 3 pp 749ndash758 2008

[12] A A Soliman and K R Raslan ldquoCollocation method usingquadratic B-spline for the RLW equationrdquo International Journalof Computer Mathematics vol 78 no 3 pp 399ndash412 2001

[13] I Dag B Saka and D Irk ldquoApplication of cubic B-splines fornumerical solution of the RLW equationrdquo Applied Mathematicsand Computation vol 159 no 2 pp 373ndash389 2004

[14] A A Soliman and M H Hussien ldquoCollocation solution forRLW equation with septic splinerdquo Applied Mathematics andComputation vol 161 no 2 pp 623ndash636 2005

[15] S U Islam S Haq and A Ali ldquoA meshfree method for thenumerical solution of the RLW equationrdquo Journal of Compu-tational and Applied Mathematics vol 223 no 2 pp 997ndash10122009

[16] S Haq S U Islam andA Ali ldquoA numerical meshfree techniquefor the solution of the MEW equationrdquo Computer Modeling inEngineering amp Sciences vol 38 no 1 pp 1ndash23 2008

[17] L Mei and Y Chen ldquoExplicit multistep method for thenumerical solution of RLW equationrdquoAppliedMathematics andComputation vol 218 no 18 pp 9547ndash9554 2012

[18] L Zhang ldquoAfinite difference scheme for generalized regularizedlong-wave equationrdquo Applied Mathematics and Computationvol 168 no 2 pp 962ndash972 2005

[19] MMohammadi and RMokhtari ldquoSolving the generalized reg-ularized longwave equation on the basis of a reproducing kernelspacerdquo Journal of Computational and Applied Mathematics vol235 no 14 pp 4003ndash4014 2011

[20] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004

[21] T Roshan ldquoA Petrov-Galerkin method for solving the gener-alized regularized long wave (GRLW) equationrdquo Computers ampMathematics with Applications vol 63 no 5 pp 943ndash956 2012

[22] A A Soliman ldquoNumerical simulation of the generalizedregularized long wave equation by Hersquos variational iterationmethodrdquoMathematics and Computers in Simulation vol 70 no2 pp 119ndash124 2005

[23] R Mokhtari and M Mohammadi ldquoNumerical solution ofGRLW equation using Sinc-collocation methodrdquo ComputerPhysics Communications vol 181 no 7 pp 1266ndash1274 2010

[24] A H A Ali A A Soliman and K R Raslan ldquoSoliton solutionfor nonlinear partial differential equations by cosine-functionmethodrdquo Physics Letters A vol 368 no 3-4 pp 299ndash304 2007

[25] L R T Gardner G A Gardner F A Irk and N K AmeinldquoApproximations of solitary waves of the MRLW equation byB-spline finite elementsrdquo The Arabian Journal for Science andEngineering A vol 22 no 2 pp 183ndash193 1997

[26] A K Khalifa K R Raslan and H M Alzubaidi ldquoA finite dif-ference scheme for the MRLW and solitary wave interactionsrdquoApplied Mathematics and Computation vol 189 no 1 pp 346ndash354 2007

[27] A K Khalifa K R Raslan and H M Alzubaidi ldquoA collocationmethod with cubic B-splines for solving the MRLW equationrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 406ndash418 2008

[28] K R Raslan and S M Hassan ldquoSolitary waves for the MRLWequationrdquo Applied Mathematics Letters vol 22 no 7 pp 984ndash989 2009

[29] K R Raslan and T S EL-Danaf ldquoSolitary waves solutions of theMRLW equation using quintic B-splinesrdquo Journal of King SaudUniversity Science vol 22 no 3 pp 161ndash166 2010

[30] A K Khalifa K R Raslan and H M Alzubaidi ldquoNumericalstudy using ADM for the modified regularized long waveequationrdquo Applied Mathematical Modelling vol 32 no 12 pp2962ndash2972 2008

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 11

[4] J Lin Z Xie and J Zhou ldquoHigh-order compact differencescheme for the regularized long wave equationrdquo Communica-tions in Numerical Methods in Engineering vol 23 no 2 pp135ndash156 2007

[5] P Avilez-Valente and F J Seabra-Santos ldquoA Petrov-Galerkinfinite element scheme for the regularized long wave equationrdquoComputational Mechanics vol 34 no 4 pp 256ndash270 2004

[6] A Esen and S Kutluay ldquoApplication of a lumped Galerkinmethod to the regularized long wave equationrdquo Applied Mathe-matics and Computation vol 174 no 2 pp 833ndash845 2006

[7] Y Liu H Li Y Du and J Wang ldquoExplicit multistep mixedfinite element method for RLW equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 768976 12 pages 2013

[8] B Saka and I Dag ldquoA numerical solution of the RLW equationby Galerkin method using quartic B-splinesrdquo Communicationsin Numerical Methods in Engineering with Biomedical Applica-tions vol 24 no 11 pp 1339ndash1361 2008

[9] I Dag ldquoLeast-squares quadratic B-spline finite elementmethodfor the regularised long wave equationrdquo Computer Methods inApplied Mechanics and Engineering vol 182 no 1-2 pp 205ndash215 2000

[10] I Dag andM N Ozer ldquoApproximation of the RLW equation bythe least square cubic B-spline finite element methodrdquo AppliedMathematical Modelling vol 25 no 3 pp 221ndash231 2001

[11] H Gu and N Chen ldquoLeast-squares mixed finite elementmethods for the RLW equationsrdquoNumericalMethods for PartialDifferential Equations vol 24 no 3 pp 749ndash758 2008

[12] A A Soliman and K R Raslan ldquoCollocation method usingquadratic B-spline for the RLW equationrdquo International Journalof Computer Mathematics vol 78 no 3 pp 399ndash412 2001

[13] I Dag B Saka and D Irk ldquoApplication of cubic B-splines fornumerical solution of the RLW equationrdquo Applied Mathematicsand Computation vol 159 no 2 pp 373ndash389 2004

[14] A A Soliman and M H Hussien ldquoCollocation solution forRLW equation with septic splinerdquo Applied Mathematics andComputation vol 161 no 2 pp 623ndash636 2005

[15] S U Islam S Haq and A Ali ldquoA meshfree method for thenumerical solution of the RLW equationrdquo Journal of Compu-tational and Applied Mathematics vol 223 no 2 pp 997ndash10122009

[16] S Haq S U Islam andA Ali ldquoA numerical meshfree techniquefor the solution of the MEW equationrdquo Computer Modeling inEngineering amp Sciences vol 38 no 1 pp 1ndash23 2008

[17] L Mei and Y Chen ldquoExplicit multistep method for thenumerical solution of RLW equationrdquoAppliedMathematics andComputation vol 218 no 18 pp 9547ndash9554 2012

[18] L Zhang ldquoAfinite difference scheme for generalized regularizedlong-wave equationrdquo Applied Mathematics and Computationvol 168 no 2 pp 962ndash972 2005

[19] MMohammadi and RMokhtari ldquoSolving the generalized reg-ularized longwave equation on the basis of a reproducing kernelspacerdquo Journal of Computational and Applied Mathematics vol235 no 14 pp 4003ndash4014 2011

[20] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004

[21] T Roshan ldquoA Petrov-Galerkin method for solving the gener-alized regularized long wave (GRLW) equationrdquo Computers ampMathematics with Applications vol 63 no 5 pp 943ndash956 2012

[22] A A Soliman ldquoNumerical simulation of the generalizedregularized long wave equation by Hersquos variational iterationmethodrdquoMathematics and Computers in Simulation vol 70 no2 pp 119ndash124 2005

[23] R Mokhtari and M Mohammadi ldquoNumerical solution ofGRLW equation using Sinc-collocation methodrdquo ComputerPhysics Communications vol 181 no 7 pp 1266ndash1274 2010

[24] A H A Ali A A Soliman and K R Raslan ldquoSoliton solutionfor nonlinear partial differential equations by cosine-functionmethodrdquo Physics Letters A vol 368 no 3-4 pp 299ndash304 2007

[25] L R T Gardner G A Gardner F A Irk and N K AmeinldquoApproximations of solitary waves of the MRLW equation byB-spline finite elementsrdquo The Arabian Journal for Science andEngineering A vol 22 no 2 pp 183ndash193 1997

[26] A K Khalifa K R Raslan and H M Alzubaidi ldquoA finite dif-ference scheme for the MRLW and solitary wave interactionsrdquoApplied Mathematics and Computation vol 189 no 1 pp 346ndash354 2007

[27] A K Khalifa K R Raslan and H M Alzubaidi ldquoA collocationmethod with cubic B-splines for solving the MRLW equationrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 406ndash418 2008

[28] K R Raslan and S M Hassan ldquoSolitary waves for the MRLWequationrdquo Applied Mathematics Letters vol 22 no 7 pp 984ndash989 2009

[29] K R Raslan and T S EL-Danaf ldquoSolitary waves solutions of theMRLW equation using quintic B-splinesrdquo Journal of King SaudUniversity Science vol 22 no 3 pp 161ndash166 2010

[30] A K Khalifa K R Raslan and H M Alzubaidi ldquoNumericalstudy using ADM for the modified regularized long waveequationrdquo Applied Mathematical Modelling vol 32 no 12 pp2962ndash2972 2008

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