Numerical Solutions of Coupled Nonlinear Evolution ... · British Journal of Mathematics & Computer...

British Journal of Mathematics & Computer Science 5(3): 310-332, 2015, Article no.BJMCS.2015.021

ISSN: 2231-0851

SCIENCEDOMAIN international www.sciencedomain.org

______________________________________________________________________________________________________________________

_____________________________________

*Corresponding author: [email protected], [email protected]

Numerical Solutions of Coupled Nonlinear Evolution

Equations via El-gendi Nodal Galerkin Method

M. El-Kady

1*, Salah M. El-Sayed

2 and Heba. E. Salem

3

1Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt.

2Department of Scientific Computing, Faculty of Computers and Informatics, Benha University,

Egypt. 3Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt.

Article Information

DOI: 10.9734/BJMCS/2015/8245

Editor(s):

(1) Raducanu Razvan, Department of Applied Mathematics, Al. I. Cuza University, Romania.

(2) Chin-Chen Chang, Department of Information Engineering and Computer Science, Feng Chia University, Taiwan. (3) Sheng Zhang, Department of Mathematics, Bohai University, Jinzhou, China.

(4) Qiankun Song, Department of Mathematics, Chongqing Jiaotong University, China.

(5) Kai-Long Hsiao, Taiwan Shoufu University, Taiwan.

(6) Paul Bracken, Department of Mathematics, University of Texas-Pan American Edinburg, TX 78539, USA.

Reviewers:

(1) Anonymous, University, Elazg, Turkey. (2) Anonymous, HITEC University Taxila Cantt, Pakistan.

(3) Anonymous, Namık Kemal University, Turkey.

(4) Anonymous, King Mongkut's University of Technology Thonburi, Thailand.

(5) Anonymous, Taif University, Taif, Saudi Arabia.

Complete Peer review History: http://www.sciencedomain.org/review-history.php?iid=727&id=6&aid=6753

Received: 06 December 2013

Accepted: 19 February 2014

Published: 04 November 2014

_______________________________________________________________________

Abstract

In this research the solution of coupled modified Korteweg-de Vries equation (mKdV) and the

generalized Hirota–Satsuma coupled KdV equation by using El-gendi nodal Galerkin (EGG)

approaches are presented. El-gendi nodal Galerkin (EGG) (EGG) approaches consist of two

approaches, the first is El-gendi Chebyshev nodal Galerkin (ECG) and the second approach is

called El-gendi Legendre nodal Galerkin (ELG). In these new approaches spaces of the solution

and the weak form to the system are presented. The resulted systems of ODES are solved by the

fourth order Runge-Kutta solver. The convergence and the stability of these new methods are

analyzed numerically. Numerical results are presented and compared with the results obtained

by pseudo-spectral method.

Original Research Article

El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021

311

Keywords: Coupled mKdV equation; generalized Hirota-Satsuma Coupled KdV equation;

El-gendi nodal Galerkin method; Legendre and Chebyshev cardinal functions.

1 Introduction

The effort in finding exact solution to a nonlinear equation is important for understanding of the

most nonlinear physical phenomena. For instances, the nonlinear wave phenomena observed in

fluid dynamics, plasma and optical fibers are often modeled by the bell shaped sech solutions and

the kink shaped tanh solutions.

In this paper, we consider coupled mKdV and a generalized Hirota-Satsuma coupled KdV

equations. In [1] the authors introduced a 44× matrix spectral problem with three potentials and

they proposed a corresponding hierarchy of nonlinear equations; two typical equations in

hierarchy are coupled mKdV equation which is given by:

,3)(32

33

2

1 2

xxxxxxxxt uuvvuuuu λ−++−=

,3333 2

xxxxxxxxt vvuvuvvvv λ++−−−= ],0[, Ttbxa ∈<< , (1)

where λ is a real constant with the initial conditions:

),()0,( 1 xfxu = ),()0,( 2 xfxv = (2)

and the boundary conditions are given in the following form

],0[),(),(),(),( 21 Tttqtbutqtau ∈== ,

],0[),(),(),(),( 21 Tttgtbvtgtav ∈== . (3)

and the generalized Hirota-Satsuma coupled KdV equation is given as follows

,)(332

1xxxxxt vzuuuu +−=

,3 xxxxt uvvv +−=

,3 xxxxt uzzz +−= ],0[, Ttbxa ∈<< , (4)

with the initial conditions:

),()0,( 1 xfxu = ),()0,( 2 xfxv = ),()0,( 3 xfxz = (5)

and the boundary conditions are given in the following form

],0[),(),(),(),( 21 Tttqtbutqtau ∈== ,


312

],0[),(),(),(),( 21 Tttgtbvtgtav ∈== ,

],0[),(),(),(),( 21 Ttthtbzthtaz ∈== . (6)

Equations (1-3) become a generalized KdV equation for u = 0 and the mKdV equation for 0=v ,

respectively. The soliton solutions of these problems describe various phenomena in nature, such

as vibrations, solutions and propagation with a finite speed. Particularly, coupled KdV system,

describes interactions of two long waves with different dispersion relations [2].

Solitary solutions for various nonlinear wave equations have been investigated using different

methods which can only solve special kind of nonlinear problems due to the limitations or

shortcomings in the methods. Many studies of generalized Hirota-Satsuma coupled KdV and

coupled mKdV equations have been done by many authors via different approaches. In [3] the

author used the extended tanh-function method and symbolic computation to obtain respectively

four kinds of soliton solutions for a new coupled mKdV and new generalized Hirota–Satsuma

coupled KdV equations, which were introduced recently by Wu et al. [1].

In [4] the author presented the decomposition method to obtain the Soliton solution for

generalized Hirota–Satsuma coupled KdV equations and coupled mKdV equation. In this paper

the algorithm is illustrated by studying an initial value problem. The obtained results are

presented, and only few terms are required to obtain an approximate solution that is found to be

accurate and efficient.

In [5] the authors applied the variational iteration method to obtain approximate analytic solutions

of coupled mKdV and generalized Hirota–Satsuma coupled KdV equations. This method provides

a sequence of functions and is based on the use of the Lagrange multiplier for the identification of

optimal values of parameters in a functional.

Also, in [6] the authors demonstrated the feasibility and validity of the differential transform

method, namely DTM. Therefore, the method has been applied to solve coupled mKdV and

generalized Hirota–Satsuma coupled KdV equations with initial conditions of two types. The

DTM method is based on the Taylor series, but the series coefficients are calculated in an iterative

manner by the help of T-transform.

In [7] the authors presented the numerical solution of systems of Hirota-Satsuma coupled KdV

and coupled mKdV equations by means of Homotopy Analysis Method (HAM). The HAM can

extremely minimize the volume of computations with respect to traditional techniques and yields

the analytical solution of the desired problem in the form of a rapidly convergent series with easily

computable components.

In [8] the Homotopy Analysis Method (HAM) is presented for obtaining the approximate solution

of new coupled modified Korteweg-de Vries (mKdV) system. The approximate analytical solution

is obtained by using this method in the form of a convergent power series with components that

are easily computable.

In this work, we aim to introduce two reliable techniques in order to solve coupled mKdV and

generalized Hirota–Satsuma coupled KdV equations. The new techniques depend on El-gendi-

nodal Galerkin (EGG) methods. Nodal Galerkin methods start from a weak form of the equations,

but replace hard to evaluate integrals by quadrature. Gauss quadrature is always used in the week


313

form. In this research, we use El-gendi quadrature formula [9]. This formula has a symmetric

property and this property leads us to reduce the number of operations to 50% and hence reducing

the rounding error. In addition, this formula is an alternating series which converges as the number

of grid points tends to infinity.

Although, we use the differentiation matrix in these methods as the pseudo-spectral method which

is a very well-known method, we find that (EGG) methods are more accurate for a long time and

converge faster than the pseudo-spectral method.

Also, we aim to discuss numerically in this case the stability and convergence of the new

approaches and make a comparison with the pseudo-spectral error where the question of stability

of the spectral approximations tends to be critical for solving hyperbolic conservation laws. The

most important reason is that the nonlinear mixing of the Gibbs oscillations with the approximate

solution will eventually cause the scheme to become unstable [10]. In other words, Trefethen and

Trummer determined the relation between the eigenvalues of the differentiation matrix and the

allowable time step in explicit time integration. On a grid of N points per space dimension,

machine rounding leads to errors in the eigenvalues of size )( 2NO . This phenomenon may lead

to inconsistency between predicted and observed time step restriction [11, 12,13]. Contrary, when

we try to solve the model problem )(uLut = where dxdL /= by the new techniques the

resulted system was insensitive to round-off errors and the condition number scaled sub-linearly

with N .

This paper is organized as follows: In section 2, we employ El-gendi Chebyshev nodal Galerkin

(ECG) method for solving (1) and (4) and we present the approximate solutions at the extrema

points of the Chebyshev polynomial. In section 3, we employ El-gendi Legendre Nodal Galerkin

(ECG) method for solving (1). In section 4, we discuss the eigenvalues and the time step for

(EGG) and pseudo-spectral methods. In section 5, we present the numerical solutions with

graphics and we compare the results with pseudo-spectral solutions at different times and for

enough large grid points.

2. EL-gendi Nodal Galerkin Method for Coupled Mkdv

In this section we present the numerical solution of the coupled mKdV by using El-gendi

Chebyshev nodal Galerkin (ECG) method and by using El-gendi Legendre nodal Galerkin (ELG)

method as follow:

2.1 El-gendi chebyshev nodal galerkin method for coupled mkdv

In this section we will explain El-gendi Chebyshev nodal Galerkin method and illustrate how it

used to solve equations (1-3) in case 1−=a and 1=b . In this method, the trail and the test

spaces are identical, so that we define for 0≥m the space )1,1(−mH to be a vector space of

functions )1,1(2 −∈ Lv such that all distributional derivatives of v is of order up to m and it can

be represented by functions in )1,1(2 −L . Since the functions of )1,1(1 −H are continuous up to


314

the boundary by Sobolev imbedding theorem, it is meaningful to introduce the following solution

subspace of )1,1(1 −H :

}0),1(),1(:)1,1({)1,1( 11

0 ==−−∈=− tutuHuH .

The weak form of equation (1) is given by multiplying both sides in equation (1) by a test

function, and then we have:

>−∈∀+

+−−−=

>−∈∀−

++−=

−∈−∈

∫

∫∫∫∫∫

∫

∫∫∫∫∫

−

−−−−−

−

−−−−−

.0),1,1(,3

33)(2

3

,0),1,1(,3

)(32

3)(

2

1

,s.t.)1,1(),1,1(find

1

0

1

1

1

1

21

1

1

1

21

1

1

1

1

0

1

1

1

1

1

1

1

1

31

1

1

1

1

0

1

0

tHGdxGv

dxGvudxGvudxGvdxGvdxGv

tHYdxYu

dxYuvdxYvdxYudxYudxYu

HvHu

x

xxxxxxxt

x

xxxxxxxt

N

λ

λ

By using the integration by parts we have:

>−∈∀−

+−+=

>−∈∀+

−−+−=

−∈−∈

∫

∫∫∫∫∫

∫

∫∫∫∫∫

−

−−−−−

−

−−−−−

.0),1,1(,3

33)(2

3

,0),1,1(,3

)(32

3)(

2

1

,s.t.)1,1(),1,1(find

1

0

1

1

1

1

21

1

1

1

21

1

1

1

1

0

1

1

1

1

1

1

1

1

31

1

1

1

1

0

1

0

tHGdxGv

dxGvudxGvudxGvdxGvdxGv

tHYdxYu

dxYuvdxYvdxYudxYudxYu

HvHu

x

xxxxxxxt

x

xxxxxxxt

N

λ

λ

(7)

Let us denote the finite dimensional subspace of )1,1(1

0 −H to be given as follows:

}0),1(),1(:P{ ==−∈= tftffX N

N,

where NP is the space of polynomials with degree N . Let kT be the Chebyshev polynomial of

degree k , then the cardinal function which is based on the Chebyshev polynomial is defined as

follows:

NixTxTN

xN

k

kikki

i ...,,1,0,)()(2

)(0

== ∑=

θθ

ϕ , (8)

where we have for all 1=kθ , except 2/10 == Nθθ and . The grid points l

x are

the so-called Chebyshev Gauss–Lobatto points,

ll ii x δϕ =)(


315

)/cos( Nkxk π= , Nk ...,,1,0= ,

which are the extrema points of the Chebyshev polynomial ( )N

T x . Consequently, we have the

following spaces [14]:

,

)}(),...,(),({ 121 xxxspanX N

N

v −= ψψψ ,

where )(xiψ has the form (8). Now, we give the approximate solution in the form

∑=

=N

j

jj

NxtUu

0

)()( ϕ , ∑=

=N

j

jj

NxtVv

0

)()( ψ , (9)

to ensure the approximations satisfy the boundary conditions, we set 00 == NUU and

00 == NVV . Also, we can set test functions )(xY and )(xG as a function of thN order

polynomials so we can write these polynomials in an equivalent cardinal form

∑=

=N

i

ii xxY0

)()( ϕα , ∑=

=N

i

ii xxG0

)()( ψβ , (10)

where the nodal values ii βα , are arbitrary except that 00 == Nαα and 00 == Nββ , to

ensure that )(xY and )(xG satisfy the boundary conditions. Now, the nodal Galerkin

approximation to equations (7) is

( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( )

( )

>∈∀−

+−+=

>∈∀+

−−+−=

∈∈

,0,,,3

,)(3,3,)(2

3,,

,0,,,3

,3,2

3,)(,

2

1,

,s.t.,find

22

3

tXGGv

GvuGvuGvGvGv

tXYYu

YvuYvYuYuYu

XvXu

N

vNx

N

N

N

x

N

N

N

x

N

xNx

N

Nx

N

xxN

N

t

N

uNx

N

Nx

NN

Nx

N

xNx

N

Nx

N

xxN

N

t

N

v

NN

u

N

λ

λ

(11)

where

( ) ∑=

=N

j

jjNjNxgxfbgf

0

)()(, , (12)

with Nkb are given by:[9]

∑= −

=2/

02

2cos

14

4 N

j

sNk

N

jk

jNb

πθ, 1...,,2,1 −= Nk ,

)}(),...,(),({ 121 xxxspanX N

N

u −= ϕϕϕ


316

1

120

−==

Nbb NNN

. (13)

Now, we evaluate the terms in (11) as follows:

( ) =N

N

t vu , ∑=

N

j

jjNjUb0

α& , (14)

and the second term is

( ) =Nx

N

xx Yu , ∑∑==

′′′N

k

kjkNk

N

j

j xub00

)(ϕα , (15)

where the first derivative of the cardinal functions )(xjϕ at the points lx have the entries of the

differentiation matrix: [15]

NjxTxTc

k

Nx

N

k

lnjk

n

kk

oddknn

j

lj ...,,1,0,)()(4

)(1

1

)(0

==′ ∑ ∑=

−

+=

θθϕ , (16)

the second derivative is:

NjxTxTc

nkk

Nx

N

k

lnjk

n

kk

evenknn

j

lj ...,,1,0,)()()(4

)(2

222

)(0

=−

=′′ ∑ ∑=

−

+=

θθϕ . (17)

and the rest terms are evaluated by the same fashion. Then (11) can be written in the form

====

−=′−′+

′′−′+′′′=

′+′′−

′−′+′′′−=

∑ ∑∑

∑ ∑∑∑∑

∑ ∑∑

∑ ∑∑∑∑

= ==

= ====

= ==

= ====

.0,0

,1...,,2,1,)(3)(3

)()(3)(2

3)()(

,)(3)()(2

3

)(3)()()(2

1

00

0 00

2

0 00

2

00

0 00

0 00

3

00

NN

N N

N

N

m

mNmm

j

N N

m

jmNjm

N

jN

N

k

kjkNK

N

jNj

N N

jN

N

k

kjkNk

N N

m

mjNmm

N

jN

N

k

kjkNK

N

jNj

VVUU

NjxbVxbUV

xxbUVxbVxxbVVb

xbUxxbV

xbVUxbUxxbUUb

l l

lllll

l

l

l

l

llll

l

l

l l

lllll

l

l

l

llll

l

l

&

&

ψλψ

ψϕψψψ

ϕλϕψ

ϕϕϕϕ

(18)

Now, we put ][ VUW = where

)()( tUtW ii = , )()( tVtW iNi =+ , Ni ...,,1,0= . (19)


317

with the boundary,

00 == NWW , 021 ==+ NN WW . (20)

Now, we can rewrite (18) by using (19-20) then the resulted system of ODEs has been solved by

using fourth order Runge-Kutta solver.

2.2 El-gendi Chebyshev Nodal Galerkin Method for Generalized Hirota-

Stasuma KdV

In this section, we have the following space

)}(),...,(),({ 121 xxxspanX N

N

z −= χχχ ,

where )(xiχ has the form (8). Now, we give the approximate solution of )(xz in the following

form

∑=

=N

j

jj

N xtZz0

)()( χ , (21)

also, we set 00 == NZZ to ensure the approximations satisfy the boundary conditions. Let

)(xV be a test function of thN order polynomials and we can write it in the equivalent cardinal

form

∑=

=N

i

ii xxV0

)()( χη , (22)

where the nodal values iη are arbitrary except that 00 == Nηη to ensure that )(xV satisfy the

boundary conditions. Now, the nodal Galerkin approximations to equations (4-6) are

======

−=′+′′′=

′+′′′=

′−′+′′′−=

∑ ∑∑∑

∑ ∑∑∑

∑ ∑∑∑∑

= ===

= ===

= ====

.0,0,0

,1...,,2,1),(3)()(

),(3)()(

,)(3)(2

3)()(

2

1

000

0 000

0 000

0 00

2

00

NNN

j

N N

m

Nmm

N

k

kjkNK

N

jNj

j

N N

m

Nmm

N

k

kjkNK

N

jNj

N N

m

mjNmm

N

jN

N

k

kjkNK

N

jNj

ZZVVUU

NjxbUZxxbZZb

xbUVxxbVVb

xbVUxbUxxbUUb

l

l

ll

l

l

l

l

ll

l

l

l

l

l

llll

l

l

&

&

&

χχχ

ψψψ

ϕϕϕϕ

(23)

Now, we put ][ ZVUW = where

)()( tUtW ii = , )()( tVtW iNi =+ , Ni ...,,1,0= . (24)

with the boundary conditions


318

00 == NWW , 021 ==+ NN WW . (25)


using fourth order Runge-Kutta solver.

3. EL-gendi Legendre Nodal Galerkin Method for Coupled

Mkdv In this section we consider the Legendre cardinal functions based on Chebyshev Gauss-Lobatto

(CGL) nodes and we present the approximate solution as a linear combination of these functions

which have the following form:

∑=

+=N

k

kiki xLxLkN

x0

)()()12(2

)(~ πϕ , Ni ...,,1,0= , (26)

where jx are the Chebyshev Gauss-Lobatto points and ll ii x δϕ =)(~

. Also, the spaces of the

solutions are constructed as before. Now, we give the approximate solution in the following form

∑=

=N

j

jj

N xtUu0

)(~)(~

ϕ , ∑=

=N

j

jj

N xtVv0

)(~)(~

ψ , (27)

Also, we set 0~~

0 == NUU and 0~~

0 == NVV . The test functions )(~

xY and )(~

xG are thN order

polynomials so we can write these polynomials in the equivalent cardinal form

∑=

=N

i

ii xxY0

)(~~)(~

ϕα , ∑=

=N

i

ii xxG0

)(~~)(

~ψβ , (28)

where the nodal values ii βα~

,~are arbitrary except that 0~~

0 == Nαα and 0~~

0 == Nββ . Now,

the nodal Galerkin approximation to equations (6-7):

====

−=′−′+

′′−′+′′′=

′+′′−

′−′+′′′−=

∑ ∑∑

∑ ∑∑∑∑

∑ ∑∑

∑ ∑∑∑∑

= ==

= ====

= ==

= ====

.0~~

,0~~

,1...,,2,1,)(~~3)(~~~

3

)(~)(~~~3)(~~

2

3)(~)(~~~

,)(~~3)(~)(~~

2

3

)(~~~3)(~~

)(~)(~~

2

1~

00

0 00

2

0 00

2

00

0 00

0 00

3

00

NN

N N

N

N

m

mNmm

j

N N

m

jmNmm

N

jN

N

k

kjkNk

N

jNj

N N

jN

N

k

kjkNk

N N

m

mjNmm

N

jN

N

k

kjkNK

N

jNj

VVUU

NjxbVxbUV

xxbUVxbVxxbVVb

xbUyxbV

xbVUxbUxxbUUb

l l

lllll

l

l

l

l

llll

l

l

l l

lllll

l

l

l

llll

l

l

&

&

ψλψ

ψϕψψψ

ϕλϕψ

ϕϕϕϕ

(29)


319

where

( ) ∑=

=N

j

jjNjNxgxfbgf

0

)()(, . (30)

and ( )lxxj =

′ψ~ is the first order differentiation matrix that depends on Legendre polynomial at the

CGL nodes and have the entries given by: [16]

( )lxxj =

′ψ ∑ ∑=

−

=

−−+=

N

mj

j

k

kj

l

j

kij xaxLN

j]2/)1[(

0

12)(

1,)(2

)12(π, (31)

where

!)!2()!(2

)!2()!22()1()(

1,kkjkj

kjkja

j

kj

k−−

−−−= ,

Also, ( )lxxj =

′′ψ has the following formula:

( )lxxj =

′′ψ ∑ ∑=

−

=

−−+=

N

mj

j

k

kj

l

j

kij xaxLN

j]2/)2[(

0

22)(

2,)(2

)12(π, (32)

where

!)!2()!(2

)!12()!2()!22()1()(

2,kkjkj

kjkjkja

j

kj

k−−

−−−−−= .

Also, as above we present the vector solution of the system as follows

]~~

[~

VUW = ,

where

)(~

)(~

tUtW ii = , )(~

)(~

tVtW iNi =+ , Ni ...,,1,0= . (33)

with

0~~

0 == NWW , 0~~

21 ==+ NN WW . (34)


using forth order Runge-Kutta solver.

4. Eigenvalues and Time Step

The aim of this section is to show where such stability restrictions come from so we consider the

first-order hyperbolic initial boundary value problem

xt uu = , )1,1(−∈x , 0>t , (35)


320

with the initial and boundary conditions

)()0,( xfxu = , 0),1( =tu , 0>t . (36)

which can be viewed as a model of more general hyperbolic system of equations with

appropriately specified boundary conditions. Now, El-gendi Galerkin approximation to this model

is given as follows:

)1()1(),(),( −−+−= ρρρ N

Nx

N

N

N

t uuu , NP∈∀ρ , (37)

where )(xiϕρ = in Chebyshev case or )(~ yiϕρ = in Legendre case. The associated

)1()1( +×+ NN matrix that represents the left hand side of (37) is

)0...,,0,1(diagMDB T

CC +−= ,

)0...,,0,1(diagMDB T

LL +−= ,

where )( jiC xD ϕ′= is the differentiation matrix in Chebyshev case, )(~jiL xD ϕ′= is the

differentiation matrix in Legendre case and TD is the transpose of the matrix,

)...,,,( 10 NNNN bbbdiagM = is the diagonal mass matrix and the resulted pseudo-spectral

matrix of (35-36) is ):1,:1(~

NNDD = after imposing the boundary condition.

Our concern is with the eigenvalues of D~

and B in the Chebyshev and Legendre cases. Since we

will use an explicit fourth order Runge-Kutta method to integrate in time, there will be a limit on

the size of the time step that depends on the size of the maximum eigenvalue of the D~

and B

matrices i.e. The approximation to be stable in time, the maximum eigenvaluemaxλ must fall

within the region of absolute stability of the time integration method. Analytic representations for

the eigenvalues of D~

and B matrices are not known, so we will find the eigenvalues

numerically. Now, for each time step t∆ , a stability region in the complex plane, defined as the

set of all C∈λ for which it reduces to a stable recurrence relation when applied to the model

problem uut λ= [12]. So the solutions of (35-36) will be bounded as ∞→t for a fixed time

step if and only if the eigenvalues of D~

and B lie in this stability region. Also, the stability region

expands in proportion to1)( −∆ t . Therefore if the eigenvalues of the matrix are of size )(NO , the

result is that the stability restriction )( 1−=∆ NOt , while if they are of size )( 2NO , the

restriction becomes )( 2−=∆ NOt .

In Fig. 1 we illustrate the stability region of the fourth order Rung-Kutta method and it is clear

from Figs. 2, 3 that the eigenvalues have large imaginary parts and some negative real parts. The

presence of the real parts indicates that the approximations are dissipative. Hence, some energy is

lost as the computations proceed. Dissipation is important for the stability of variable coefficient

and nonlinear problems. Moreover, it is clear that the eigenvalues are produced by round-off error

effects and the structure of the eigenvalues differs from the Chebyshev and Legendre


321

approximations. The largest eigenvalues of the Legendre approximation are very near the

imaginary axis, while the corresponding eigenvalues of the Chebyshev approximation have

significantly larger real parts. The time step will be limited by the location in the complex plane of

the largest eigenvalue. In Figs. 4 and 5 the eigenvalues of CB , LB are in the region of stability

and scaled sub-linearly with N and relatively insensitive to round-off errors where the algorithm

is said to be stable if its outcome is relatively insensitive to round-off error.

Fig. 1. The stability region of the fourth-order Runge-Kutta method in the complex plane.

Fig. 2. Distribution of the eigenvalues of CD~

for 100,32=N .

It is clear from Fig. 6 (a) that the condition numbers in the 2-norm of differentiation matrices scale

)(2

NO whereas (EGG) matrices scale sub-linearly with N . The maximum, minimum eigenvalues

and condition numbers of D~

and B matrices are indistinguishable graphically in the two cases

(Chebyshev and Legendre). We see from the Fig. 6 (b) that the largest eigenvalue of D~

grows

asymptotically as )( 2NO whereas the maximum eigenvalue of (EGG) matrices scale sub-linearly

with N .

-4 -3 -2 -1 0 1-3

-2

-1

0

1

2

3

R e a l

I m

a g

i n

a r

y

2.780

2.927i

-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2-100

-80

-60

-40

-20

0

20

40

60

80

100

i m

a g

i n

a r

y

r e a l

N=32

-250 -200 -150 -100 -50 0-1000

-800

-600

-400

-200

0

200

400

600

800

1000

i m

a g

i n

a r

y

r e a l

N=100


322

Fig. 3. Distribution of the eigenvalues of LD~

for 100,32=N .

Fig. 4. Distribution of the eigenvalues of CB for 100,32=N .

Fig. 5. Distribution of the eigenvalues of LB for 100,32=N .

-14 -12 -10 -8 -6 -4 -2 0-100

-80

-60

-40

-20

0

20

40

60

80

100i

m a

g i

n a

r y

r e a l

N=32

-160 -140 -120 -100 -80 -60 -40 -20 0-800

-600

-400

-200

0

200

400

600

800

i m

a g

i n

a r

y

r e a l

N=100

-3 -2 -1 0 1 2 3 4

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

R e a l

I m

a g

i n

a r

y

N=32

-3 -2 -1 0 1 2 3 4-3

-2

-1

0

1

2

3

R e a l

I m

a g

i n

a r

y

N=100

-3 -2 -1 0 1 2 3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

R e a l

I m

a g

i n

a r

y

N=32

-4 -3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

R e a l

I m

a g

i n

a r

y

N=100


323

(a) (b)

Fig. 6. (a) Spectral condition numbers of D~

and B , (b) maximum and minimum moduli of

eigenvalues of D~

and B .

When we solved the following problem

xxt uu = , )1,1(−∈x , 0>t , (38)

with the initial and boundary conditions

0),1(),1( ==− tutu , 0>t . (39)

El-gendi Galerkin approximation to this model is given as follows:

Nx

N

xN

N

t uu ),(),( ρρ −= , N

P∈∀ρ ,(40)

The associated )1()1( −×− NN matrix that represents the left hand side of (35) is

C

T

CC MDDB −=2, L

T

LL MDDB −=2, (41)

and the resulted pseudo-spectral matrix after imposing the boundary conditions is

):2,:2(2~ 2

NNDD = . In [12], experiments show that the second-order spectral differentiation,

with zero boundary conditions at both endpoints have real and negative eigenvalues and its

maximum magnitude is of )( 4NO for both Chebyshev and Legendre. Moreover, it is noted that

the maximum eigenvalue of LD2~

approximately one half that of CD2~

.Although, their

eigenvalues are less sensitive to perturbations than in the first-order case.

Fig. 7 (b) illustrates the size of the maximum and minimum eigenvalues of the two matrices ( 2D

and 2B ) for different numbers of grid points N . It is clear that the maximum eigenvalues of 2B

101

102

101

102

103

104

105

N

cond (D)

cond (B)

101

102

10-2

10-1

100

101

102

103

104

105

106

N

D-min

D-max

B-min

B=max


324

which is a symmetric positive definite matrix grow as )( 2NO , whereas the minimum eigenvalues

decay as )(1−

NO but the maximum eigenvalues of 2D grow as )(4

NO for both the Chebyshev

and the Legendre approximations and in computations we see that the maximum eigenvalues of

the Legendre methods is typically only half as large as that of the corresponding Chebyshev

method. On the other hand, the condition number of 2B grows like )( 3NO which is cleared from

Fig. 7 (a). Thus, the time step in the EGG will be of order )( 2−NO and the time step in pseudo-

spectral method will be of order )( 4−NO , so the EGG methods will be faster than the pseudo-

spectral method.

(a) (b)

Fig .7. (a) Spectral condition numbers of 2D and 2B and (b) maximum and minimum

moduli of eigenvalues of 2D and 2B .

5. Numerical Experiments

In this section we give two examples and we use MATLAB 7.0 softwaretoobtain the numerical

results.

Example 1. Consider the coupled mKdV equations are given in the following form:

,3)(32

33

2

1 2

xxxxxxxxt uuvvuuuu λ−++−=

,3333 2

xxxxxxxxt vvuvuvvvv λ++−−−= ],0[, Ttbxa ∈<< ,

with the solitary wave solutions,

)tanh(),( ξkktxu = ,

)(tanh4)4(2

1),( 222 ξλ kkktxv −+= , ],0[, Ttbxa ∈<< ,

101

102

10-4

10-2

100

102

104

106

N

Cond (D2)

Cond (B2)

101

102

10-4

10-2

100

102

104

106

N

D2-min

D2-max

B2-min

B2-max


325

where tkx )32(2

1 2 λξ −−+= , k and λ are arbitrary constants.

The following error notations are defined:

Absolute error )()( iappiex xuxu −= , 1,...,1 −= Ni ,

and

Maximum error )()(max iappiexi

xuxu −= , 1,...,1 −= Ni .

where ( )ex i

u x and ( )app i

u x are the exact and approximate solutions, respectively.

In Table (1), the problem solved in the domain ],[ ba and since Gauss-Lobatto points are in

interval[ 1,1]− , therefore, the interval ]1,1[− is mapped to ],[ ba by a linear mapping defined

by:

++

−=

22

ababX ii η , Ni ,...,0= ,

whereiη are the Gauss-Lobatto nodes. Now, we introduce the u -ELG, v -ELG, u -ECG, v -

ECG, u -pseudo and v -pseudo (i.e. the absolute error of the u -solution in the interior points by

ELG, ECG and pseudo-spectral methods respectively) in Table 1.

Table 1. The solution of mKdV at T=0.5 and 1.0=∆t , 1.0=k , 1.0=λ , 10=N ,

100−=a and 100=b .

X u-ELG u-ECG u-pseudo v-ELG v-ECG v-pseudo

X(2)

X(3)

X(4)

X(5)

X(6)

X(7)

X(8)

X(9)

5.4101E-12

1.7438E-10

3.5669E-08

2.2957E-05

2.3747E-05

3.6976E-08

1.8077E-10

5.6084E-12

8.0966E-12

3.0086E-10

5.5464E-08

2.6850E-05

2.7763E-05

5.7497E-08

3.1189E-10

8.3934E-12

1.0003E-04

1.0156E-04

1.4105E-04

2.4272E-04

2.4255E-04

1.4043E-04

1.0121E-04

9.9921E-05

2.1640E-12

6.9752E-11

1.4266E-08

8.6397E-06

8.9170E-06

1.4789E-08

7.2309E-11

2.2434E-12

3.2386E-12

1.2034E-10

2.2182E-08

9.9930E-06

1.0306E-05

2.2995E-08

1.2476E-10

3.3574E-12

1.2566E-06

1.3906E-06

3.4664E-06

4.0934E-05

4.0431E-05

3.4915E-06

1.3984E-06

1.2527E-06

Figs. 8 and 9 present the numerical solutions of coupled mKdV by pseudo-spectral and (EGG)

methods respectively 1.0=∆t , 1.0=k , 1=λ , 40−=a , 40=b and for 30=N . It is clear

that the v -solution by the pseudo-spectral method for long time is unstable in the boundary but

the v -solution by (EGG) methods is stable at the boundary for a long time.


326

Fig. 8. The approximate solution of the coupled mKdV by the pseudo-spectral method.

Fig. 9. The approximate solution of the coupled mKdV by (EGG) methods.

Fig. 10 displays the infinity norm of the pseudo-spectral method. The maximum absolute error

increased linearly from 0≈t to reach its maximum at 09.0≈t and then the error increases

linearly as a function of t .

Fig. 10. The infinity norm error of the pseudo-spectral method as a function of time for

10=N .

-40 -30 -20 -10 0 10 20 30 400

0.5

1

-0.1

-0.05

0

0.05

0.1

X

t

u

-40-20

020

40 0

0.2

0.4

0.6

0.8

1

0.5

0.505

0.51

0.515

t

X

v

-40 -30 -20 -10 0 10 20 30 400

0.5

1

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

x

t

u

-40 -30 -20 -10 0 10 20 30 400

0.5

1

0.502

0.504

0.506

0.508

0.51

0.512

0.514

0.516

0.518

t

x

v

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7x 10

-3

t

m a

x e

r r

U (

t )

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7x 10

-3

t

m a

x e

r r

V (

t )


327

Fig. 11 displays the infinity norm error of (EGG) methods. The infinity error oscillates

periodically with decreasing amplitude from 0≈t to reach their minimum at 3.0≈t and then

increases oscillatory in the same fashion to reach the same type of periodicity at 6.0≈t after that

amplitude from 67.0≈t will be fixed until 1≈t but the maximum error will increase.

Fig. 11. The infinity norm error of El-gendi Galerkin method as a function of time for

10=N .

Figs. 12 and 13 illustrate that the point-wise absolute error at each point in the grid when the

number of grid points is 30=N . We observe that (EGG) methods are better than the pseudo-

spectral method around the boundary but in case u -solution the absolute error will be maximum

around and at 0=x . In contrast, in case v -solution the absolute error is very small at 0=x and

very large around 0=x .

We observe from Fig. 14 that (EGG) methods outperform the pseudo-spectral method in terms of

accuracy. It is clear from Fig. 15 that for sufficiently large N (100 to 200) that the logarithm base

as a function of grid points N the logarithm decreases while N increases. So (EGG) methods are

converging faster and more accurate than the pseudo-spectral method.

Fig. 12. Absolute error of u and v by using the pseudo-spectral method for N=30 and

1.0=∆t , 5.0=t .

0 0.2 0.4 0.6 0.8 10

1

2

x 10-4

t

m a

x e

r r

U (

t )

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9x 10

-5

t

m a

x e

r r

V (

t )

-100 -50 0 50 1000

1

2

3

4

5

6

7

8x 10

-4

X

a b

s e

r r

U

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

-3

X

a b

s e

r r

V


328

Fig. 13. Absolute error of u and v by using El-gendi Galerkin method for N=30 and

1.0=∆t , 5.0=t .

Fig. 14. Logarithm of the maximum error at 5.0=t .

Fig. 15. Logarithm of the maximum error at 1=t and for large N .

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

-4

x

a b

s e

r r

U

-100 -50 0 50 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

-4

X

a b

s e

r r

V

0 20 40 60 80 100-12

-10

-8

-6

-4

-2

0

2

4

6

8

N

L o

g 1

0 (

m a

x -

n o

r m

) U

p s e u d o

EGG

0 20 40 60 80 100-15

-10

-5

0

5

10

15

N

L o

g 1

0 (

m a

x -

n o

r m

) V

p s e u d o

EGG

100 110 120 130 140 150 160 170 180 190 200-4.048

-4.046

-4.044

-4.042

-4.04

-4.038

-4.036

-4.034

-4.032

N

L o

g 1

0 (

m a

x -

n o

r m

) V

E G G V

F i t

100 110 120 130 140 150 160 170 180 190 200-3.232

-3.231

-3.23

-3.229

-3.228

-3.227

-3.226

-3.225

-3.224

-3.223

N

L o

g 1

0 (

m a

x -

n o

r m

) U


329

A practical way to verify that a wave solution is stable is to check if the maximum absolute error

remains, for long times, less than2)( tO ∆ [17]. So, if the errorincreases over this value,

oscillations will soon grow and become unbounded after relatively short times, not only because

of the numerical scheme, but also due to the nonlinear nature of the equations. This is clear from

Fig. 13 that the error of (EGG) methods are still less than 2)( tO ∆ in case N=100 and time step

)(10 1−=∆ Nt . In contrast, the pseudo-spectral method the error becomes very large and the

method fails to satisfy good accuracy. In other words, for a long time 10=t , N=40 and

)(0025.0 1−=∆ Nt we find the maximum error of the El-gendi Galerkin in the u-solution to be

(1.356445e-002) and the maximum error of the pseudo-spectral method is (5.888195e+005) which

is a very big error and refers to the instability of the method.

Example 2. Consider the generalized Hirota-Satsuma coupled KdV equation [18]

,)(332

1xxxxxt uzuuuu +−=

,3 xxxxt uvvv +−=

,3 xxxxt uzzz +−= ],0[, Ttbxa ∈<< ,

with the following exact solution

,3/3/8)(tanh4),( 2

22

2

2 cqkqktxu −−= ξ

,3/43/2)(tanh2),( 02

22

2

2 cqkqktxv −−−= ξ

,2)(tanh2),( 02

22

2

2 cqkqktxz +−= ξ

where )(2 ctxkq −=ξ . In Table (2) we solve this example for 30−=a , 30=b , 1.0=c ,

1.00 =c , 1.02 =q , 1.0=k , 01.0=∆t and 10=N .

Table 2. Comparison between the Maximum errors (Mer) of the pseudo-spectral methodand

ECG method for solving the generalized Hirota-Satsuma coupled KdV equation

N T Method 0.5 1.0 2.0

Mer-pseudo Mer-ECG Mer-pseudo Mer-ECG Mer-pseudo Mer-ECG

10 8.6790E-05 4.8267E-06 1.7359E-04 9.655 e-06 3.4720e-04 1.9318e-05 20 8.7593E-05 4.8664E-06 1.7519E-04 9.733 e-06 3.5040e-04 1.9468e-05 30 8.7482E-05 4.8604E-06 1.7497E-04 9.721 e-06 3.4996e-04 1.9446e-05 40 8.7638E-05 4.8686E-06 1.7528E-04 9.737 e-06 3.5056e-04 1.9474e-05 50 8.7562E-05 4.8647E-06 8.2176E-04 9.730 e-06 6.2498e+11 1.9462e-05 60 6.5429E-03 4.8686E-06 8.3951E+12 9.737 e-06 7.6754e+11 1.9474e-05 70 1.2814E+12 4.8676E-06 1.1556E+12 9.735 e-06 1.1302e+12 1.9472e-05


330

It is clear from Table (2) that the error which resulted from pseudo-spectral method is unbounded

when the number of grid points increases. The error of pseudo-spectral method becomes

unbounded for 70=N at a time 5.0=t and when the time increases the unbounded error will

appear for small number of grids. In El-gendi Chebyshev Galerkin (ECG) method the errors are

bounded for large number of grid points N and have the same order. On the other hand, the figures

16, 17 and 18 display the numerical solutions of the proposed problem where the left column

presents the behavior of the solution by the pseudo-spectral method for 45=N and the right

column illustrate the behavior of the numerical solution by using ECG method for the same

number of nodes.

(a) (b)

Fig. 16. (a) Presents the pseudo-spectral of u solution for 45=N and (b) presents the

nodal Galerkin of u solution for 45=N .

(a) (b)

Fig. 17. (a) Presents the pseudo-spectral of v solution for 45=N and (b) presents the

nodal Galerkin of v solution for 45=N .

(a) (b)

Fig. 18. (a) Presents the pseudo-spectral of z solution for 45=N and (b) presents the

nodal Galerkin of z solution for 45=N .

-30-20

-100

1020 30 0

0.5

1

1.5

2

-0.036

-0.0355

-0.035

-0.0345

-0.034

t

x

u p

se

ud

o

-30-20

-100

1020

30 0

0.5

1

1.5

2

-0.036

-0.0355

-0.035

-0.0345

-0.034

t

x

u G

al

-30 -20 -10 0 10 20 30 0

0.5

1

1.5

2

-1.434

-1.433

-1.432

-1.431

t

x

v p

se

ud

o

-30-20

-100

1020

30 0

0.5

1

1.5

2

-1.434

-1.4335

-1.433

t

x

v a

p p

-30-20

-100

1020

30 0

0.5

1

1.5

2

0.098

0.0982

0.0984

0.0986

0.0988

0.099

t

x

z a

p p

-30-20

-100

1020

30 0

0.5

1

1.5

2

0.098

0.0982

0.0984

0.0986

0.0988

0.099

t

x

z G

al


331

6. Conclusion

In this paper, two efficient methods depend on nodal Galerkin method are employed to solve the

coupled mKdV and the generalized Hirota-Satsuma coupled KdV equations. A study of the

stability of these methods and the pseudo-spectral method is presented. Numerical results are

given for long times and for a large number of grid points. EGG methods have smaller errors than

pseudo-spectral method and lead to stable approximations. Stability guarantees that the solution

remains bounded as N approaches infinity.

Acknowledgements

The authors are very grateful to the referees for carefully reading the paper and for their comments

and suggestions which have improved the paper.

Competing Interests The authors declare that no competing interests exist.

References

[1] Wu YT, Geng XG, Hu XB, Zhu SM. A generalized Hirota–Satsuma coupled Korteweg–de

Vries equation and Miura transformations, Phys. Lett. A. 1999;255:259-264.

[2] Zayed EM, Nofal TA, Gepreel KA. On using the homotopy perturbation method for finding

the travelling wave solutions of generalized nonlinear Hirota- Satsuma Coupled KdV

equations. International Journal of Nonlinear Science. 2009;7:159-166.

[3] Raslan KR. The decomposition method for a Hirota-Satsuma coupled KdV equation and

acoupled MKdV equation, Intern. J. Compu. Math. 2004;81:1497-1505.

[4] Fan EG. Soliton solutions for a generalized Hirota Satsuma coupled KdV equation and a

coupled MKdV equation, Physics Letters A. 2001;282:18-22.

[5] Guo-Zhong Z, Xi-Jun Y, Yun X, Jiang Z, Di W. Approximate analytic solutions for

ageneralized Hirota Satsuma coupled KdV equation and a coupled mKdV equation, Chin.

Phys. B. 2010;19.

[6] Kangalgil F, Ayaz F. Solitary wave solutions for Hirota-Satsuma coupled KdV equation

and coupled mKdV equation by differential transformation method. The Arabian Journal

for Science and Engineering. 2010;35:203-213.

[7] Arife AS, Vanani SK, Yildirim A. Numerical solution of Hirota-satsuma couple Kdv and a

coupled MKDV equation by means of homotopy analysis method. World Applied Sciences

Journal. 2011;13: 2271-2276.


332

[8] Ghoreishi M, Ismail A, Rashid A. The solution of coupled modified KDV system by

thehomotopy analysis method, TWMS Jour. Pure Appl. Math. 2012;3:122-134.

[9] El-gendi SE. Chebyshev solutions of Differential, Integral, and Integro-Differential

Equations. Computer Journal. 1969;12:282-287.

[10] Gottlieb D, Hesthaven JS. Spectral methods for hyperbolic problems. Journal of

Computational and Applied Mathematics. 2002;128:186-221.

[11] Canuto C, Hussaini M, Quarteroni A, Zang T. Spectral Methods Fundamentals in Single

Domains, Springer Verlag; 2006.

[12] Trefethen LN, Trummer MR. An instability phenomenon in spectral methods, SIAM J.

Numer. Anal. 1987;24:1008-1023.

[13] Trefethen LN. Spectral Methods in MATLAB, SIAM, Philadelphia; 2000.

[14] Chen F, Shen J. Efficient spectral-Galerkin methods for systems of coupled second-order

equations and their applications. Journal of Computational Physics. 2012;231:5016-5028.

[15] Elbarbary ME, El-Sayed M. Higher order pseudospectral differentiation matrices. Applied

Numerical Mathematics. 2005;55:425-438.

[16] El-Kady M, Biomy M. Interactive Chebyshev – Legendre Approach for Linearquadratic

optimal regulator systems. IJWMIP. 2011;9:459-483.

[17] Tzirtzilakis EE, Skokos CD, Bountis TC. Numerical solution of the Boussinesq equation

using spectral methods and stability of solitary wave propagation. 1st International

Conference “From Scientific Computing to Computational Engineering”, (III). 2005;928-

933.

[18] Khater AH, Temsah RS, Callebaut DK. Numerical solutions for some coupled nonlinear

evolution equations by using spectral collocation method. Mathematical and Computer

Modelling. 2008;48:1237-1253.

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Numerical Solutions of Coupled Nonlinear Evolution ... · British Journal of Mathematics & Computer...

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